Computational Methods for Differential Equations
https://cmde.tabrizu.ac.ir/
Computational Methods for Differential Equationsendaily1Wed, 01 May 2024 00:00:00 +0330Wed, 01 May 2024 00:00:00 +0330Numerical solution of Burgers' equation with nonlocal boundary condition: Use of Keller-Box scheme
https://cmde.tabrizu.ac.ir/article_17162.html
In this paper, we transform the given nonlocal boundary condition problem into a manageable local equation. By introducing an additional transformation of the variables, we can simplify this equation into conformable Burgers&rsquo; equation. Thus, the Keller Box method is used as a numerical scheme to solve the equation. A comparison is made between numerical results and the analytic solution to validate the results of our proposed method.Upper and lower solutions for fractional integro-differential equation of higher-order and with nonlinear boundary conditions
https://cmde.tabrizu.ac.ir/article_17165.html
This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\`a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations.Efficiency of vaccines for COVID-19 and stability analysis with fractional derivative
https://cmde.tabrizu.ac.ir/article_16926.html
The objectives of this study are to develop the SEIR model for COVID-19 and evaluate its main parameters such as therapeutic vaccines, vaccination rate, and effectiveness of prophylactic. Global and local stability of the model and numerical simulation are examined. The local stability of equilibrium points was classified. A Lyapunov function is constructed to analyze the global stability of the disease-free equilibrium. The simulation part is based on two situations, including the USA and Iran. Our results provide a good contribution to the current research on this topic.An efficient algorithm for computing the eigenvalues of conformable Sturm-Liouville problem
https://cmde.tabrizu.ac.ir/article_17114.html
In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order $2 \alpha$, $\frac{1}{2}&lt;\alpha \leq 1$, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the $n$th eigenvalue, the correction technique reduces the error order from $O(n^4h^2)$ to $O(n^2h^2)$.Wong-Zakai approximation of stochastic Volterra integral equations
https://cmde.tabrizu.ac.ir/article_17331.html
This study aims to investigate a stochastic Volterra integral equation driven by fractional Brownian motion with Hurst parameter $H\in (\frac 12, 1)$. We employ the Wong-Zakai approximation to simplify this intricate problem, transforming the stochastic integral equation into an ordinary integral equation. Moreover, we consider the convergence and the rate of convergence of the Wong-Zakai approximation for this kind of equation.Some delta q−fractional linear dynamic equations and a generalized delta q−Mittag-Leffler function
https://cmde.tabrizu.ac.ir/article_17017.html
In this paper, we introduce a generalized delta q&minus;Mittag-Leffler function. Also, we solve some Caputo delta q&minus;fractional dynamic equations and these solutions are expressed by means of the newly introduced delta q&minus;Mittag-Leffler function.&nbsp;On fractional linear multi-step methods for fractional order multi-delay nonlinear pantograph equation
https://cmde.tabrizu.ac.ir/article_17115.html
This paper presents the development of a series of fractional multi-step linear finite difference methods (FLMMs) designed to address fractional multi-delay pantograph differential equations of order $0 &lt; \alpha \leq 1$. These $p$ FLMMs are constructed using fractional backward differentiation formulas of first and second orders, thereby facilitating the numerical solution of fractional differential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed $p$-FLMMs. To substantiate our theoretical findings, we offer numerical examples that corroborate the efficacy and reliability of our approach.An efficient computational method based on exponential B-splines for a class of fractional sub-diffusion equations
https://cmde.tabrizu.ac.ir/article_17747.html
The primary objective of this research is to develop and analyze a robust computational method based on exponential B-splines for solving fractional sub-diffusion equations. The fractional operator includes the Mittag-Leffler function of one parameter in the form of a kernel that is non-local and non-singular in nature. The current approach is based on an effective finite difference method for discretizing in time, and the exponential B-spline functions for discretizing in space. The proposed scheme is proven to be unconditionally stable and convergent. Also, unique solvability of the method is established. Numerical simulations conducted for multiple test examples validate the agreement between the obtained theoretical results and the corresponding numerical outcomes.A mathematical study on the non-linear boundary value problem of a porous fin
https://cmde.tabrizu.ac.ir/article_17132.html
An analytical study of two different models of rectangular porous fins are investigated using a new approximate analytical method, the Ananthaswamy-Sivasankari method. The obtained results are compared with the numerical solution, which results in a very good agreement. The impacts of several physical parameters involved in the problem are interlined graphically. Fin efficiency and the heat transfer rate are also calculated and displayed. The result obtained by this method is in the most explicit and simple form. The convergence of the solution determined is more accurate as compared to various analytical and numerical methods.Existence and Uniqueness of Positive Solutions for a Hadamard Fractional Integral Boundary Value problem
https://cmde.tabrizu.ac.ir/article_17736.html
&lrm;In this paper&lrm;, &lrm;a kind of boundary value problem including Hadamard type fractional differential equations with an integral boundary condition is investigated&lrm;. &lrm;Using the method of upper and lower solutions and Schauder's fixed point theorem&lrm;, &lrm;the existence and uniqueness of positive solutions of this problem is proved&lrm;. &lrm;Illustrated example is presented to explain the proved theorems&lrm;.A new perspective for the Quintic B-spline collocation method via the Lie-Trotter splitting algorithm to solitary wave solutions of the GEW equation
https://cmde.tabrizu.ac.ir/article_17759.html
A hybrid method utilizing the collocation technique with B-splines and Lie-Trotter splitting algorithm applied for 3 model problems which include a single solitary wave, &nbsp;two solitary wave interaction, and a Maxwellian initial condition is designed for getting the approximate solutions for the generalized equal width (GEW) equation. Initially, the considered problem has been split into 2 sub-equations as linear $U_t=\hat{A}(U)$ and nonlinear $U_t=\hat{B}(U)$ in the &nbsp;terms of time. &nbsp;After, numerical schemes have been constructed for these sub-equations utilizing the finite element method (FEM) together with quintic B-splines. Lie-Trotter splitting technique $\hat{A}o\hat{B}$ has been &nbsp;used to generate approximate solutions of the main equation. The stability analysis of acquired numerical schemes has been examined by the Von Neumann method. Also, the error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$, $I_2$, and $I_3$, respectively are calculated to illustrate how perfect solutions this new algorithm applied to the problem generates and the ones produced are compared with those in the literature. These new results exhibit that the algorithm presented in this paper is more accurate and successful, and easily applicable to other non-linear partial differential equations (PDEs) as the present equation.Solitary waves with two new nonlocal Boussinesq types equations using a couple of integration schemes
https://cmde.tabrizu.ac.ir/article_17767.html
The Boussinesq equation and its related types are able to provide a significant explanation for a variety of different physical processes that are relevant to plasma physics, ocean engineering, and fluid flow. Within the framework of shallow water waves, the aim of this research is to find solutions for solitary waves using newly developed nonlocal models of Boussinesq&rsquo;s equations. The extraction of bright and dark solitary wave solutions along with bright&ndash;dark hybrid solitary wave solutions is accomplished through the implementation of two integration algorithms. The general projective Riccati equations method and the enhanced Kudryashov technique are the ones that have been implemented as techniques. The enhanced Kudryashov method combines the benefits of both the original Kudryashov method and the newly developed Kudryashov method, which may generate bright, dark, and singular solitons. The Projective Riccati structure is determined by two functions that provide distinct types of hybrid solitons. The solutions get increasingly diverse as these functions are combined. The techniques that were applied are straightforward and efficient enough to provide an approximation of the solutions discovered in the research. Furthermore, these techniques can be utilized to solve various kinds of nonlinear partial differential equations in mathematical physics and engineering. In addition, plots of the selected solutions in three dimensions, two dimensions, and contour form are provided.Plasma particles dispersion based on Bogoyavlensky-Konopelchenko mathematical model
https://cmde.tabrizu.ac.ir/article_17740.html
An optimal system of Lie infinitesimals has been used in an investigation to find a solution to the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation (BKE). This investigation was conducted to characterize certain fantastic characteristics of plasma-particle dispersion. A careful investigation into the Lie space with an unlimited number of dimensions was carried out to locate the relevant arbitrary functions. When developing accurate solutions for the BKE, it was necessary to establish an optimum system that could be employed in single, double, and triple combination forms.There were some fantastic wave solutions developed, and these were depicted visually. The Optimal Lie system demonstrates that it can obtain many accurate solutions to evolution equations.Innovative Computational Approach for fuzzy space-time fractional Telegraph equation via the New Iterative Transform Method
https://cmde.tabrizu.ac.ir/article_17737.html
In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and Physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results.The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence.A numerical approach for solving Caputo-Prabhakar distributed-order time-fractional partial differential equation
https://cmde.tabrizu.ac.ir/article_17417.html
In this paper, we proposed a numerical method based on the shifted fractional order Jacobi and trapezoid methods to solve a type of distributed partial differential equations. The fractional derivatives are considered in the Caputo-Prabhakar type. By shifted fractional-order Jacobi polynomials our proposed method can provide highly accurate approximate solutions by reducing the problem under study to a set of algebraic equations which is technically simpler to handle. In order to demonstrate the error estimates, several lemmas are provided. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.A NUMERICAL APPROACH FOR SOLVING THE FRACTAL ORDINARY DIFFERENTIAL EQUATIONS
https://cmde.tabrizu.ac.ir/article_17734.html
In this paper, fractal differential equations are solved numerically. Here, the typical fractal equation isconsidered as follows:du(t) /dt = f ft; u(t)g ; &gt; 0:f can be a nonlinear function and the main goal is to get u(t). The continuous and discrete modes ofthis method have differences, so that subject must be carefully studied. How to solve fractal equations in their discrete form will be another goal of this research and also its generalization to higher dimensions than other aspects of this research.Convolutional neural network-based high capacity predictor estimation for reversible data embedding in cloud network
https://cmde.tabrizu.ac.ir/article_17735.html
This paper proposes a reversible data embedding algorithm in encrypted images of cloud storage where the embedding was performed by detecting a predictor that provides a maximum embedding rate. Initially, the scheme generates trail data which are embedded using the prediction error expansion in the encrypted training images to obtain the embedding rate of a predictor. The process is repeated for different predictors from which the predictor that offers the maximum embedding rate is estimated. Using the estimated predictor as the label the Convolutional neural network (CNN) model is trained with the encrypted images. The trained CNN model is used to estimate the best predictor that provides the maximum embedding rate. The estimation of the best predictor from the test image does not use the trail data embedding process. The evaluation of proposed reversible data hiding uses the datasets namely BossBase and BOWS-2 with the metrics such as embedding rate, SSIM, and PSNR. The proposed predictor classification was evaluated with the metrics such as classification accuracy, recall, and precision. The predictor classification provides an accuracy, recall, and precision of 92.63\%, 91.73\%, and 90.13\% respectively. The reversible data hiding using the proposed predictor selection approach provides an embedding rate of 1.955 bpp with a PSNR and SSIM of 55.58dB and 0.9913 respectively.EIGENVALUE INTERVALS OF PARAMETERS FOR ITERATIVE SYSTEMS OF NONLINEAR HADAMARD FRACTIONAL BOUNDARY VALUE PROBLEMS
https://cmde.tabrizu.ac.ir/article_17788.html
This study uses a classic fixed point theorem of cone type in a Banach space to identify the eigenvalueintervals of parameters for which an iterative system of a Hadamard fractional boundary value problem has at least one positive solution. To the best of our knowledge, no attempt has been made to obtain such results for Hadamard-type problems in the literature. We provided an example to illustrate the feasibility of our findings in order to show how effective they are.Efficient family of three-step with-memory methods and their dynamics
https://cmde.tabrizu.ac.ir/article_17020.html
In this work, we have proposed a general manner to extend some two-parametric with-memory methods to obtain simple roots of nonlinear equations. Novel improved methods are two-step without memory and have two self-accelerator parameters that do not have additional evaluation. The methods have been compared with the nearest competitions in various numerical examples. Anyway, the theoretical order of convergence is verified. The basins of attraction of the suggested methods are presented and corresponded to explain their interpretation.Higher-order multi-step Runge-Kutta-Nystrom methods with frequency dependent coefficients for second-order initial value problem u''=f(x,u,u')
https://cmde.tabrizu.ac.ir/article_17800.html
In this study, for the numerical solution of general second-order ordinary differential equations (ODEs) that exhibit oscillatory or periodic behavior, fifth- and sixth-order explicit multi-step Runge-Kutta-Nystrom (MSGRKN) methods, respectively, are constructed. The parameters of the proposed methods rely on the frequency $\omega$ of each problem whose solution is a linear combination of functions $\{e^{ (i \omega x)},\quad e^{ (-i \omega x)}\}$ or $\{\cos (\omega x),\quad \sin (\omega x)\}$. The study also includes an analysis of the linear stability of the suggested methods. The numerical results indicate the efficiency of the proposed methods in solving such problems compared to methods with similar characteristics in the literature.Approximate solutions of inverse Nodal problem with conformable derivative
https://cmde.tabrizu.ac.ir/article_16924.html
Our research is about the Sturm-Liouville equation which contains conformable fractional derivatives of order $\alpha \in (0,1]$ in lieu of the ordinary derivatives. First, we present the eigenvalues, eigenfunctions, and nodal points, and the properties of nodal points are used for the reconstruction of an integral equation. Then, the Bernstein technique was utilized to solve the inverse problem, and the approximation of solving this problem was calculated. Finally, the numerical examples were introduced to explain the results. Moreover, &nbsp;the analogy of this technique is shown in a numerical example with the Chebyshev interpolation technique .A Chebyshev pseudo-spectral based approach for solving Troesch's problem with convergence analysis
https://cmde.tabrizu.ac.ir/article_17805.html
In this article, the Chebyshev pseudo-spectral (CPS) method is presented for solving Troesch's problem, which is a singular, highly sensitive, and nonlinear boundary problem and occurs in an consideration of the confinement of a plasma column by radiation pressure. Here, a continuous time optimization (CTO) problem corresponding to Troesch's problem is first proposed. Then, the Chebyshev pseudo-spectral method is used to convert the CTO problem to a discrete time optimization problem that its optimal solution can be find by nonlinear programming methods. The feasibility and convergence of the generated approximate solutions are analyzed. The proposed method is used to solve various kinds of Troesch's equation. The obtained results have been compared with approximate solutions resulted from well-known numerical methods. It can be confirmed that the numerical solutions resulted from this method are completely acceptable and accurate, compared with other techniques.The complex SEE transform technique in difference equations and differential difference equations
https://cmde.tabrizu.ac.ir/article_17828.html
Differential equations are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the difference and differential-difference equations by benefiting from the properties of this complex transform to solve some problems related to difference and differential-difference equations. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-difference problems with initial conditions.Generalization of Katugampola fractional kinetic equation involving incomplete H-function
https://cmde.tabrizu.ac.ir/article_17756.html
In this article, Katugampola fractional kinetic equation (KE) has been expressed in terms of polynomial along with incomplete $H$-function, incomplete Meijer's $G$-function, incomplete Fox-Wright function and incomplete generalized hypergeometric function, weighing the novel significance of the fractional KE that appear in a variety of scientific and engineering scenarios. $\tau$-Laplace transform is used to solve the Kathugampola fractional KE. The obtained solutions have been presented with some real values and the simulation done via MATLAB. Furthermore, the numerical and graphical interpretations are also mentioned to illustrate the main results. Each of the obtained conclusions is of a general nature and is capable of generating the solutions to several fractional KE.The use of Sinc-collocation method for solving steady–state concentrations of carbon dioxide absorbed into phenyl glycidyl ether
https://cmde.tabrizu.ac.ir/article_17745.html
In this paper&lrm;, &lrm;the Sinc-collocation method is applied to solve a system of coupled nonlinear differential equations that report the chemical reaction &lrm;&lrm;&lrm;of carbon dioxide CO$_2$ and phenyl glycidyl ether in solution&lrm;. &lrm;The model has Dirichlet and Neumann boundary conditions&lrm;. &lrm;The given &lrm;scheme has &lrm;transformed &lrm;this &lrm;problem &lrm;into&lrm; some algebraic equations&lrm;. &lrm;The approach is quite simple to handle and the new numerical solutions are compared with some known solutions, which shows that the new technique is accurate and efficient&lrm;.Extended hyperbolic function method for the model having cubic-quintic-septimal nonlinearity in weak nonlocal
https://cmde.tabrizu.ac.ir/article_17774.html
Optical solitons are self-trapped light beams that maintain their shape and transverse dimensionduring propagation. This paper investigates the propagation of solitons in an optical materialwith a weak nonlocal media, modeled by a cubic-quintic-septimal nonlinearity. The extendedhyperbolic function method is used to derive the exact traveling wave solutions of the equationexpressed in hyperbolic, rational and trigonometric functions multiplied by exponential functionsin the form of the periodic, bright, kink and singular type solitons. These solutions provideexplicit expressions for the behavior of optical waves in media. Our findings provide betterunderstanding of the dynamics of the nonlinear waves in optical media and may have practicalapplications in optical communication and signal processing. The role of nonlocal nonlinearityand time constant on soliton solutions is also discussed with the help of graphs.Designing an efficient algorithm for fractional partial integro-differential viscoelastic equations with weakly singular kernel
https://cmde.tabrizu.ac.ir/article_17816.html
In this paper, the discretization method is developed by means of Mott-fractional Mott functions (MFM-Fs) for solving fractional partial integro-differential viscoelastic equations with weakly singular kernels. By taking into account the Riemann-Liouville fractional integral operator and operational matrix of integration, we convert the proposed problem to fractional partial integral equations with weakly singular kernels. It is necessary to mention that the operational matrices of integration are obtained with new numerical algorithms. These changes effectively affect the solution process and increase the accuracy of the proposed method. Besides, we investigate the error analysis of the approach. Finally, several examples are solved applying the discretization method combining MFM-Fs and the gained results are compared with the methods available in the literature.A unified Explicit form for difference formulas for fractional and classical derivatives and applications
https://cmde.tabrizu.ac.ir/article_17827.html
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivative with a desired order of accuracy at any nodal point in the computational domain. It also gives Gr\"unwald type approximations for fractional derivatives with arbitrary order of approximation at any point. Thus, this explicit unifies approximations of both types of derivatives. Moreover, for classical derivatives, it provides various finite difference formulas such as forward, backward, central, staggered, compact, non-compact, etc. Efficient computations of the coefficients of the difference formulas are also presented that lead to automating the solution process of differential equations with a given higher order accuracy. Some basic applications are presented to demonstrate the usefulness of this unified formulation.Analysis of the effect of isolation on the transmission dynamics of COVID-19: a mathematical modelling approach
https://cmde.tabrizu.ac.ir/article_17830.html
COVID-19 was declared a pandemic on March 11, 2020, after the global cases and mortalities in more than 100 countries surpassed 100 000 and 3 000, respectively. Because of the role of isolation in disease spread and transmission, a system of differential equations was developed to analyse the effect of isolation on the dynamics of COVID-19. The validity of the model was confirmed by establishing the positivity and boundedness of its solutions. Equilibria analysis was conducted, and both zero and nonzero equilibria were obtained. The effective and basic reproductive ratios were also derived and used to analyse the stability of the equilibria. The disease-free equilibrium is stable both locally and globally if the reproduction number is less than one; otherwise, it is the disease-endemic equilibrium that is stable locally and globally. A numerical simulation was carried out to justify the theoretical results and to visualise the effects of various parameters on the dynamics of the disease. Results from the simulations indicated that COVID-19 incidence and prevalence depended majorly on the effective contact rate and per capita probability of detecting infection at the asymptomatic stage, respectively. The policy implication of the result is that disease surveillance and adequate testing are important to combat pandemics.A Green's function-based computationally efficient approach for solving a kind of nonlocal BVPs
https://cmde.tabrizu.ac.ir/article_17831.html
This study attempts to find approximate numerical solutions for a kind of second-order nonlinear differential problem subject to some Dirichlet and mixed-type nonlocal (specifically three-point) boundary conditions, appearing in various realistic physical phenomena, such as bridge design, control theory, thermal explosion, thermostat model, and the theory of elastic stability. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of nonlinear differential problems with nonlocal boundary conditions. Picard's iterative technique and quasilinearization method are the basis for the proposed coupled iterative methodology. In order to convert nonlinear boundary value problems to linearized form, the quasilinearization approach (with convergence controller parameters) is implemented. Making use of the Picard's iteration method with the assistance of Green's function, an equivalent integral representation for the linearized problems is derived. Discussion is also had over the proposed method's convergence analysis. In order to determine its efficiency and effectiveness, the coupled iterative technique is tested on some numerical examples. Results are also compared with the existing techniques and documented (in terms of absolute errors) to validate the accuracy and precision of the proposed iterative technique.ON THE AN EFFICIENT METHOD FOR THE FRACTIONAL NONLINEAR NEWELL–WHITEHEAD–SEGEL EQUATIONS
https://cmde.tabrizu.ac.ir/article_17833.html
In this study, the time-fractional Newell-Whitehead-Segel (NWS) equation and its different nonlinearity cases are investigated. Schemes obtained by Newtonian linearization method are used to numerically solve different cases of the time-fractional Newell-Whitehead-Segel (NWS) equation. Stability and convergence conditions of Newtonian linearization method have been determined for the related equation. The numerical results obtained as a result of the appropriate stability criteria are compared with the help of tables and graphs with exact solutions for different fractional values.SOLITON SOLUTIONS TO THE DS AND GENERALIZED DS SYSTEM VIA AN ANALYTICAL METHOD
https://cmde.tabrizu.ac.ir/article_17834.html
In this article, the exact solutions for nonlinear Drinfeld-Sokolov (DS) and generalized Drinfeld-Sokolov (gDS) equations are established. The rational Exp-function method (EFM) is used to construct solitary and soliton solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Also exact solutions with solitons and periodic structures are obtained. The obtained results are not only presented numerically but are also accompanied by insightful physical interpretations, enhancing the understanding of the complex dynamics described by these mathematical models. The utilization of the rational EFM and the broad spectrum of obtained solutions contribute to the depth and significance of this research in the field of nonlinear wave equationsAPPLICATION OF A NEW METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER ARISING IN FLUID MECHANICS
https://cmde.tabrizu.ac.ir/article_17835.html
In this work, we established some exact solutions for the (2+1)-dimensional Zakharov-Kuznetsov, KdV and K(2,2) equations which are considered based on the improved Exp-function method, by utilizing Maple software. We use the fractional derivatives with fractional complex transform. We obtained new periodic solitary wave solutions. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. Many other such types of nonlinear equations arising in fluid dynamics and nonlinear phenomenaA method of lines for solving the nonlinear time- and space-fractional Schrodinger equation via stable Gaussian radial basis function interpolation
https://cmde.tabrizu.ac.ir/article_17837.html
The stable Gaussian radial basis function (RBF) interpolation is applied to solve the time- and space-fractional Schrodinger equation (TSFSE) in one and two dimensional cases. In this regard, the fractional derivatives of stable Gaussian radial basis function interpolants are obtained. By a method of lines the computations of the TSFSE are converted to a coupled system of Caputo fractional ODEs. To solve the resulted system of ODEs, a high order finite difference method is proposed, and the computations are reduced to a coupled system of nonlinear algebraic equations, in each time step. Numerical illustrations are performed to certify the ability and accuracy of the new method. Some comparisons are made with the results in other literature.COMPARISON OF FEATURE-BASED ALGORITHMS FOR LARGE-SCALE SATELLITE IMAGE MATCHING
https://cmde.tabrizu.ac.ir/article_17841.html
Using different algorithms to extract, describe, and match features requires knowing their capabilities and weaknesses in various applications. Therefore, it is a basic need to evaluate algorithms and understand their performance and characteristics in various applications. In this article, classical local feature extraction and description algorithms for large-scale satellite image matching are discussed. Eight algorithms, SIFT, SURF, MINEIGEN, MSER, HARRIS, FAST, BRISK and, KAZE, have been implemented, and the results of their evaluation and comparison have been presented on two types of satellite images. In previous studies, comparisons have been made between local feature algorithms for satellite image matching. However, the difference between the comparison of algorithms in this article and the previous comparisons is in the type of images used, which both reference and query images are large-scale, and the query image covers a small part of the reference image. The experiments were conducted in three criteria: time, repeatability, and accuracy. The results showed that the fastest algorithm was Surf, and in terms of repeatability and accuracy, Surf and Kaze got the first rank, respectively.Solving Initial Value Problems Using Multilayer Perceptron Artificial Neural Networks
https://cmde.tabrizu.ac.ir/article_17842.html
This research introduces a novel approach using artificial neural networks (ANNs) to tackle ordinary differential equations (ODEs) through an innovative technique called enhanced back-propagation (EBP)&lrm;. &lrm;The ANNs adopted in this study&lrm;, &lrm;particularly multilayer perceptron neural networks (MLPNNs)&lrm;, &lrm;are equipped with tunable parameters such as weights and biases&lrm;. &lrm;The utilization of MLPNNs with universal approximation capabilities proves to be advantageous for ODE problem-solving&lrm;. &lrm;By leveraging the enhanced back-propagation algorithm&lrm;, &lrm;the network is fine-tuned to minimize errors during unsupervised learning sessions&lrm;. &lrm;To showcase the effectiveness of this method&lrm;, &lrm;a diverse set of initial value problems for ODEs are solved and the results are compared against analytical solutions and conventional techniques&lrm;, &lrm;demonstrating the superior performance of the proposed approach&lrm;The use of Technological Intelligence Model in Solving Terrorism Dynamics: A Case Study of Nigeria
https://cmde.tabrizu.ac.ir/article_17843.html
Nigeria is one of the most populated countries in West Africa and is in seventh position globally. The issue of terrorism has become a common problem in Nigeria, and the government has been applying local strategies to address the situation but has yet to produce good results. The challenges necessitate the effort in this paper to develop a new deterministic model to curb terrorism and insurgency through technology intelligence in Nigeria. This analysis indicates that unmanned aerial vehicles (UAV) and the transmission rate per capita are the most sensitive parameters. Also pictured from the graphs in Figures 2, 3, and 4 were drone used to reduced the number of informants of both the terrorist and kidnapper individuals in Nigeria. Finally, this paper recommended the model adopted for controlling terrorism in Nigeria.An Efficient High-Order Compact Finite Difference Scheme for Lane-Emden Type Equations
https://cmde.tabrizu.ac.ir/article_17844.html
&lrm;In this paper&lrm;, &lrm;an efficient high&lrm;-order &lrm;compact finite difference (HOCFD) scheme is introduced for solving generalized Lane-Emden equations&lrm;. &lrm;For nonlinear types&lrm;, &lrm;it is shown that a combined quasilinearization and HOCFD scheme gives excellent results while a few quasilinear iterations is needed&lrm;. &lrm;Then the proposed method is developed for solving the system of linear and nonlinear Lane-Emden equations&lrm;. &lrm;Some numerical examples are provided&lrm;, &lrm;and obtained results of the proposed method are then compared with previous well-established methods&lrm;. &lrm;The numerical experiments show the accuracy and efficiency of the proposed method&lrm;.Alternating Direction Implicit Method for Approximation Solution of the HCIR Model, including Transaction Costs in a Jump-Diffusion Model
https://cmde.tabrizu.ac.ir/article_17845.html
The standard model, which determines option pricing, is the well-known Black-Scholes formula. Heston in addition to Cox-Ingersoll-Ross which is called CIR, respectively, implemented the models of stochastic volatility and interest rate to the standard option pricing model. The cost of transaction, which the Black-Scholes method overlooked, is another crucial consideration that must be made when trading a service or production. It is acknowledged that by employing the log-normal stock diffusion hypothesis with constant volatility, the Black-Scholes model for option pricing departs from reality. The standard log-normal stock price distribution used in the Black-Scholes model is insufficient to account for the leaps that regularly emerge in the discontinuous swings of stock prices. A jump-diffusion model, which combines a jump process and a diffusion process, is a type of mixed model in the Black-Scholes model belief. Merton developed a jump model as a modification of jump models to better describe purchasing and selling behavior. In this study, the Heston-Cox-Ingersoll-Ross (HCIR) model with transaction costs is solved using the alternating direction implicit (ADI) approach and the Monte Carlo simulation assuming the underlying asset adheres to the jump-diffusion case, then the outcomes are compared to analytical solution. In addition, the consistency of the numerical method is proven for the model.q-Exponential Fixed Point Theorem for Mixed Monotone Operator with q-Fractional Problem
https://cmde.tabrizu.ac.ir/article_17850.html
In this work, we examine the existence and uniqueness(EU) of q-Exponential positive solution(q-EPS) of the hybrid q-fractional boundary value problem (q-FBVP).We prove the q-Exponential fixed point theorem (q-EFPT) with a new set &rho;h,e1in the Banachspace E to check the EU of q-EPS of the q-FBVP. In the long run, an exemplum is given toshow the correctness of our results.Symmetries of the minimal Lagrangian hypersurfaces on Cylindrically Symmetric Static Space-Times
https://cmde.tabrizu.ac.ir/article_17851.html
In this work, we study a hypersurface immersed in specific types of cylindrically symmetric static space-times, then we identify the gauge fields of the Lagrangian that minimizes the area besides the Noether symmetries. We show that these symmetries are part of the Killing algebra of cylindrically symmetric static space-times. By using Noether's theorem, we construct the conserved vector fields for the minimal hypersurface.Unveiling Traveling Waves and Solitons of Dirac Integrable System Via Homogenous Balance and Singular Manifolds Methods
https://cmde.tabrizu.ac.ir/article_17861.html
This study utilizes two robust methodologies to examine the precise solutions of the Dirac integrable system. The Homogeneous Balance Method (HB) is initially employed to generate an accurate solution. The system of equations for the quasi-solution is solved, where all the equations are of the same nature. The quasi-solution of the traveling wave results in the solitary wave solution of the system. The singular manifold method (SMM) is utilized following the Lie reduction of the Dirac system in order to search for the traveling wave solutions of the system. Both approaches demonstrate the existence of traveling wave solutions inside the system. The precise solutions of the Dirac system are shown in three-dimensional graphs. We have created solutions to the examined problem, including bright solutions, periodic soliton solutions, and complicated solutions.Solving a Class of Volterra Integral Equations With M-Derivative
https://cmde.tabrizu.ac.ir/article_17862.html
This article presents the development and analysis of the M-fractional Neumann method (MFNM) for solving a class of time M-fractional Volterra integral equations. The MFNM is based on the well-known Neumann method and is shown to provide efficient solutions for the considered equations. The proposed approach's existence, uniqueness, and convergence are studied using several theorems. The results demonstrate the MFNM's effectiveness in solving time M-fractional Volterra integral equations. Numerical examples are provided to illustrate the capabilities and adequacy of the MFNM for a class of fractional integral equations.FINITE-DIFFERENCE METHOD FOR HYGROTHERMOELASTIC BOUNDARY VALUE PROBLEM
https://cmde.tabrizu.ac.ir/article_17865.html
A two-dimensional coupled hygrothermoelastic medium boundary problem using Finite difference method is discussed in the present work. Explicit and Implicit finite difference schemes for this problem are formed. The solutions of these schemes are carried out using numerical methods of finite difference. These solutions are compared of and analyzed and exciting similarities were found as result.Exploring Novel Solutions for the Generalized q-Deformed Wave Equation
https://cmde.tabrizu.ac.ir/article_17866.html
Our primary goal is to address the $q$-deformed wave equation, which serves as a mathematical framework for characterizing physical systems with symmetries that have been violated. By incorporating a $q$-deformation parameter, this equation expands upon the traditional wave equation, introducing non-commutativity and non-linearity to the dynamics of the system. In our investigation, we explore three distinct approaches for solving the generalized $q$-deformed wave equation: the reduced $q$-differential transform method (R$q$DTM) \cite{1}, the separation method (SM), and the variational iteration method (VIM). The R$q$DTM is a modified version of the differential transform method specially designed to handle $q$-deformed equations. The SM aims to identify solutions that can be expressed as separable variables, while the VIM employs an iterative scheme to refine the solution. We conduct a comparative analysis of the accuracy and efficiency of the solutions obtained through these methods and present numerical results. This comparative analysis enables us to evaluate the strengths and weaknesses of each approach in effectively solving the $q$-deformed wave equation, providing valuable insights into their applicability and performance. Additionally, this paper introduces a generalization of the $q$-deformed wave equation, as previously proposed in \cite{2}, and investigates its solution using two different analytical methods: R$q$DTM, SM, and an approximation method known as VIM.Application of general Lagrange scaling functions for evaluating the approximate solution time-fractional diffusion-wave equations
https://cmde.tabrizu.ac.ir/article_17867.html
This manuscript provides an efficient technique for solving time-fractional diffusion-wave equations using general Lagrange scaling functions (GLSFs). In GLSFs, by selecting various nodes of Lagrange polynomials, we get various kinds of orthogonal or non-orthogonal Lagrange scaling functions. General Riemann-Liouville fractional integral operator (GRLFIO) of GLSFs is obtained generally. General Riemann-Liouville fractional integral operator of the general Lagrange scaling function is calculated exactly using the Hypergeometric functions. The operator extraction method is precisely calculated and this has a direct impact on the accuracy of our method. The operator and optimization method are implemented to convert the problem to a set of algebraic equations. Also, error analysis is discussed. To demonstrate the efficiency of the numerical scheme, some numerical examples are examined.On Unique Solutions of Integral Equations by Progressive Contractions
https://cmde.tabrizu.ac.ir/article_17869.html
The authors consider Hammerstein type integral equations for the purpose of obtaining new results on the uniqueness of solutions on an infinite interval. The approach used in the proofs is based on the technique called progressive contractions due to T. A. Burton. Here, the authors apply Burton's method to a general Hammerstein type integral equation that also yields existence of solutions. In much of the existing literature, investigators prove uniqueness of solutions of integral equations by applying some type of fixed point theorem. This can prove to be a difficult process that sometimes involves patching together solutions on short intervals and perhaps involving translations. In this paper, using progressive contractions in three simple short steps, each one being an elementary contraction mapping on a short interval, the authors improve the technique due to Burton by considering a general Hammerstein type integral equation, and they obtain the uniqueness of solutions on an infinite interval.Hopf bifurcation and Turing instability in a cross-diffusion prey-predator system with group defense behavior
https://cmde.tabrizu.ac.ir/article_17870.html
This paper is concerned with a cross-diffusion prey-predator system in which the prey species is equipped with the group defense ability under the Neumann boundary conditions. The tendency of the predator to pursue the prey is expressed in the cross-diffusion coefficient, which can be positive, zero, or negative. We first select the environmental protection of the prey population as a bifurcation parameter. Next, we discuss the Turing instability and the Hopf bifurcation analysis on the proposed cross-diffusion system. We show that the system without cross-diffusion is stable at the constant positive stationary solution but it becomes unstable when the cross-diffusion appears in the system. Furthermore, the stability of bifurcating periodic solutions and the direction of Hopf bifurcation are examined.Modeling and Simulation of COVID-19 Disease Dynamics via Caputo Fabrizio Fractional Derivative
https://cmde.tabrizu.ac.ir/article_17871.html
The motive of this paper is to investigate the SEIQRD model of COVID-19 outbreak in Indonesia with the help of fractional modeling approach. The model is described by the nonlinear system of six fractional order differential equations (DE) incorporating Caputo Fabrizio Fractional derivative (CFFD) operator. The existence and uniqueness of model are proved by applying the well-known Banach contraction theorem. The reproduction number ($ R_0$) is calculated, and its sensitivity analysis is conducted concerning each parameters of the model for the prediction and persistence of the infection. Moreover, the numerical simulation for various fractional orders is performed using Adams-Bashforth technique to analyze the transmission behavior of disease and to get the approximated solutions. At last, we represent our numerical simulation graphically to illustrate our analytical findings.A NEW APPROXIMATE ANALYTICAL METHOD FOR SOLVING SOME NON-LINEAR BOUNDARY VALUE PROBLEMS IN REACTION-DIFFUSION MODEL
https://cmde.tabrizu.ac.ir/article_17872.html
The applications of a Reaction-Diffusion boundary value problem are found in science, biochemical applications, and chemical applications. The Ananthaswamy-Sivasankari method (ASM) is employing to solve the considered specific models like non-linear reaction-diffusion model in porous catalysts, spherical catalysts pellet, and catalytic reaction-diffusion process in a catalyst slab. An accurate semi-analytical expression for the concentrations and effectiveness factors are given in explicit form. Graphical representations are used to display the impacts of several parameters, including the Thiele modulus, characteristic reaction rate, concentration of half saturation, reaction order and dimensionless constant in Langmuir-Hinshelwood kinetics. The impact of numerous parameters namely the Langmuir-Hinshelwood kinetics and Thiele modulus on effectiveness factors are display graphically. Our semi-analytical findings shows good match in all parameters when compared to numerical simulation using MATLAB. Many non-linear problems in chemical science especially, the Reaction-Diffusion equations, Michaelis-Menten kinetic equation, can be resolved with the aid of the new approximate analytical technique, ASM.On the dynamics of newly generated analytical solutions and conserved vectors of a generalized 3D KP-BBM equation
https://cmde.tabrizu.ac.ir/article_17925.html
This paper examines a high-dimensional non-linear partial differential equation called the generalized Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation existing in three dimensions. The Lie symmetry analysis of the equation is carried out step-by-step. In consequence, we found symmetries from which various group-invariant solutions results from which numerous solutions of interest that satisfy the KP-BBM equation are obtained. Solutions of interest secured include hyperbolic functions as well as elliptic functions with the latter being the more general of the two solutions. Besides, a good number of algebraic solutions with arbitrary functions are also achieved. Moreover, the dynamics of the solutions are further explored diagrammatically using computer software. In the concluding part, various conservation laws of the underlying model are constructed via the multiplier method as well as the Noether theorem.Convergence analysis for piecewise Lagrange interpolation method of fractal fractional model of tumor-immune interaction with two different kernels
https://cmde.tabrizu.ac.ir/article_17926.html
Ahmad et al. [Alex. Eng. J. (2022) 61, 5735-5752.] presented a piecewise Lagrange interpolation method for solving tumor-immune interaction model with fractal fractional operators using a power law and exponential kernel. We suggest a convergence analysis for this method and we obtain the order of convergence. Of course there are some mistakes in this numerical method that were corrected. Furthermore, Numerical illustrations are demonstrated to show the effectiveness of the corrected numerical method.Integrated Pests Management and Food Security: A Mathematical Analysis
https://cmde.tabrizu.ac.ir/article_17927.html
The basic necessities of life are food, shelter and clothing. Food is more necessary because the existence of life depends on food. In order to foster global food security, integrated pest management (IPM), an environmentally-friendly programme, was designed to maintain the density of pest population in the equilibrium level below the economic damage. For years, mathematics has been an ample tool to solve and analyse various real life problems in science, engineering, industry and so on but the use of mathematics to quantify ecological phenomena is relatively new. While efforts have been made to study various methods of pest control, the extent to which pests' enemies as well as natural treatment can reduce crop damage is new in the literature. Based on this, deterministic mathematical models are designed to investigate the prey-predator dynamics on a hypothetical crop field in the absence or presence of natural treatment. The existence and uniqueness of solutions of the models are examined using Derrick and Grossman theorem. The equilibria of the models are derived and the stability analysed following stability principle of differential equations and Bellman and Cooke's theorem. The theoretical results of the models are justified by a means of numerical simulations based on a set of reasonable hypothetical parameter values. Results from the simulations reveal that the presence of pests' enemies on a farm without application of natural treatment may not avert massive crop destruction. It is also revealed that the application of natural treatment may not be enough to keep the density of pest population below the threshold of economic damage unless the rate of application of natural treatment exceeds the growth rate of pest.Interpolating MLPG method to investigate predator-prey population dynamic with complex characters
https://cmde.tabrizu.ac.ir/article_17928.html
&nbsp;The predator-prey model is a pair of first-order nonlinear differential equations which is used to explain the dynamics of biological systems. These systems contain&nbsp;two species interact, one as a predator and the other as prey. This work proposes a meshless local Petrov-Galerkin (MLPG) method based upon the interpolating moving least squares (IMLS) approximation, &nbsp;for numerical solution of the predator-prey systems. With this aim, the space derivative is discretized by MLPG technique in which the test and trial functions are chosen from the shape functions of IMLS approximation. In the next, a semi-implicit finite difference approach is utilized to discrete the time derivative. The main aim of this work is to bring forward a flexible numerical procedure &nbsp;to solve predator-prey systems on the complicated geometries.THE DYNAMICAL SYSTEMS IN PRODUCT LUKASIEWICZ SEMIRINGS
https://cmde.tabrizu.ac.ir/article_18005.html
This paper studies about dynamical systems in prod-uct Lukasiewicz semirings and we generalize the results of Marke-chova and Riecan concerning the logical entropy. Also, the notionof logical entropy of a product Lukasiewicz semiring is introducedand it is shown that entropy measure is invariant under isomor-phism.Multi-Step DTM simulation of Transesterification reactions model
https://cmde.tabrizu.ac.ir/article_18018.html
The multi-step differential transform method (DTM) adopted from the standard DTM was employed in this case study to solve a model of the transesterification reaction. The DTM is considered in a sequence of time intervals. The accuracy of the proposed method was confirmed by comparing its results with those of the fourth order RungeKutta (RK4) method. In addition, the experimental results were investigated with the Multi-step DTM to demonstrate the efficiency and effectiveness of these chemical reactions obtained in the laboratory. The present findings confirmed the effectiveness of using the multi-step DTM in validating the chemical models obtained in laboratories.Highly accurate spline collocation technique for numerical solution of generalized Burgers-Fisher's problem
https://cmde.tabrizu.ac.ir/article_18047.html
This study employs the cubic B-spline collocation strategy to address the solution challenges posed for the nonlinear generalized Burgers-Fisher's equation (gBFE), with some improvisation. This approach incorporates refinements within the spline interpolants, resulting in enhanced convergence rates along the spatial dimension. Temporal integration is achieved through the Crank Nicolson methodology. The stability of the technique is assessed using the rigorous von Neumann method. Convergence analysis based on Green's function reveals a fourth-order convergence along space domain and second-order convergence along temporal domain. The results are validated by taking number of examples. MATLAB 2017 is used for computational work.Numerical solution of different population balance models using operational method based on Genocchi polynomials
https://cmde.tabrizu.ac.ir/article_18065.html
Genocchi polynomials have exciting properties in the approximation of functions. Their derivative and integral calculations are simpler than other polynomials and, in practice, they give better results with low degrees. For these reasons, in this article, after introducing the important properties of these polynomials, we use them to approximate the solution of different population balance models. In each case, we first discuss the solution method and then do the error analysis. Since we do not have an exact solution, we compare our numerical results with those of other methods. The comparison of the obtained results shows the efficiency of our method. The validity of the presented results is indicated using MATLAB-Simulink.Optimal control of fractional differential equations with interval uncertainty
https://cmde.tabrizu.ac.ir/article_18066.html
The purpose of this paper is to obtain numerical solutions of fractional interval optimal control problems. To do so, first, we obtain a system of fractional interval differential equations through necessary conditions for the optimality of these problems, via the interval calculus of variations in the presence of interval constraint arithmetic. Relying on the trapezoidal rule, we obtain a numerical approximation for the interval Caputo fractional derivative. This approach causes the obtained conditions to be converted to a set of algebraic equations which can be solved using an iterative method such as the interval Gaussian elimination method and interval Newton method. Finally, we solve some examples of fractional interval optimal control problems in order to evaluate the performance of the suggested method and compare the past and present achievements in this manuscript.Limit cycles in piecewise smooth differential systems of focus-focus and saddle-saddle dynamics
https://cmde.tabrizu.ac.ir/article_18067.html
In this paper, we obtained the Poincare return maps for the planar piecewise linear differential systems of the type focus-focus. Normal forms for planar piecewise smooth systems with two zones of the type focus-focus and saddle-saddle, separated by a straight line and with a center at the origin, are obtained. Upper bounds for the number of limit cycles bifurcated from the period annulus of these normal forms due to perturbation by polynomial functions of any degree are established.Application of new Kudryashov method to Sawada-Kotera and Kaup-Kupershmidt equations
https://cmde.tabrizu.ac.ir/article_18072.html
In this article, with the help of the new Kudryashov method, we examine general solutions to the (2+1)-dimensional Sawada-Kotera equation (SKE) and Kaup&ndash;Kupershmidt (KK) equation. Using Maple, a symbolic computing application, it was shown that all obtained solutions are given by hyperbolic, exponential and logaritmic function solutions which obtained solutions are useful for fluid dynamics, optics and so on. Finally, we have presented some graphs for general solutions of these equations with special parameter values. The reliability and scope of programming provide eclectic applicability to high-dimensional nonlinear evolution equations for the development of this method. The results found gave us important information regarding the applicability of the new Kudrashov method.Exploring High-Frequency Waves and Soliton Solutions of Fluid Turbulence through Relaxation Medium Modeled By Vakhnenko-Parkes Equation
https://cmde.tabrizu.ac.ir/article_18073.html
One of the most important natural phenomena that has been studied extensively in engineering, oceanography, meteorology and other fields is called fluid turbulence (FT). FT stands for irregular flow of fluid. Scientists detected models to describe this phenomenon, among these models is the (3+1)-dimensional Vakhnenko-Parkes (VP) equation. In this research, the high-frequency waves&rsquo; dynamical behavior through the relaxation medium is explored considering two semi-analytic methods, the (G^'/G) and the tanh-coth (TC) expansion methods. Nineteen different solutions have been detected and some of these solutions have been illustrated graphically. Figures show a range of degenerate, periodic, and complex propagating soliton wave solutions.A Novel Hybrid Approach to Approximate fractional Sub-Diffusion Equation
https://cmde.tabrizu.ac.ir/article_18075.html
This article introduces a new numerical hybrid approach based on an operational matrix and spectral technique to solve Caputo fractional sub-diffusion equations. This method transform the model into a set of nonlinear algebraic equation system. Chebyshev polynomials are used as basis function. The study includes theoretical analysis to demonstrate the convergence and error bounds of the proposed method. Two test problems are conducted to illustrate the method's accuracy. The results indicate the efficiency of the proposed method.NUMERICAL STUDY OF ASTROPHYSICS EQUATIONS USING BESSEL COLLOCATION METHODS OF FIRST KIND
https://cmde.tabrizu.ac.ir/article_18146.html
A hybrid computational procedure of Newton Raphson method and orthogonal collocation has been applied to study the behaviour of non-linear astrophysics equations. The non-linear Lane Emden equation has been discretized using the orthogonal collocation method using $n^{th}$-order Bessel polynomial as $J_n(\xi)$ as base function. The system of collocation equations has been solved numerically using Newton Raphson's method. Numerical examples have been discussed to check the reliability and efficiency of the scheme. Numerically calculated results have been compared to the exact values as well as the values already given in the literature to check the compatibility of the scheme. Error analysis has been studied by calculating the absolute error, $L_2- norm$ and $L_{\infty}- norm$. Computer codes have been prepared using MATLAB.Generalized Solutions for Conformable Schrödinger Equations with Singular Potentials
https://cmde.tabrizu.ac.ir/article_18147.html
This paper employs Colombeau algebra as a mathematical framework to establish both the existence and uniqueness of solutions for the fractional Schr&ouml;dinger equation when subjected to singular potentials. A noteworthy contribution lies in the introduction of the concept of a generalized conformable semigroup, marking the first instance of its application. This innovative approach plays a pivotal role in demonstrating the sought-after results within the context of the fractional Schr&ouml;dinger equation. The utilization of Colombeau algebra, coupled with the introduction of the generalized conformable semigroup, represents a novel and effective strategy for addressing challenges posed by singular potentials in the study of this particular type of Schr&ouml;dinger equation.Existence and Uniqueness Theorems for Fractional Differential Equations with Proportional Delay
https://cmde.tabrizu.ac.ir/article_18167.html
In this paper, we applied successive approximation method (SAM) to deal with the solution of non-linear differential equations (DEs) with proportional delay. Utilizing SAM we derived the results about existence and uniqueness. The differential equations (DEs) with proportional delay are a particular case of the time-dependent delay differential equations (DDEs). In this sense, we demonstrated that the equilibrium solution of time-dependent DDEs is asymptotically stable on finite time intervals. We obtained a series solution of pantograph and Ambartsumian equations and proved its convergence. Further, we proved that the zero solution of pantograph and Ambartsumian equations are asymptotically stable. The outcomes of integer order obtained for DEs with proportional delay and time-dependent DDEs have been extended to initial value problem (IVP) for fractional DDEs and a system of fractional DDEs involving Caputo fractional derivative. Finally, we illustrate the efficacy of the SAM by considering particular non-linear DEs with proportional delay. The results obtained for non-linear DEs with proportional delay by SAM are compared with exact solutions and other iterative methods. It is noted that SAM is easier to use than other techniques and the solutions obtained using SAM are consistent with the exact solution.FITTED MESH CUBIC SPLINE TENSION METHOD FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITION
https://cmde.tabrizu.ac.ir/article_18170.html
The cubic spline in tension method is taken into consideration to solve the singularly perturbed delay differential equations of convection diffusion type with integral boundary condition. Simpson&rsquo;s 1/3 rule is used to the non-local boundary condition and two model problems are examined for numerical treatment and are addressed using a variety of values for the perturbation parameter ϵ and the mesh size to verify the scheme&rsquo;s applicability. The computational results and rate of convergence are given in tables, and it is seen that the proposed method is more precise and improves the methods used in the literature.Numerical study of the non-linear time fractional Klein-Gordon equation using the Pseudo-spectral Method
https://cmde.tabrizu.ac.ir/article_18171.html
This paper presents a numerical scheme for solving the non-linear time fractional Klein-Gordon equation. To approximate spatial derivatives, we employ the pseudo-spectral method based on Lagrange polynomials at Chebyshev points, while using the finite difference method for time discretization. Our analysis demonstrates that this scheme is unconditionally stable, with a time convergence order of $\mathcal{O}({3-\alpha})$. Additionally, we provide numerical results in one, two, and three dimensions, highlighting the high accuracy of our approach. The significance of our proposed method lies in its ability to efficiently and accurately address the non-linear time fractional Klein-Gordon equation. Furthermore, our numerical outcomes validate the effectiveness of this scheme across different dimensions.Computational Approaches for Analyzing Fractional Impulsive Systems in Differential Equations
https://cmde.tabrizu.ac.ir/article_18178.html
This research introduces an algorithmically efficient framework for analyzing the fractional impulsive system, which can be seen as specific instances of the broader fractional Lorenz impulsive system. Notably, these systems find pertinent applications within the financial domain. To this end, the utilization of cubic splines is embraced to effectively approximate the fractional integral within the context of the system. The outcomes derived from this method are subsequently compared with those yielded by alternative techniques documented in existing literature, all pertaining to the integration of functions.Furthermore, the proposed methodology is not only applied to the resolution of fractional impulsive system, but also extended to encompass scenarios involving the fractional Lorenz system with impulsive characteristics. The discernible effects stemming from the selection of disparate impulse patterns are meticulously demonstrated. In synthesis, this paper endeavors to present a pragmatic and proficient resolution to the intricate challenges posed by impulsive systems.Mechanics of nanofluidic flow induced nonlinear vibrations of single and multi-walled branched nanotubes in a thermal-magnetic environment
https://cmde.tabrizu.ac.ir/article_18239.html
The nonlinear vibration analysis of embedded multi-walled branching nanotubes with integrated nanofluids that are resting on a Winkler-Pasternak foundation in a thermal-magnetic environment is the main emphasis of this work. The coupled equations of motion controlling the transverse and longitudinal vibrations of the nanotube are derived using the Euler-Bernoulli theory, the Hamilton&rsquo;s principle, and nonlocal elasticity theory. Additionally, the pressure variation in the tubes and the equation for the deformation of the nanotubes are derived. Furthermore, the vibration models are coupled with the Navier-Stokes equation and the energy equation for the fluid and nanotube. Since the dynamics of multi-walled carbon nanotubes differ from the typical assumption of plug flow, careful investigation is needed when combining them with Navier-Stokes and energy equations. Thus, the generated coupled systems of nonlinear partial differential equations in this work are solved using multi-dimensional differential transformation method. With the aid of the analytical solution, parametric studies are performed. The findings show that the system's stability reduces as the downstream angle increases. Furthermore, the system's dynamic behavior yielded results that show the magnetic effect has a 20% attenuating or damping effect. Additionally, there is a more than 11% discrepancy between the plug flow assumption and real functioning procedures. Existing analytical, numerical, and experimental results were used to verify and validate the analytical method. It is hoped that this study will provide further understanding of the design of nanotubes and act as a reference for further research in the fieldAnalysis of a chaotic and a non-chaotic 3D dynamical system: The Quasi-Geostrophic Omega Equation and the Lorenz-96 model
https://cmde.tabrizu.ac.ir/article_18240.html
This paper delves into the analysis of two 3D dynamical systems of ordinary differential equations (ODEs), namely the Quasi-Geostrophic Omega Equation and the Lorenz-96 Model. The primary objective of this paper is to analyze the chaotic and non-chaotic behavior exhibited by the QG Omega Equation and the Lorenz-96 Model in three dimensions. Through numerical simulations and analytical techniques, the author aimed to characterize the existence and properties of attractors within these systems and explore their implications for atmospheric dynamics. Also, the author has investigated how changes in initial conditions and system parameters influence the behavior of the dynamical systems. Furthermore, we investigate how changes in initial conditions and system parameters influence the behavior of the dynamical systems. Employing a combination of numerical simulations and analytical methods, including stability analysis and Lyapunov function, we uncover patterns and correlations that shed light on the mechanisms driving atmospheric phenomena. This analysis contributes to the understanding of atmospheric dynamics and has implications for weather forecasting and climate modeling, offering insights into the predictability and stability of atmospheric systems. Finally, We draw the phase portrait of the chaotic system and visualizations of the attractors of both systems. Concluding how the rigorous proofs of attractor existence as long as the matrix transformation of the system with the numerical methods followed may pave the way for future research in understanding and analyzing the behavior of chaotic weather models.Some existence and nonexistence results for a class of Kirchhoff-double phase systems in bounded domains
https://cmde.tabrizu.ac.ir/article_18256.html
In this paper, the existence and nonexistence of multiple solutions for a class of Kirchhoff-double phase systems depending on one parameter in bounded domains are considered. Our main tools are essentially based on variational techniques. To our best knowledge, there seems to be few results on Kirchhoff-double phase type systems in the existing literature.A higher order kernel approach for linear fourth order boundary value problems
https://cmde.tabrizu.ac.ir/article_18261.html
This paper aims at finding high order convergent numerical approach to solve fourth order linear boundary value problems (BVPs). By employing the good property of reproducing kernel functions (RKFs), a new collocation technique is proposed. The present approach can give highly accurate numerical solutions to fourth order BVPs. Some numerical experiments are performed and compared with other approaches to indicate the validity of the proposed technique.A nonlinear mathematical model of the delayed predator-prey system that includes the factors of intraspecific rivalry among predator species, the fear effect, and the Holling type-IV functional response
https://cmde.tabrizu.ac.ir/article_18269.html
In ecological systems, predator-prey contact is seen as something that happens naturally. How does the density of prey populations effect predators? This is a naturally occurring issue in ecosystems. Even though it plays a little role in population dynamics, predators in most ecological models lower prey numbers by direct killing. Research on vertebrates has shown that predator aversion may impact prey population dynamics and reproductive rates. There has been new research on mathematical models of predator-prey systems that include a range of predator functional responses that include the fear effect. Researchers in these research failed to account for the impact of fear on prey mortality rates. In light of the above, our study focuses on analysing a predator-prey system that incorporates the cost of perceived fear into reproductive processes using a Holling type-IV functional response. The scheme also includes intraspecific competition within the predators and a gestation delay to make the interactions more realistic and natural. The increase of the predator population is constrained for high predator to prey density ratios by this extra intraspecific competition term. These dynamic model's fundamental aspects such as non-negative, boundedness of solutions, and viability of equilibria are investigated, and adequate conditions are discovered. Both the local and global stability of the system are obtained with sufficient conditions on its functionals and parameters. This study makes a major impact in that it creates a novel technique to quantify some important, regulating system resilience parameters, and it studies the presence of Hopf bifurcation when the time lag parameters exceed the critical values by looking at the relating characteristic equation. Furthermore, we addressed how time delay factors reaching thresholds causes the Hopf bifurcation. Numerous numerical examples are used to validate all of these theoretical inferences, and simulations are given to help visualise the examples.A Study on the Fractional Ebola Virus Model by the Semi-Analytic and Numerical Approach
https://cmde.tabrizu.ac.ir/article_18270.html
In this study, An Ebola virus model involving the fractional derivatives in the Caputo sense is considered and studied through three different techniques called the Homotopy analysis method (HAM), Haar wavelet method (HWM), and Runge-Kutta method (RKM). HAM is a semi-analytical approach proposed for solving fractional order nonlinear systems of ordinary differential equations (ODEs), the Haar wavelet technique (HWT) is a numerical approach for both fractional and integer order, and the RK method is a numerical method used to solve the system of ODEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for the model. First, we solved the model through HAM by choosing the preferred control parameter. Secondly, HWT is considered; through this technique, the operational matrix of integration is used to convert the given FDEs into a set of algebraic equation systems, and then the RK method is applied. The model is studied through all three methods, and the solutions are juxtaposed with ND Solver solutions. The nature of the model is analyzed with different parameters, and the calculations are performed using Scilab and Mathematica software. The Obtained results are expressed in graphs and tables. Theorems on convergence have been discussed in terms of theorems.A HIERARCHICAL METHOD TO SOLVE ONE MACHINE MULTICRITERIA SEQUENCING PROBLEM
https://cmde.tabrizu.ac.ir/article_18294.html
Abstract. The problem of minimizing a function of three criteria maximumearliness, total of square completion times and total lateness in a hierarchical(lexicographical) method is proposed in this article. On one machine, n independently tasks (jobs) must planned. It is always available starting at timezero and can only do mono task (job) at time period. Processing for task (job)j(j = 1, 2, ..., nj) is necessary meantime the allotted positively implementationtime ptj . For the problem of three criteria maximization earliness, total of squarecompletion times, and total lateness in a hierarchy instance, the access of limitation that which is desired sequence is hold out. The Generalized Least DeviationMethod (GLDM), a robust technique for analyzing historical data to projectfuture trends is analyzed.SOLVING THE OPTIMIZING PARAMETERS PROBLEM FOR NON-LINEAR DATASETS USING THE HIGH-ORDER GENERAL LEAST DEVIATIONS METHOD (GLDM) ALGORITHM
https://cmde.tabrizu.ac.ir/article_18295.html
This study presents an innovative approach to determining the coefficients of a high-order quasilinear autoregressive model using the Generalized Least Deviations Method (GLDM). The model aims to capture the dynamics of observed state variables over time, employing a set of given functions to relate past observations to current values. The errors in the observations are considered unknown. The core innovation lies in addressing the Cauchy problem within the GLDM framework, which enhances the robustness and precision of parameter estimation for non-linear datasets. GLDM is achieved by incorporating a loss function based on the arctangent function, improving resilience against outliers and non-standard error distributions. Comprehensive computational experiments and statistical validation determine optimal model orders for various datasets, including small NDVI (Normalized Difference Vegetation Index) time series, extensive temperature time series, and large wind speed datasets. The second-order model is most effective for small NDVI datasets, while the fifth-order model excels for large temperature datasets. For wind speed data, despite its large size, the second-order GLDM model demonstrates superior performance due to its ability to balance model complexity with the need for capturing essential dynamics without overfitting. Furthermore, a comparative analysis of GLDM-based models with classical forecasting models demonstrates the superior adaptability and accuracy of GLDM models across different dataset characteristics. This highlights their robustness against outliers and data anomalies. The study underscores the versatility and efficacy of high-order GLDM models as powerful tools in predictive modeling, offering significant improvements over traditional methods.Kernel density estimation applications in vessel extraction for MRA images
https://cmde.tabrizu.ac.ir/article_18296.html
Vascular-related diseases have become increasingly significant as public health concerns. The analysis of blood vessels plays an important role in detecting and treating diseases. Extraction of vessels is a very important technique in vascular analysis. Magnetic Resonance Angiography (MRA) is a medical imaging technique used to visualize the blood vessels and vascular system in three-dimensional images. These images provide detailed information about the size and shape of the vessels, any narrowing or stenosis, as well as blood supply and circulation in the body. Tracing vessels from medical images is an essential step in diagnosing and treating vascular-related diseases. Many different techniques and algorithms have been proposed for vessel extraction. In this paper, we present a vessel extraction method based on Kernel density estimation (KDE). Numerical experiments on real 2D MRA images demonstrate that the presented method is very eﬀicient. The effectiveness of the proposed method has been proven through comparative analysis with validated existing methods.SEIaIsQRS EPIDEMIC MODEL FOR COVID-19 BY USING COMPARTMENTAL ANALYSIS AND NUMERICAL SIMULATION
https://cmde.tabrizu.ac.ir/article_18301.html
In this paper, we developed a SEIaIsQRS epidemic model for COVID-19 by using compartmentalanalysis. In this article, the dynamics of COVID-19 are divided into six compartments: susceptible,exposed, asymptomatically infected, symptomatically infected, quarantined, and recovered.The positivity and boundedness of the solutions have been proven. We calculated the basic reproduction number for our model and found both disease-free and endemic equilibria.It is shown that the disease-free equilibrium is globally asymptotically stable. We explained under what conditions, the endemic equilibrium point is locally asymptotically stable. Additionally, the center manifold theorem is applied to examine whether our model undergoes a backward bifurcation at R0 = 1 or not. To finish, we have confirmed our theoretical results by numerical simulation.Existence of solutions of Caputo fractional integro-differential equations
https://cmde.tabrizu.ac.ir/article_18302.html
In this paper, by using the techniques of measures of non-compactness and the Petryshyn fixedpoint theorem, we investigate the existence of solutions of a Caputo fractional functional integro-differential equation and obtain some new results. These existence results involve particular resultsgained from earlier studies under weaker conditions.Mathematical modelling of the epidemiology of Corona Virus Infection with constant spatial diffusion term in Ghana.
https://cmde.tabrizu.ac.ir/article_18320.html
The purpose of this study is to develop a mathematical model that incorporates diffusion term in one dimension in the dynamics of Corona Virus Disease-19 (Covid-19) in Ghana. A reaction-diffusion model is derived by applying the law of conservation of matter and Fick's law which are fundamental theorems in fluid dynamics. Since Covid-19 is declared to be pandemic, most African countries are affected by the negative impacts of the disease. However, controlling the spread becomes a challenge for many developing countries like Ghana. A lot of studies about the dynamics of the infection do not consider the fact that since the disease is pandemic, its model should be spatially dependent, therefore fail to incorporate the diffusion aspect. In this study, the local and global stability analysis are carried out to determine the qualitative solutions to the SEIQRF model. Significant findings are made from these analysis as well as the numerical simulations and results. The basic reproduction number ($R_o$) calculated at the disease-free fixed point is obtained to be $R_o\approx2.5$ implying that, an infectious individual is likely to transmit the corona virus to about three susceptible persons. A Lyapunov functional constructed at the endemic fixed point also explains that, the system is globally asymptotically stable, meaning that Covid-19 will be under control in Ghana for a long period of time.LQR Technique Based SMC Design for a Class of Uncertain Time-delay Conic Nonlinear Systems
https://cmde.tabrizu.ac.ir/article_18321.html
&lrm;In this paper&lrm;, &lrm;the finite-time sliding mode controller design problem of a class of conic-type nonlinear systems with time-delays&lrm;, &lrm;mismatched external disturbance and uncertain coefficients is investigated&lrm;. &lrm;The time-delay conic nonlinearities are considered to lie in a known hypersphere with an uncertain center&lrm;. &lrm;Conditions have been obtained to design a linear quadratic regulator based on sliding mode control&lrm;. &lrm;For this purpose&lrm;, &lrm;by applying Lyapunov&lrm;- &lrm;Krasovskii stability theory and linear matrix inequality approach&lrm;, &lrm;sufficient conditions are derived to ensure the finite-time boundedness of the closed-loop systems over the finite-time interval&lrm;. &lrm;Thereafter&lrm;, &lrm;an appropriate control strategy is constructed to drive the state trajectories onto the specified sliding surface in a finite time&lrm;. &lrm;Finally&lrm;, &lrm;an example related to the time-delayed Chua's circuit is given to demonstrate the effectiveness of the suggested method&lrm;. &lrm;Also&lrm;, &lrm;the efficiency of the suggested method is compared with other methods by using an another numerical example&lrm;.Modelling the transmission of Mpox with case study in Nigeria and Democratic Republic of the Congo (DRC)
https://cmde.tabrizu.ac.ir/article_18322.html
This paper focuses on the dynamics of Mpox, a viral disease, in Nigeria and the Democratic Republic of Congo (DRC), employing mathematical modeling and parameter estimation techniques. Utilizing optimization methods, the model parameters were calibrated to match the observed Mpox cases and deaths. The basic reproduction number Ro was calculated for each region, indicating the disease's transmission potential, and a sensitivity analysis was conducted to identify key parameters influencing disease outcomes. Subsequently, numerical simulations were performed to assess the impact of intervention scenarios on Mpox cases and deaths. The primary goal is to create mathematical methods that can evaluate the risk of Mpox transmission and implement control measures in Nigeria and DRC, potentially extending the findings to other countries. Results show that reducing parameters related to transmission and progression significantly decreases disease burden, highlighting the importance of preventive measures. These findings provide valuable insights for policymakers and public health officials in designing effective strategies to mitigate Mpox's impact on human populations.Two-Dimensional Nonlinear Schrödinger Equation Using an Alternating Direction Implicit Method
https://cmde.tabrizu.ac.ir/article_18368.html
This paper introduces an alternating direction implicit (ADI) finite difference method to solve the two-dimensional time-dependent nonlinear Schr&ouml;dinger equation. The method involves linearizing the nonlinear term by utilizing the wave function values from the previous time step at each iteration. The resulting block tridiagonal system of equations is solved using the Gauss-Seidel method through sparse matrix computation. Stability analysis employing matrix techniques reveals the scheme to be conditionally stable. Numerical examples are provided to demonstrate the effectiveness, stability, and accuracy of the proposed numerical approach. The computed results are in good agreement with exact solutions, further confirming the validity of the proposed method.HYERS-ULAM AND EXPONENTIAL STABILITIES OF AUTONOMOUS AND NON-AUTONOMOUS DIFFERENCE EQUATIONS
https://cmde.tabrizu.ac.ir/article_18375.html
In this manuscript, we studied the Hyers-Ulam and exponential stabilitiesof autonomous and non-autonomous difference equations of first and secondorder. At the end we provide some examples to support the results.Two-dimensional temporal fractional advection-diffusion problem resolved through the Sinc-Galerkin method
https://cmde.tabrizu.ac.ir/article_18383.html
Applying the Sinc-Galerkin method, even for problems that include infinity and semi-infinite intervals, is known as exponential fading errors and in certain conditions as the optimal convergence rate.Additionally, this approach does not suffer from the normal instability issues that often arise in other methods. Therefore, a numerical technique based on the Sinc-Galerkin method is devised in this study to solve the two-dimensional time fractional advection-diffusion problem. To be precise, the spatial and temporal discretizations of the Sinc-Galerkin and finite difference methods are coupled to provide the suggested approach. Additionally, the suggested method's convergence is looked at. Two numerical examples are provided in depth in the conclusion to demonstrate the effectiveness and precision of the suggested approach.APPLICATION OF tan(ϕ/2)-EXPANSION METHOD FOR SOLVING THE FRACTIONAL BISWAS-MILOVIC EQUATION FOR KERR LAW NONLINEARITY
https://cmde.tabrizu.ac.ir/article_18431.html
In this paper, the improved tan (&Phi;(&xi;)/2)-expansion method (ITEM)is proposed to obtaining the fractional Biswas-Milovic equation. The exact particular solutions containing four types hyperbolic function solution, trigonometricfunction solution, exponential solution and rational solution. We obtained thefurther solutions comparing with other methods as [7]. Recently this method isdeveloped for searching exact travelling wave solutions of nonlinear partial differential equations. These solutions might play important role in nonlinear opticand physics fields. It is shown that this method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solvingproblems in nonlinear optic.