Computational Methods for Differential Equations
https://cmde.tabrizu.ac.ir/
Computational Methods for Differential Equationsendaily1Sat, 01 Apr 2023 00:00:00 +0430Sat, 01 Apr 2023 00:00:00 +0430Optimal fractional order PID controller performance in chaotic system of HIV disease: particle swarm and genetic algorithms optimization method
https://cmde.tabrizu.ac.ir/article_15397.html
The present study aims to investigate the optimal fractional order PID controller performance in the chaotic system of HIV disease fractional order using the Particle Swarm optimization and Genetic algorithm method. Differential equations were used to represent the chaotic behavior associated with HIV. The optimal fractional order of the PID controller was constructed, and its performance in the chaotic system with HIV fractional order was tested. Optimization methods were used to get PID control coefficients from particle swarm and genetic algorithms. Findings revealed that the equations for the HIV disease model are such that the system&rsquo;s behavior is greatly influenced by the number of viruses produced by infected cells, such that if the number of viruses generated by infected cells exceeds 202, the disease&rsquo;s behavior is such that the virus and disease spread. For varying concentrations of viruses, the controller created for this disease does not transmit the disease.New approximations of space- time fractional Fokker-Planck equations
https://cmde.tabrizu.ac.ir/article_15256.html
The present study focus on the two new hybrid methods: variational iteration J-transform technique (J-VIT)and J-transform method with optimal homotopy analysis (OHAJTM) for analytical assessment of space-timefractional Fokker-Planck equations (STF-FPE), appearing in many realistic physical situations, e.g., in ultra-slowkinetics, Brownian motion of particles, anomalous diffusion, polymerases of Ribonucleic acid, deoxyribonucleicacid, continuous random movement and formation of wave patterns. OHAJTM is developed via optimal homotopyanalysis after implementing the properties of J-transform while (J-VIT) is produced by implementing propertiesof the J-transform and the theory of variational iteration.Banach approach is utilized to analyze convergence of these methods. In addition, it is demonstrated that J-VITis T-stable. Computed new approximations are reported as closed form expression of Mittag-Leffler function, andin addition, effectiveness/validity of the proposed new approximations is demonstrated via three test problems ofSTF-FPE by computing the error norms: L2 and absolute errors. The numerical assessment demonstrates thatthe developed techniques perform better for STF-FPE and for a given iteration, and OHAJTM produces newapproximations with better accuracy as compared to J-VIT as well as the techniques developed recently.Non-polynomial cubic spline method for solution of higher order boundary value problems
https://cmde.tabrizu.ac.ir/article_15002.html
In this paper, a new algorithm based on non-polynomial spline is developed for the solution of higher order boundary value problems(BVPs). Employment of the method is done by decomposing the higher order BVP into a system of third order BVPs. Convergence analysis of the developed method is also discussed. The method is tested on higher order linear as well as non-linear BVPs which shows the accuracy and efficiency of the method and also compared our results with some existing fourth order methods.A modified split-step truncated Euler-Maruyama method for SDEs with non-globally Lipschitz continuous coefficients
https://cmde.tabrizu.ac.ir/article_15520.html
In this paper we propose an explicit diffuse split-step truncated Euler-Maruyama (DSSTEM) methodfor stochastic differential equations with non-global Lipschitz coefficients. We investigate the strongconvergence of the new method under local Lipschitz and Khasiminskii-type conditions. We show that the newly proposed method achieves a strong convergence rate arbitrarily close to half under someadditional conditions. Finally, we illustrate the efficiency and performance of the proposed method with numerical results.A numerical method for solving the Duffing equation involving both integral and non-integral forcing terms with separated and integral boundary conditions
https://cmde.tabrizu.ac.ir/article_15001.html
This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results.&nbsp;Improved Residual Method for Approximating Bratu Problem
https://cmde.tabrizu.ac.ir/article_15638.html
In this study, firstly, the residual method, which was developed for initial value problems, is improved to find unknown coefficients without the need for any system solution. Later, the adaptation of improved residual method is given to find approximate solutions of boundary value problems. Finally, the method improved and adapted for boundary value problems is used to find both critical eigenvalue and eigenfunctions of the one-dimensional Bratu problem. Most significant advantage of the method is finding approximate solutions of nonlinear problems without any linearization or solving any system of equations. Error analysis of the adapted method is given and an upper bound is obtained for the error while approximating to eigenfunctions. The numerical results obtained are compared with the theoretical findings. Comparisons and theoretical observations show that the improved and adapted method is very convenient and successful in solving boundary value problems and eigenvalue problems approximately with high accuracy.Steady state bifurcation in a cross diffusion prey-predator model
https://cmde.tabrizu.ac.ir/article_15401.html
In this paper, we study the bifurcation of nontrivial steady state solutions for a cross-diffusion prey-predator model with homogeneous Neumann boundary conditions. The existence of positive steady state solutions near a bifurcation point is proved using a crossing curve bifurcation theorem. We consider a situation where the transversality condition is not satisfied. Unlike the case in saddle-node bifurcation, the solution set is a pair of transversally intersecting curves.An efficient numerical approach for solving nonlinear Volterra integral equations
https://cmde.tabrizu.ac.ir/article_15640.html
This study deals with a numerical solution of a nonlinear Volterra integral equation of the first kind. The method of this research is based on a new kind of orthogonal wavelets, called the Chebyshev cardinal wavelets. These wavelets known as new basis functions contain numerous beneficial features like orthogonality, spectral accuracy, and cardinality. In addition, we assume an expansion of the terms of Chebyshev cardinal wavelets within unknown coefficients as a substitute for an unknown solution. Relatively, considering the mentioned expansion and the cardinality feature within the generated operational matrix of the introduced wavelets, a system of nonlinear algebraic equations is extracted for the stated problem. Finally, by solving the yielded system, the estimated solution results.A novel scheme for SMCH equation with two different approaches
https://cmde.tabrizu.ac.ir/article_15225.html
In this study, the unified and improved F-expansion methods are applied to derive exact traveling wave solutions of the simplified modified Camassa-Holm (SMCH) equation. The current methods can calculate all branches of solutions at the same time, even if several solutions are quite near and therefore impossible to identify via numerical methods. All obtained solutions are given by hyperbolic, trigonometric, and rational function solutions which obtained solutions are useful for real-life problems in fluid dynamics, optical fibers, plasma physics and so on. The two-dimensional (2D) and three-dimensional (3D) graphs of the obtained solutions are plotted. Finally, we can state that these strategies are extremely successful, dependable, and simple. These ideas might potentially be applied to many nonlinear evolution models in mathematics and physics.&nbsp;A novel analytical approximation approach for strongly nonlinear oscillation systems based on the energy balance method and He's Frequency-Amplitude formulation
https://cmde.tabrizu.ac.ir/article_15660.html
Nonlinear oscillations are an essential fact in physical science, mechanical structures, and other engineering problems. Some of the popular analytical solutions to analyze nonlinear differential equations governing the behavior of strongly nonlinear oscillators are the Energy Balance Method (EBM), and He's Amplitude Frequency Formulation (HAFF). The lack of precision and accuracy despite needing several computational steps to resolve the system frequency is the main demerit of these methods. This research creates a novel analytical approximation approach with a very efficient algorithm that can perform the calculation procedure much easier and with much higher accuracy than classic EBM and HAFF. The presented method's steps rely on Hamiltonian relations described in EBM and the defined relationship between frequency and amplitude in HAFF. This paper demonstrates the substantial precision of the presented method compared to common EBM and HAFF applied in different and well-known engineering phenomena. For instance, the approximate solutions of the equations govern some strongly nonlinear oscillators, including the two-mass&ndash;spring systems, buckling of a column, and duffing relativistic oscillators, are presented here. Subsequently, their results are compared with the Runge-Kutta method and exact solutions obtained from the previous research. The proposed novel approach resultant error percentages show an excellent agreement with the numerical solutions and illustrate a very quickly convergent and more precise than mentioned typical methods.Haar wavelet-based valuation method for pricing European options
https://cmde.tabrizu.ac.ir/article_15258.html
A numerical method based on the Haar wavelet is introduced in this study for solving the partial differential equation which arises in the pricing of European options. In the first place, and due to the change of variables, the related partial differential equation (PDE) converts into a forward time problem with a spatial domain ranging from 0 to 1. In the following, the Haar wavelet basis is used to approximate the highest derivative order in the equation concerning the spatial variable. Then the lower derivative orders are approximated using the Haar wavelet basis. Finally, by substituting the obtained approximations in the main PDE and doing some computations using the finite differences approach, the problem reduces to a system of linear equations that can be solved to get an approximate solution. The provided examples demonstrate the effectiveness and precision of the method.An optimum solution for multi-dimensional distributed-order fractional differential equations .....
https://cmde.tabrizu.ac.ir/article_15682.html
This manuscript investigates a computational method based on fractional-order Fibonacci functions (FFFs) for solving distributed-order (DO) fractional differential equations and DO time-fractional diffusion equations. Extra DO fractional derivative operator and pseudo-operational matrix of fractional integration for FFFs are proposed. To evaluate the unknown coefficients in the FFF expansion, utilizing the matrices, an optimization problem relating to considered equations is formulated. This approach converts the original problems into a mathematical programming one. The approximation error is proposed. Several problems are proposed to investigate the applicability and computational efficiency of the scheme. The approximations achieved by some existing schemes are also tested conforming to the efficiency of the present method. Also, the model of the motion of the DO fractional oscillator is solved, numerically.Numerical computation of exponential functions in frame of Nabla fractional calculus
https://cmde.tabrizu.ac.ir/article_15239.html
Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for non-homogeneous nabla fractional difference equations.A method for second-order linear fuzzy two-point boundary value problems based on the Hukuhara differentiability
https://cmde.tabrizu.ac.ir/article_15707.html
In this paper, we deal with second-order fuzzy linear two-point boundary value problems (BVP) In this paper, we deal with second-order fuzzy linear two-point boundary value problems (BVP) underHukuhara derivatives. Considering the first-order and second-order Hukuhara derivatives, four typesof fuzzy linear two-point BVPs can be obtained where each may or may not have a solution.Therefore a fuzzy two-point (BVP) may have one, two, three, or four different kinds of solutions concerning this kind of derivative. To solve this fuzzy linear two-point (BVP), we convert eachto two cases of crisp boundary value problems. We apply a standard method(numerical or analytical)to solve crisp two-point BVPs in their domain. Subsequently, the crisp solutions are combined to obtain a fuzzy solution to the fuzzy problems, and the solutions are checked to see if they satisfy the fuzzy issues. Conditions are presented under which fuzzy problems have the fuzzy solution and illustrated with some examplesA numerical method for KdV equation using rational radial basis functions
https://cmde.tabrizu.ac.ir/article_15500.html
In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Korteweg-de Vries (KdV) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourth-order Runge-Kutta method to advance the resulting semi-discrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions.&nbsp;Spectral collocation method based on special functions for solving nonlinear high-order pantograph equations
https://cmde.tabrizu.ac.ir/article_15777.html
In this paper, a spectral collocation method for solving nonlinear pantograph type delay differential equations is presented. The basis functions used for the spectral analysis are based on Chebyshev, Legendre and Jacobi polynomials. By using the collocation points and operations matrices of required functions such as derivative functions and delays of unknown functions, the method transforms the problem into a system of nonlinear algebraic equations. The solutions of this nonlinear system determine the coefficients of assumed solution. The method is explained by numerical examples and the results are compared with the available methods in literature. It is seen from the applications that our method gives efficient results than that of the reported methods.On the existence of periodic solutions of third order delay differential equations
https://cmde.tabrizu.ac.ir/article_14812.html
This work deals with the existence of periodic solutions (EPSs) to a third order nonlinear delay differential equation (DDE) with multiple constant delays. For the considered DDE, a theorem is proved, which includes sufficient criteria related to the EPSs. The technique of the proof depends on Lyapunov-Krasovskiˇı functional (LKF) approach. The obtained result extends and improves some results that can be found in the literature. In a particular case of the considered DDE, an example is provided to show the applicability of the main result of this paper.&nbsp;Numerical solution of third-Order boundary value problems using non-classical sinc-collocation method
https://cmde.tabrizu.ac.ir/article_15786.html
In this work, a non-classical sinc-collocation method is used to nd numerical solution of third-orderboundary value problems. The novelty of this approach is based on using the weight functions in thetraditional sinc-expansion. The properties of sinc-collocation are used to reduce the boundary valueproblems to a nonlinear system of algebraic equations which can be solved numerically. In addition to convergence of the proposed method is discussed by preparing the theorems which show exponential convergence and guarantee its applicability. Several examples are solved and the numerical results show the efficiency and applicability of the method.To study existence of unique solution and numerically solving for a kind of three-point boundary fractional high-order problem subject to Robin condition
https://cmde.tabrizu.ac.ir/article_15257.html
In this paper, we prove the existence and uniqueness of the solutions for a non-integer high order boundary value problem which is subject to the Caputo fractional derivative. The boundary condition is a non-local type. Analytically, we introduce the fractional Green&rsquo;s function. The main principle applied to simulate our results is the Banach contraction fixed point theorem. We deduce this paper by presenting some illustrative examples. Furthermore, it is presented a numerical based semi-analytical technique to approximate the unique solution to the desired order of precision.&nbsp;Numerical methods for m-polar fuzzy initial value problems
https://cmde.tabrizu.ac.ir/article_15844.html
An m-polar fuzzy set model is a useful mathematical tool for addressing uncertainty in multipolar information. In this paper, we study differential equations in m-polar fuzzy environment. We introduce the concept gH-derivative of the m-polar fuzzy-valued function. We present some properties of gH-differentiability of m-polar fuzzy-valued function by considering different types of differentiability. We consider m-polar fuzzy Taylor expansion. By using Taylor expansion, the Euler method and modified Euler method are presented for solving m-polar fuzzy initial value problems. We discuss the convergence analysis of these methods. Some numerical examples are described to see the convergence and stability of proposed methods. We make a comparison of these methods by computing global truncation errors. From numerical results, we see that the modified Euler method converges fastly to exact solution as compared to the Euler method.
&nbsp;Solution of time-fractional equations via Sumudu-Adomian decomposition method
https://cmde.tabrizu.ac.ir/article_15000.html
This paper investigates the semi-analytical solutions of linear and non-linear Time Fractional Klein-Gordon equations with appropriate initial conditions to apply the New Sumudu-Adomian Decomposition method (NSADM). This paper shows the semi-analytical as well as a graphical interpretation of the solution by using mathematical software &ldquo;Mathematica Wolform&rdquo; and considering Caputo&rsquo;s sense derivatives to semi-analytical results obtained by NSADM.&nbsp;Fully fuzzy initial value problem of Caputo-Fabrizio fractional differential equations
https://cmde.tabrizu.ac.ir/article_15924.html
&lrm;We aim at presenting results including analytical solutions to linear fully fuzzy Caputo-Fabrizio fractional differential equations. In such linear equations, the coefficients are fuzzy numbers and, as a useful approach, the cross product has been considered as a multiplication between the fuzzy data. This approach plays an essential role in simplifying of computation of analytical solutions of linear fully fuzzy problems. The obtained results have been applied for deriving explicit solutions of linear Caputo-Fabrizio differential equations with fuzzy coefficients and of the corresponding initial value problems. Some of the topics which are needed for the results of this study from the point of view of the cross product of fuzzy numbers have been explained in detail. We illustrate our technique and compare the effect of uncertainty of the coefficients and initial value on the related solutions.A pseudospectral Sinc method for numerical investigation of the nonlinear time-fractional Klein-Gordon and sine-Gordon equations
https://cmde.tabrizu.ac.ir/article_15481.html
In this paper, a pseudospectral method is proposed for solving the nonlinear time-fractional Klein-Gordon and sine-Gordon equations. The method is based on the Sinc operational matrices. A finite difference scheme is used to discretize the Caputo time-fractional derivative, while the spatial derivatives are approximated by the Sinc method. The convergence of the full discretization of the problem is studied. Some numerical examples are presented to confirm the accuracy and efficiency of the proposed method. The numerical results are compared with the analytical solution and the reported results in the literature.&nbsp;Fitted mesh numerical scheme for singularly perturbed delay reaction diffusion problem with integral boundary condition
https://cmde.tabrizu.ac.ir/article_15925.html
This article presents a numerical treatment of the singularly perturbed delay reaction diffusion problem with an integral boundary condition. In the considered problem, a small parameter ", is multiplied on the higher order derivative term. The presence of this parameter causes the existence of boundary layers in the solution. The solution also exhibits an interior layer because of the large spatial delay. Simpson&rsquo;s 1/3 rule is applied to approximate the integral boundary condition given on the right end of the domain. A standard finite difference scheme on piecewise uniform Shishkin mesh is proposed to discretize the problem in the spatial direction, and the Crank-Nicolson method is used in the temporal direction. The developed numerical scheme is parameter uniformly convergent, with orders of convergence almost two in space and two in time. Two numerical examples are considered to validate the theoretical results.Numerical solution of nonlinear Sine-Gordon equation using modified cubic B-spline-based differential quadrature method
https://cmde.tabrizu.ac.ir/article_15238.html
In this article, we discuss the numerical solution of the nonlinear Sine-Gordon equation in one and two dimensions and its coupled form. A differential quadrature technique based on a modified set of cubic B-splines has been used. The chosen modification possesses the optimal accuracy order four in the spatial domain. The spatial derivatives are approximated by the differential quadrature technique, where the weight coefficients are calculated using this set of modified cubic B-splines. This approximation will lead to the discretization of the problem in the spatial domain that gives a system of first-order ordinary differential equations. This system is then solved using the SSP-RK54 scheme to progress the solution to the next time level. The convergence of this numerical scheme solely depends on the differential quadrature and is found to give a stable solution. The order of convergence is calculated and is observed to be four. The entire computation is performed up to a large time level with an efficient speed. It is found that the computed solution is in good agreement with the exact one and the error comparison with similar works in the literature indicates the scheme outperforms.&nbsp;Analytical solutions of the fractional (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation
https://cmde.tabrizu.ac.ir/article_15970.html
This paper addresses the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with fractional derivative definition. Initially, conformable derivative definitions and their features are presented. Then, by submitting exp(&ndash;&phi;(&xi;))-expansion, generalized (G&prime;/G)-expansion and Modified Kudryashov methods, exact solutions of this equation are generated. The 3D, contour, and 2D surfaces, as well as the related contour plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The physical meaning of the geometrical structures for some of these solutions is discussed. For the observation of the physical activities of the problem, achieved exact solutions are vital. The acquired results can help to demonstrate the physical application of the investigated models and other nonlinear physical models found in mathematical physics. Therefore, it would appear that these approaches might yield noteworthy results in producing the exact solutions to fractional differential equations in a wide range.Jacobi wavelets method for numerical solution of fractional population growth model
https://cmde.tabrizu.ac.ir/article_14470.html
This paper deals with the generalization of the fractional operational matrix of Jacobi wavelets. The fractional population growth model was solved by using this operational matrix and compared with other existing methods to illustrate the applicability of the method. Then, convergence and error analysis of this procedure were studied.&nbsp;Solving Abel's equations with the shifted Legendre polynomials
https://cmde.tabrizu.ac.ir/article_15975.html
In this article, a numerical method is presented to solve Abel's equations. In the given method, the solution of the equation is found as a finite expansion of the shifted Legendre polynomials. To this end, the integral and differential parts of the equation are converted to vector-matrix representations. Therefore the equation is converted to an algebraic system of the equations and by solving it, the solution of the equation is obtained. Further, the numerical example is given to illustrate the method's efficiency.Applying moving frames to finding conservation laws of the nonlinear Klein-Gordon equation
https://cmde.tabrizu.ac.ir/article_14999.html
In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the one-dimensional nonlinear Klein-Gordon equation. Noether&rsquo;s First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the Klein-Gordon equation, we first present a Lagrangian whose Euler-Lagrange equation is the Klein-Gordon equation, and then according to Gon&cedil;calves and Mansfield&rsquo;s method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action.&nbsp;Infinitely smooth multiquadric RBFs combined high-resolution compact discretization for nonlinear 2D elliptic PDEs on a scattered grid network
https://cmde.tabrizu.ac.ir/article_15926.html
Multiquadric radial basis functions combined with compact discretization to estimate solutions of two dimensions nonlinear elliptic type partial differential equations are presented. The scattered grid network with continuously varying step sizes helps tune the solution accuracies depending upon the location of high oscillation. The radial basis functions employing a nine-point grid network are used to improve the functional evaluations by compact formulation, and it saves memory space and computing time. A detailed description of convergence theory is presented to estimate the error bounds. The analysis is based on a strongly connected graph of the Jacobian matrix, and their monotonicity occurred in the scheme. It is shown that the present strategy improves the approximate solution values for the elliptic equations exhibiting a sharp changing character in a thin zone. Numerical simulations for the convection- diffusion equation, Graetz-Nusselt equation, Schro ̈dinger equation, Burgers equation, and Gelfand-Bratu equation are reported to illustrate the utility of the new algorithm.On approximating eigenvalues and eigenfunctions of fractional order Sturm-Liouville problems
https://cmde.tabrizu.ac.ir/article_15240.html
In this paper, the eigenvalues and corresponding eigenfunctions of afractional order Sturm-Liouville problem (FSLP) are approximated byusing fractional differential transform method (FDTM), which is ageneralization of differential transform method (DTM). FDTM reducesthe proposed fourth order FSLP to a system of algebraic equations.The resulting coefficient matrix defines a characteristic polynomialwhich its roots correspond to the eigenvalues of FSLP. The obtainednumerical results which are compared with the results of otherpapers confirm the efficiency of the method.Improved new qualitative results on stochastic delay differential equations of second order
https://cmde.tabrizu.ac.ir/article_15681.html
This paper deals with a class of stochastic delay differential equations (SDDEs) of second order with multiple delays. Here, two main and novel results are proved on stochastic asymptotic stability and stochastic boundedness of solutions of the considered SDDEs. In the proofs of results, the Lyapunov-Krasovskii functional (LKF) method is used as the main tool. A comparison between our results and that are available in the literature shows that the main results of this paper have new contributions to the related ones in the current literature. Two numerical examples are given to show the applications of the given results.An explicit split-step truncated Milstein method for stochastic differential equations
https://cmde.tabrizu.ac.ir/article_15639.html
In this paper, we propose an explicit split-step truncated Milstein method for stochastic differential equations (SDEs) with commutative noise. We discuss the mean-square convergence properties of the new method for numerical solutions of a class of highly nonlinear SDEs in a finite time interval. As a result, we show that the strong convergence rate of the new method can be arbitrarily close to one under some additional conditions. Finally, we use an illustrative example to highlight the advantages of our new findings in terms of both stability and accuracy compared to the results in Guo et al. (2018).