Computational Methods for Differential Equations
https://cmde.tabrizu.ac.ir/
Computational Methods for Differential Equationsendaily1Fri, 01 Mar 2024 00:00:00 +0330Fri, 01 Mar 2024 00:00:00 +0330Boundary controller design for stabilization of stochastic nonlinear reaction-diffusion systems with time-varying delays
https://cmde.tabrizu.ac.ir/article_16752.html
This paper is focused on studying the stabilization problems of stochastic nonlinear reaction-diffusion systems (SNRDSs) with time-varying delays via boundary control. Firstly, the boundary controller was designed to stabilization for SNRDSs. By utilizing the Lyapunov functional method, Ito&rsquo;s differential formula, Wirtinger&rsquo;s inequality, Gronwall inequality, and LMIs, sufficient conditions are derived to guarantee the finite-time stability (FTS) of proposed systems. Secondly, the basic expressions of the control gain matrices are designed for the boundary controller. Finally, numerical examples are presented to verify the efficiency and superiority of the proposed stabilization criterion.&nbsp;Prolific new M-fractional soliton behaviors to the Schrödinger type Ivancevic option pricing model by two efficient techniques
https://cmde.tabrizu.ac.ir/article_16763.html
The principal purpose of this research is to study the M-fractional nonlinear quantum-probability grounded Schr&ouml;dinger kind Ivancevic option pricing model (IOPM). This well-known economic model is an alternative of the standard Black-Scholes pricing model which represents a controlled Brownian motion in an adaptive setting with relation to nonlinear Schr&ouml;dinger equation. The exact solutions of the underlying equation have been derived through the well-organized extended modified auxiliary equation mapping and generalized exponential rational function methods. Different forms of optical wave structures including dark, bright, and singular solitons are derived. To the best of our knowledge, verified solutions using Maple are new. The results obtained will contribute to the enrichment of the existing literature of the model under consideration. Moreover, some sketches are plotted to show more about the dynamic behavior of this model.Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval
https://cmde.tabrizu.ac.ir/article_16638.html
Most of fractional differential equations are considered on a fixed interval. In this paper, we consider a typical fractional differential equation on a symmetric interval $[-\alpha,\alpha]$, where $\alpha$ is the order of fractional derivative. For a positive real number &alpha; we prove that the solutions are &nbsp;$T_{n,\alpha}(x)=(\alpha+x)^\frac{1}{2}Q_{n,\alpha}(x)$ where $Q_{n,\alpha}(x)$ produce a family of orthogonal polynomials with respect to the weight function$w_\alpha(x)=(\frac{\alpha+x}{\alpha-x})^{\frac{1}{2}}$ on $[-\alpha,\alpha]$. For integer case $\alpha = 1 $, we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional differential equations.&nbsp;Rumor spread dynamics and its sensitivity analysis under the influence of the Caputo fractional derivatives
https://cmde.tabrizu.ac.ir/article_16911.html
Rumor spreading is the circulation of doubtful messages on the social network. Fact retrieving process that aims at preventing the spread of the rumor, appears to have a significant global impact. In this research, we have investigated a mathematical model projecting rumor spread by considering six groups of individuals namely ignorant, exposed, intentional rumor spreader, unintentional rumor spreader, stifler, and fact retriever. To represent the current abnormal and fast pattern of the message spread around various platforms, in the projected model, we have implemented the fractional derivative in the Caputo context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the rumor-free and rumor equilibrium points, and the global stability of the rumor-free equilibrium point. Computing basic reproduction numbers, we have analyzed the existence and stability of points of equilibrium. The sensitivity of basic reproduction numbers is also examined. Importance of the fact retrieving drive is highlighted by relating it to the basic reproduction number. Finally, by applying the Adams-Bashforth-Moulton method, we have presented the numerical results by capturing the profile of each of the groups under the influence of fractional derivative and investigated the impact of rumor verification rate and contact rate in controlling and preventing the rumor. With the Caputo fractional operator in the projected model, the current research highlights the significance of the fact retriever and the curb in individual contact and captures the relevant consequences.Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions
https://cmde.tabrizu.ac.ir/article_16921.html
We present a novel computational approach for computing invariant manifolds that correspond to equilibrium solutions of nonlinear parabolic partial differential equations (or PDEs). Our computational method combines Lie symmetry analysis with the parameterization method. The equilibrium solutions of PDEs and the solutions of eigenvalue problems are exactly obtained. As the linearization of the studied nonlinear PDEs at equilibrium solutions yields zero eigenvalues, these solutions are non-hyperbolic, and some invariant manifolds are center manifolds. We use the parameterization method to model the infinitesimal invariance equations that parameterize the invariant manifolds. We utilize Lie symmetry analysis to solve the invariance equations. We apply our framework to investigate the Fisher equation and the Brain Tumor growth differential equation.&nbsp;Exponentially fitted IMEX peer methods for an advection-diffusion problem
https://cmde.tabrizu.ac.ir/article_16641.html
In this paper, Implicit-Explicit (IMEX) Exponential Fitted (EF) peer methods are proposed for the numerical solution of an advection-diffusion problem exhibiting an oscillatory solution. Adapted numerical methods both in space and in time are constructed. The spatial semi-discretization of the problem is based on finite differences, adapted to both the diffusion and advection terms, while the time discretization employs EF IMEX peer methods. The accuracy and stability features of the proposed methods are analytically and numerically analyzed.&nbsp;Inverse coefficient problem in hyperbolic partial differential equations: An analytical and computational exploration
https://cmde.tabrizu.ac.ir/article_16709.html
This investigation centers on the analysis of an inverse hyperbolic partial differential equation, specifically addressing a coefficient inverse problem that emerges under the imposition of an over-determination condition. In order to address this challenging problem, we employ the well-established homotopy analysis technique, which has proven to be an effective and reliable approach in similar contexts. By utilizing this technique, our primary objective is to achieve an efficient and accurate solution to the inverse problem at hand. To substantiate the effectiveness and reliability of the proposed method, we present a numerical example as a practical illustration, demonstrating its applicability in real-world scenarios.&nbsp;Finite element solution of a class of parabolic integro-differential equations with inhomogeneous jump conditions using FreeFEM++
https://cmde.tabrizu.ac.ir/article_16639.html
The finite element solution of a class of parabolic integro&ndash;partial differential equations with interfaces is presented. The spatial discretization is based on the triangular element while a two-step implicit scheme together with the trapezoidal method is employed for time discretization. For the spatial discretization, the elements in the neighborhood of the interface are more refined such that the interface is at $\sigma$-distance from the approximate interface. The convergence rate of optimal order in L2-norm is analyzed with the assumption that the interface is arbitrary but smooth. Examples are given to support the theoretical findings with implementation on FreeFEM++.A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions
https://cmde.tabrizu.ac.ir/article_16883.html
In the current work, a new reproducing kernel method (RKM) for solving nonlinear forced Duffing equations with integral boundary conditions is developed. The proposed collocation technique is based on the idea of RKM and the orthonormal Bernstein polynomials (OBPs) approximation together with the quasi-linearization method. In our method, contrary to the classical RKM, there is no need to use the Gram-Schmidt orthogonalization procedure and only a few nodes are used to obtain efficient numerical results. Three numerical examples are included to show the applicability and efficiency of the suggested method. Also, the obtained numerical results are compared with some results in the literature.Adaptive-grid technique for the numerical solution of a class of fractional boundary-value-problems
https://cmde.tabrizu.ac.ir/article_16547.html
In this study, we numerically solve a class of two-point boundary-value-problems with a Riemann-Liouville-Caputo fractional derivative, where the solution might contain a weak singularity. Using the shooting technique based on the secant iterative approach, the boundary value problem is first transformed into an initial value problem, and the initial value problem is then converted into an analogous integral equation. The functions contained in the fractional integral are finally approximated using linear interpolation. An adaptive mesh is produced by equidistributing a monitor function in order to capture the singularity of the solution. A modified Gronwall inequality is used to establish the stability of the numerical scheme. To show the effectiveness of the suggested approach over an equidistributed grid, two numerical examples are provided.Dynamics of an SEIR epidemic model with saturated incidence rate including stochastic influence
https://cmde.tabrizu.ac.ir/article_16885.html
This paper aims to develop a stochastic perturbation into SEIR (Susceptible-Exposed-Infected-Removed) epidemic model including a saturated estimated incidence. A set of stochastic differential equations is used to study its behavior, with the assumption that each population&rsquo;s exposure to environmental unpredictability is represented by noise terms. This kind of randomness is considerably more reasonable and realistic in the proposed model. The current study has been viewed as strengthening the body of literature because there is less research on the dynamics of this kind of model. We discussed the structure of all equilibriums&rsquo; existence and the dynamical behavior of all the steady states. The fundamental replication number for the proposed method was used to discuss the stability of every equilibrium point; if $R_0&lt;1$, the infected free equilibrium is resilient, and if $R_0&gt;1$, the endemic equilibrium is resilient. The system&rsquo;s value is primarily described by its ambient stochasticity, which takes the form of Gaussian white noise. Additionally, the suggested model can offer helpful data for comprehending, forecasting, and controlling the spread of various epidemics globally. Numerical simulations are run for a hypothetical set of parameter values to back up our analytical conclusions.&nbsp;Synchronization a chaotic system with Quadratic terms using the contraction Method
https://cmde.tabrizu.ac.ir/article_16884.html
In this article, Synchronization and control methods are discussed as essential topics in science. The contraction method is an exciting method that has been studied for the synchronization of chaotic systems with known and unknown parameters. The controller and the dynamic parameter estimation are obtained using the contraction theory to prove the stability of the synchronization error and the low parameter estimation. The control scheme does not employ the Lyapunov method. For demonstrate the ability of the proposed method, we performed a numerical simulation and compared the result with the previous literature.Numerical solution of fractional Volterra integro-differential equations using flatlet oblique multiwavelets
https://cmde.tabrizu.ac.ir/article_16807.html
The presented paper investigates a new numerical method based on the characteristics of flatlet oblique multiwavelets for solving fractional Volterra integro-differential equations, in this method, first using the dual bases of the flatlet multiwavelets, the operator matrices are made for the derivative of fractional order and Volterra integral. Then, the fractional Volterra integro-differential equation reduces to a set of algebraic equations which can be easily solved. The error analysis and convergence of the presented method are discussed. Also, numerical examples will indicate the acceptable accuracy of the proposed method, which is compared with the methods used by other researchers.&nbsp;Novel traveling wave solutions of generalized seventh-order KdV equation and related equation
https://cmde.tabrizu.ac.ir/article_16546.html
In this paper, we acquire novel traveling wave solutions of the generalized seventh-order Korteweg&ndash;de Vries equation and the seventh-order Kawahara equation as a special case with physical interest. Primarily, we use the advanced $\exp (-\varphi (\xi ))$-expansion method to find new exact solutions of the first equation, by considering two auxiliary equations. Then, we attain some exact solutions of the seventh-order Kawahara equation by using this method with another auxiliary equation, and also using the modified $(G^{'}/G) $-expansion method, where G satisfies a second-order linear ordinary differential equation. Additionally, utilizing the recent scientific instruments, the 2D, 3D, and contour plots are displayed. The solutions obtained in this paper include bright solitons, dark solitary wave solutions, and multiple dark solitary wave solutions. It is shown that these two methods provide an effective mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.Modified simple equation method (MSEM) for solving nonlinear (3+1)-dimensional space-time fractional equations
https://cmde.tabrizu.ac.ir/article_16886.html
In the present paper, modified simple equation method (MSEM) is implemented for obtaining exact solutions of three nonlinear (3 + 1)-dimensional space-time fractional equation, namely three types of modified Korteweg-de-Vries (mKdV) equations. Here, the derivatives are of the type of conformable fractional derivatives. The solving process produces a system of algebraic equations which is possible to be easily with no need of using software for determining unknown coefficients. Results show that this method can supply a powerful mathematical tool to construct exact solutions of mKdV equations and it can be employed for other nonlinear (3 + 1) - dimensional space-time fractional equations.Approximate solutions of inverse Nodal problem with conformable derivative
https://cmde.tabrizu.ac.ir/article_16924.html
Our research is about Sturm-Liouville equation which contains conformable fractionalderivatives of order &alpha; &isin; (0, 1] in lieu of the ordinary derivatives. First, we present theeigenvalues, eigenfunctions and nodal points and the properties of nodal points are used forthe reconstruction of an integral equation. Then, the Bernstein technique was utilized tosolve the inverse problem and the approximation of solving this problem was calculated.Finally, the numerical examples were introduced to explain the results. Moreover, theanalogy of this technique is shown in a numerical example with Chebyshev interpolationtechnique .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. Local stability of equilibrium points was classified. A Lyapunov function is constructed to analyze 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.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 $q-$Mittag-Leffler function. Also, we solve some Caputo delta $q-$fractional dynamic equations, and these solutions are expressed by means of the newly introduced delta $q-$Mittag-Leffler function.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 twoparametricwith-memory methods to obtain simple roots of nonlinear equations.Novel improved methods are two-step without memory and have twoself-accelerator parameters that do not have additional evaluation. The methodshave been compared with the nearest competitions in various numericalexamples. Anyway, the theoretical order of convergence is verified. The basinsof attraction of the suggested methods are presented and corresponded to explaintheir interpretation.An efficient algorithm for computing the eigenvalues of conformable Sturm-Liouville problem
https://cmde.tabrizu.ac.ir/article_17114.html
&lrm;In this paper&lrm;, &lrm;Computing the eigenvalues of Conformable Sturm-Liouville Problem (CSLP) of order $2 \alpha$&lrm;, &lrm;$\frac{1}{2}&lt;\alpha \leq 1$&lrm;, &lrm;and dirichlet boundary conditions is considered&lrm;. &lrm;For this aim&lrm;, &lrm;CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions&lrm;. &lrm;Then by a method based on asymptotic form of the eigenvalues we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP&lrm;. &lrm;Finally&lrm;, &lrm;some numerical examples to show the efficiency of the proposed method are given&lrm;. &lrm;Numerical results show that for the $n$th eigenvalue the correction technique reduces the error order from $O(n^4h^2)$ to $O(n^2h^2)$&lrm;.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 offractional multi-step linear finite difference methods (FLMMs)designed to address fractional multi-delay pantograph differentialequations of order $0 &lt; \alpha \leq 1$. These $p$-FLMMs areconstructed using fractional backward differentiation formulas offirst and second orders, thereby facilitating the numerical solutionof fractional differential equations. Notably, we employ accurateapproximations for the delayed components of the equation,guaranteeing the retention of stability and convergencecharacteristics in the proposed $p$-FLMMs. To substantiate ourtheoretical findings, we offer numerical examples that corroboratethe efficacy and reliability of our approach.A mathematical study on 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 is 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.Numerical Solution of Burgers' Equation with nonlocal boundary condition: Using the Keller-Box Scheme
https://cmde.tabrizu.ac.ir/article_17162.html
In this study, we transform the given non-local boundary condition problem into a managable 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.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.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 orderJacobi and trapezoid methods to solve a type of distributed partial differentialequations. The fractional derivatives are considered in the Caputo-Prabhakar type.By shifted fractional-order Jacobi polynomials our proposed method can providehighly accurate approximate solutions by reducing the problem under study to aset of algebraic equations which is technically simpler for handling. In order todemonstrate the error estimates, several lemmas are provided. Finally, numericalresults 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.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;.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.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 in order to characterise certain fantastic characteristics of plasma-particle dispersion. A careful investigation into the Lie space with an unlimited number of dimensions was carried out in order 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 demonstratesA GENERALIZED ADAPTIVE MONTE CARLO ALGORITHM BASED ON A TWO-STEP ITERATIVE METHOD FOR LINEAR SYSTEMS AND ITS APPLICATION TO OPTION PRICING
https://cmde.tabrizu.ac.ir/article_17743.html
In this paper&lrm;, &lrm;we present a &lrm;&lrm;&lrm;&lrm;generalized&lrm; adaptive Monte Carlo algorithm &lrm;using&lrm; Diagonal and Off-Diagonal Splitting (DOS) iteration method to &lrm;solve&lrm; system of linear algebraic equations &lrm;(SLAE)&lrm;&lrm;.&lrm; I&lrm;n &lrm;fact, &lrm;t&lrm;he DOS method is a generalized iterative method &lrm;which has some known iterative methods such as Jacobi&lrm;, &lrm;Gauss-Seidel and Successive Overrelaxation methods&lrm; as its special cases&lrm;. &lrm;Monte Carlo algorithms &lrm;usually&lrm; use the Jacobi method &lrm;to &lrm;solve&lrm; &lrm;SLAE&lrm;&lrm;. &lrm;In this paper&lrm;, the DOS &lrm;method &lrm;is &lrm;used&lrm; instead of the Jacobi method &lrm;w&lrm;hich transforms the Monte Carlo &lrm;algorithm&lrm; into the generalized Monte Carlo &lrm;algorithm&lrm;. &lrm;we &lrm;establish&lrm; t&lrm;heoretical &lrm;results&lrm; to justify the convergence of the algorithm&lrm;.&lrm;&lrm; &lrm;&lrm;Finally&lrm;, &lrm;numerical experiments are discussed to illustrate the accuracy and &lrm;eff&lrm;i&lrm;ciency of the theoretical results&lrm;. Furthermore, &lrm;&lrm;&lrm;the &lrm;generalized&lrm; algorithm is &lrm;implemented&lrm; to &lrm;price &lrm;options&lrm; &lrm;using&lrm; &lrm;fi&lrm;nite &lrm;diff&lrm;erence &lrm;method&lrm;. &lrm;We compare the generalized algorithm with &lrm;standard numerical and stochastic algorithms &lrm;to &lrm;show &lrm;its&lrm; &lrm;eff&lrm;i&lrm;ciency.&lrm;&lrm;&lrm;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;.Gradient estimates for a nonlinear equation under the almost Ricci soliton condition
https://cmde.tabrizu.ac.ir/article_17746.html
In this paper, we study the gradient estimate for the positive solutions of the equation$\Delta u+au(\log u)^{p}+bu=f$on an almost Ricci soliton $(M^{n},g,X,\lambda)$. In a special case, when $X=\nabla h$ for a smooth function $h$, we derive a gradient estimate for an almost gradient Ricci soliton.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.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.A New Perspective for the Quintic B-spline Collocation Method with via the Lie- Trotter Splitting Algorithm to Solitary Wave Solutions of the GEW Equation
https://cmde.tabrizu.ac.ir/article_17759.html
An 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, 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 terms of time. 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 used to generate approximate solutions of the main equation. Stability analysis of acquired numerical shemes 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 literat&uuml;re. 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.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.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.Direct and inverse problems of rod equation using finite element method and a correction technique
https://cmde.tabrizu.ac.ir/article_17789.html
The free vibrations of a rod are governed by a differential equation of the form $(a(x)y^\prime)^\prime+\lambda a(x)y(x)=0$, where $a(x)$ is the cross sectional area and $\lambda$ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form $(K-\Lambda M)u=0$ and, for given $a(x)$, we correct the eigenvalues $\Lambda$ of the matrix pair $(K,M)$ to approximate the eigenvalues of the rod equation. The results show that with step size $h$ the correction technique reduces the error from $O(h^2i^4)$ to $O(h^2i^2)$ for the $i$-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient $a(x)$ from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms.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.QUALITY THEOREMS ON THE SOLUTIONS OF QUASILINEAR SECOND ORDER PARABOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
https://cmde.tabrizu.ac.ir/article_17801.html
A class of quasilinear second order parabolic equations with discontinuous coefficients is con-sidered in this work. The analog of Harnack inequality is proved for the non-negative solutions of theseequations.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.A numerical investigation for the COVID-19 spatiotemporal lockdown-vaccination model
https://cmde.tabrizu.ac.ir/article_17808.html
The present article investigates a numerical analysis of COVID-19 (temporal and spatio-temporal) lockdown-vaccination models. The proposed models consist of six nonlinear ordinary differential equations as a temporal model and six nonlinear partial differential equations as a spatio-temporal model. The evaluation of reproduction number is a forecast spread of the COVID-19 pandemic. Sensitivity analysis is used to emphasize the importance of pandemic parameters. We show the stability regions of the disease-free equilibrium point and pandemic equilibrium point. We use effective methods such as central finite difference (CFD) and Runge-Kutta of fifth order (RK-5). We apply Von-Neumann stability and consistency of the numerical scheme for the spatio-temporal model. We examine and compare the numerical results of the proposed models under various parameters.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.THE COMPLEX SEE TRANSFORM TECHNIQUE IN DIFFERENCE EQUATIONS AND DIFFERENTIAL DIFFERENCE EQUATIONS
https://cmde.tabrizu.ac.ir/article_17828.html
Diﬀerential equations is used to represent diﬀerent scientiﬁc problems are handled eﬃciently by integral transformations, where inte-gral transforms represent an easy and eﬀective tool for solving many prob-lems in the mentioned ﬁelds. This work utilizes the integral transformof the Complex SEE integral transformation to provide an eﬃcient solu-tion method to diﬀerence and diﬀerential-diﬀerence equations by beneﬁtingfrom the properties of this complex transform to solve some problems re-lated to diﬀerence and diﬀerential-diﬀerence equations. The 3D, contour,and 2D surfaces, as well as the related density plot surfaces of some ac-quired data, are used to draw the physical aspect of the obtained ﬁndings.The proposed approach oﬀers an eﬃcient and rapid solution for address-ing the inherent complexity of diﬀerential-diﬀerence problems with initial conditions.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.