We design a new model of preconditioner for systems of linear equations. The convergence properti... more We design a new model of preconditioner for systems of linear equations. The convergence properties of the proposed methods have been analyzed and compared with the classical methods. Numerical experiments of convection-diffusion equations show a good im- provement on the convergence, and show that the convergence rates of proposed methods are superior to the other modified iterative methods.
The solutions of weakly singular fractional Volterra integro‐differential equations involving the... more The solutions of weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative typically have solutions whose derivatives are unbounded at the left end‐point of the interval of integration. In this paper, we design an algorithm to prevail on this non‐smooth behavior of solutions of the nonlinear fractional Volterra integro‐differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.
The solutions of weakly singular fractional integro-differential equations involving the Caputo de... more The solutions of weakly singular fractional integro-differential equations involving the Caputo derivative have singularity at the lower bound of the domain of integration. In this paper, we design an algorithm to prevail on this non-smooth behaviour of solutions of the nonlinear fractional integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.
International Journal of Nonlinear Analysis and Applications, 2022
This article deals with the nonlinear parabolic equation with piecewise continuous arguments (EP... more This article deals with the nonlinear parabolic equation with piecewise continuous arguments (EPCA). This study, therefore, with the aid of the $theta$ -methods, aims at presenting a numerical solution scheme for solving such types of equations which has applications in certain ecological studies. Moreover, the convergence and stability of our proposed numerical method are investigated. Finally, to support and confirm our theoretical results, some numerical examples are also presented.
International Journal of Nonlinear Analysis and Applications, 2022
The Burger‒Huxley equation as a well-known nonlinear physical model is studied numerically in the... more The Burger‒Huxley equation as a well-known nonlinear physical model is studied numerically in the present paper. In this respect, the nonstandard finite difference (NSFD) scheme in company with the Richtmyer’s (3, 1, 1) implicit formula is formally adopted to accomplish this goal. Moreover, the stability, convergence, and consistency analyses of nonstandard finite difference schemes are investigated systematically. Several case studies with comparisons are provided, confirming that the current numerical scheme is capable of resulting in highly accurate approximations.
In this paper, a one-dimensional fractional advection-diffusion equation is considered. First we ... more In this paper, a one-dimensional fractional advection-diffusion equation is considered. First we propose a numerical approximation of the Riemann-Liouville fractional derivative which is fourth-order accurate, and then a numerical method for the fractional advection-diffusion equation using a high order finite difference scheme is presented. It is proved that the scheme is convergent. The stability analysis of numerical solutions is also discussed. The method is applied in several examples and the accuracy of the method is tested in terms of error norm. Furthermore, the numerical results have been compared with some other methods.
In this paper we have studied a numerical approximation to the solution of the nonlinear Burgers&... more In this paper we have studied a numerical approximation to the solution of the nonlinear Burgers' equation. The presented scheme is obtained by using the Non-Standard Finite Difference Method (NSFD). The use of NSFD method and its approximations play an important role for the formation of stable numerical methods. The main advantage of the scheme is that the algorithm is very simple and very easy to implement.
In this paper, we introduce a new preconditioner from class of (I+S)-type in order to improve the... more In this paper, we introduce a new preconditioner from class of (I+S)-type in order to improve the convergence rates of iterative methods. Also we prove that this preconditioner compare with the others works better.Moreover, the behavior of the above preconditioners to the boundary value problems (BVP) is studied.
Abstract: The purpose of this paper is to present the efficient iterative method for solving Non-... more Abstract: The purpose of this paper is to present the efficient iterative method for solving Non-Hermitian positive definite systems of Linear complementarity problems (LCP), using splitting of the coefficient matrix and fixed-point principle . Furthermore, the global convergence properties of the proposed method have been analyzed. Numerical results show the applicability of our method. AMS Subject Classifications: 65F10, 65H10, 90C33. Keywords: Linear complementarity problem, SSOR method, Non-Hermitian positive definite matrices.
For linear equations, there are numerous stationary iterative methods. However, these methods are... more For linear equations, there are numerous stationary iterative methods. However, these methods are not applicable in some important problems such as linear system arising from the boundary element method (BEM). In this paper, we proposed two approaches for using stationary iterative methods to linear equations arising from the BEM for the Laplace and convective diffusion with first-order chemical reaction problems. Our proposed methods are simple and graceful. Finally, numerical example is given to show the efficiency of our results. Keywords: Stationary iterative methods, First-order chemical reaction, Boundary element method, FIM methods, Gauss-Seidel, Preconditioning methods.
Computational Methods for Differential Equations, 2020
This article is concerned with using a finite difference method, namely the theta-methods, to solv... more This article is concerned with using a finite difference method, namely the theta-methods, to solve the diffusion-convection equation with piecewise constant arguments.The stability of this scheme is also obtained. Since there are not many published results on the numerical solution of this sort of differential equation and because of the importance of the above equation in the physics and engineering sciences, we have decided to study and present a stable numerical solution for the above mentioned problem. At the end of article some experiments are done to demonstrate the stability of the scheme. We also draw the figures for the numerical and analytical solutions which confirm ou results.The numerical solutions have also been compared with analytical solutions.
We design a new model of preconditioner for systems of linear equations. The convergence properti... more We design a new model of preconditioner for systems of linear equations. The convergence properties of the proposed methods have been analyzed and compared with the classical methods. Numerical experiments of convection-diffusion equations show a good im- provement on the convergence, and show that the convergence rates of proposed methods are superior to the other modified iterative methods.
The solutions of weakly singular fractional Volterra integro‐differential equations involving the... more The solutions of weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative typically have solutions whose derivatives are unbounded at the left end‐point of the interval of integration. In this paper, we design an algorithm to prevail on this non‐smooth behavior of solutions of the nonlinear fractional Volterra integro‐differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.
The solutions of weakly singular fractional integro-differential equations involving the Caputo de... more The solutions of weakly singular fractional integro-differential equations involving the Caputo derivative have singularity at the lower bound of the domain of integration. In this paper, we design an algorithm to prevail on this non-smooth behaviour of solutions of the nonlinear fractional integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.
International Journal of Nonlinear Analysis and Applications, 2022
This article deals with the nonlinear parabolic equation with piecewise continuous arguments (EP... more This article deals with the nonlinear parabolic equation with piecewise continuous arguments (EPCA). This study, therefore, with the aid of the $theta$ -methods, aims at presenting a numerical solution scheme for solving such types of equations which has applications in certain ecological studies. Moreover, the convergence and stability of our proposed numerical method are investigated. Finally, to support and confirm our theoretical results, some numerical examples are also presented.
International Journal of Nonlinear Analysis and Applications, 2022
The Burger‒Huxley equation as a well-known nonlinear physical model is studied numerically in the... more The Burger‒Huxley equation as a well-known nonlinear physical model is studied numerically in the present paper. In this respect, the nonstandard finite difference (NSFD) scheme in company with the Richtmyer’s (3, 1, 1) implicit formula is formally adopted to accomplish this goal. Moreover, the stability, convergence, and consistency analyses of nonstandard finite difference schemes are investigated systematically. Several case studies with comparisons are provided, confirming that the current numerical scheme is capable of resulting in highly accurate approximations.
In this paper, a one-dimensional fractional advection-diffusion equation is considered. First we ... more In this paper, a one-dimensional fractional advection-diffusion equation is considered. First we propose a numerical approximation of the Riemann-Liouville fractional derivative which is fourth-order accurate, and then a numerical method for the fractional advection-diffusion equation using a high order finite difference scheme is presented. It is proved that the scheme is convergent. The stability analysis of numerical solutions is also discussed. The method is applied in several examples and the accuracy of the method is tested in terms of error norm. Furthermore, the numerical results have been compared with some other methods.
In this paper we have studied a numerical approximation to the solution of the nonlinear Burgers&... more In this paper we have studied a numerical approximation to the solution of the nonlinear Burgers' equation. The presented scheme is obtained by using the Non-Standard Finite Difference Method (NSFD). The use of NSFD method and its approximations play an important role for the formation of stable numerical methods. The main advantage of the scheme is that the algorithm is very simple and very easy to implement.
In this paper, we introduce a new preconditioner from class of (I+S)-type in order to improve the... more In this paper, we introduce a new preconditioner from class of (I+S)-type in order to improve the convergence rates of iterative methods. Also we prove that this preconditioner compare with the others works better.Moreover, the behavior of the above preconditioners to the boundary value problems (BVP) is studied.
Abstract: The purpose of this paper is to present the efficient iterative method for solving Non-... more Abstract: The purpose of this paper is to present the efficient iterative method for solving Non-Hermitian positive definite systems of Linear complementarity problems (LCP), using splitting of the coefficient matrix and fixed-point principle . Furthermore, the global convergence properties of the proposed method have been analyzed. Numerical results show the applicability of our method. AMS Subject Classifications: 65F10, 65H10, 90C33. Keywords: Linear complementarity problem, SSOR method, Non-Hermitian positive definite matrices.
For linear equations, there are numerous stationary iterative methods. However, these methods are... more For linear equations, there are numerous stationary iterative methods. However, these methods are not applicable in some important problems such as linear system arising from the boundary element method (BEM). In this paper, we proposed two approaches for using stationary iterative methods to linear equations arising from the BEM for the Laplace and convective diffusion with first-order chemical reaction problems. Our proposed methods are simple and graceful. Finally, numerical example is given to show the efficiency of our results. Keywords: Stationary iterative methods, First-order chemical reaction, Boundary element method, FIM methods, Gauss-Seidel, Preconditioning methods.
Computational Methods for Differential Equations, 2020
This article is concerned with using a finite difference method, namely the theta-methods, to solv... more This article is concerned with using a finite difference method, namely the theta-methods, to solve the diffusion-convection equation with piecewise constant arguments.The stability of this scheme is also obtained. Since there are not many published results on the numerical solution of this sort of differential equation and because of the importance of the above equation in the physics and engineering sciences, we have decided to study and present a stable numerical solution for the above mentioned problem. At the end of article some experiments are done to demonstrate the stability of the scheme. We also draw the figures for the numerical and analytical solutions which confirm ou results.The numerical solutions have also been compared with analytical solutions.
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