We investigate the application of Quarter-Sweep Gauss-Seidel (QSGS) iterative method for solving (SFPDE's) space-fractional partial diffusion equations with Dirichlet boundary condition. To do this, ...implicit finite difference scheme and Caputo's derivative operator are used to discretize one-dimensional linear space-fractional equation to form system of linear equations. Then basic ideas formulation and application of the suggested iterative method are also introduced. Numerical examples of tested problems were carried out to demonstrate advantages of the proposed iterative method beside to the HSGS and FSGS as control method. Based on computational results, the QSGS method is shown to be the most superior than the HSGS and FSGS iterative methods.
This paper examines the performance of the Preconditioned SOR (PSOR) method together with an unconditionally implicit Caputo's time-fractional finite difference approximation equation for solving ...time-fractional partial diffusion equations (TFPDE's). To do it, the implicit Caputo's time-fractional approximation equations and preconditioned matrix are used to construct the corresponding preconditioned linear system. In addition to that, formulation and implementation the PSOR iterative method are also presented. Based on numerical results of the proposed iterative method, it can be concluded that the proposed iterative method is superior to the basic iterative methods.
In this study, we propose approximate solution of the time-fractional diffusion equation (TFDE's) based on a quarter-sweep implicit finite difference approximation equation. To derive this ...approximation equation, the Caputo's time fractional derivative has been used to discretize the proposed problems. By using the Caputo's finite difference approximation equation, a linear system will be generated and solved iteratively. In addition to that, formulation and implementation the Quarter-Sweep Gauss-Seidel (QSGS) iterative method are also presented. Based on numerical results of the proposed iterative method, it can be concluded that the proposed iterative method is superior to the FSGS and HSGS iterative method.
This paper considers the numerical solution of a one-dimensional space-fractional diffusion equation. To obtain the solution, we use an unconditionally stable implicit finite difference approximation ...with the Caputo's space-fractional operator. We study on improving the convergence rate of the solution while solving the generated linear system through the approximation equation iteratively. In our study, we apply the preconditioning technique to construct a preconditioned linear system which eventually derives into a Full-Sweep Preconditioned AOR. From the presented results, we show that the proposed Full-Sweep Preconditioned AOR iterative method has superiority in efficiency compared to the basic Full-Sweep Preconditioned SOR and Full-Sweep Preconditioned Gauss-Seidel iterative methods.
This paper presents the application of a half-sweep iteration concept to the Grünwald implicit difference schemes with the Kaudd Successive Over-Relaxation (KSOR) iterative method in solving ...one-dimensional linear time-fractional parabolic equations. The formulation and implementation of the proposed methods are discussed. In order to validate the performance of HSKSOR, comparisons are made with another two iterative methods, full-sweep KSOR (FSKSOR) and Gauss-Seidel (FSGS) iterative methods. Based on the numerical results of three tested examples, it shows that the HSKSOR is superior compared to FSKSOR and FSGS iterative methods.
We deal with the application of an unconditionally implicit finite difference approximation equation of the one-dimensional linear time fractional diffusion equations (TFDE's) via the Caputo's time ...fractional derivative. Based on this implicit approximation equation, the corresponding linear system can be generated in which its coefficient matrix is large scale and sparse. To speed up the convergence rate in solving the linear system iteratively, we construct the corresponding preconditioned linear system. Then we formulate and implement the Preconditioned AOR (PAOR) iterative method for solving the generated linear system. One example of the problem is presented to illustrate the effectiveness of PAOR method. The numerical results of this study show that the proposed iterative method is superior to PSOR and PGS, GS iterative method.
In this paper, our main concerned is on the application of the formulation of a four-point explicit group successive over-relaxation (4EGSOR) iterative method in solving one-dimensional ...time-fractional parabolic equations based on the second-order Grünwald implicit approximation equation. The formulation of the 4EGSOR method is constructed by using the implicit approximation equation which is derived by the Grunwald derivative operator and the implicit finite difference discretization scheme. In order to access the performance results of the 4EGSOR iterative method, another block and point iterative methods which are the four-point EGGS (4EGGS) and the Gauss-Seidel (GS) were also presented as control methods. The results of three numerical experiments show substantial improvement in terms of the number of iterations and execution time.
The aim of this paper is to examine the effectiveness of Successive Over-Relaxation (SOR) iterative method for solving one-dimensional time-fractional parabolic equations. The Grünwald fractional ...derivative operator and implicit finite difference scheme have been used to discretize the proposed linear time-fractional equations to construct system of Grünwald implicit approximation equation. The basic formulation and application of the SOR iterative method are also presented. To investigate the effectiveness of the proposed iterative method, numerical experiments and comparison are made based on the iteration numbers, time execution, and maximum absolute error. Based on numerical results, the accuracy of Grünwald implicit solution obtained by proposed iterative method is in excellent agreement, and it can be concluded that the proposed iterative method requires less number of iterations and execution time as compared to the Gauss-Seidel (GS) iterative method.
In this study, system of Grünwald implicit approximation equations has been developed through the discretization of one-dimensional linear time-fractional parabolic equations using the Grünwald ...fractional derivative operator and second-order implicit finite difference scheme. The aim of this paper is to examine the effectiveness of Kaudd Successive Over-Relaxation (KSOR) iterative method, which is one of the weighted point iterative schemes for solving the proposed time-fractional parabolic equations by considering the Grünwald implicit approximation equation. To investigate the effectiveness of the proposed iterative method, numerical experiments and comparison are made in terms of number of iterations, execution time, and maximum absolute error. Based on numerical results, the accuracy of Grünwald implicit solution obtained by proposed iterative method is in excellent agreement, and it can be concluded that the proposed KSOR iterative method requires less number of iterations and execution time as compared to the existing point iterative method.
In this study, we propose approximate algorithm solution of the space-fractional diffusion equation (SFDE's) based on a quarter-sweep (QS) implicit finite difference approximation equation. To derive ...this approximation equation, the Caputo's space-fractional derivative has been used to discretize the proposed problems. By using the Caputo's finite difference approximation equation, a linear system will be generated and solved iteratively. In addition to that, formulation and implementation algorithm the Quarter-Sweep AOR (QSAOR) iterative method are also presented. Based on numerical results of the proposed iterative method, it can be concluded that the proposed iterative method is superior to the FSAOR and HSAOR iterative method.
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