It is time-memory consuming when numerically solving time fractional partial differential equations, as it requires O ( N 2 ) computational cost and O ( M N ) memory complexity with finite difference methods, where, N and M are the total number of time steps and spatial grid points, respectively. To surmount this issue, we develop an efficient hybrid method with O ( N ) computational cost and O ( M ) memory complexity in solving two-dimensional time fractional diffusion equation. The presented method is based on the Laplace transform method and a finite difference scheme. The stability and convergence of the proposed method are analyzed rigorously by the means of the Fourier method. A comparative study drawn from numerical experiments shows that the hybrid method is accurate and reduces the computational cost, memory requirement as well as the CPU time effectively compared to a standard finite difference scheme.
<abstract><p>In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.</p></abstract>
In this paper, the development of new hybrid group iterative methods for the numerical solution of a two-dimensional time-fractional cable equation is presented. We use Laplace transform method to approximate the time fractional derivative which reduces the problem into an approximating partial differential equation. The obtained partial differential equation is solved by four-point group iterative methods derived from two implicit finite difference schemes. Matrix norm analysis together with mathematical induction are utilized to investigate the stability and convergence properties. A comparative study with the recently developed hybrid standard point (HSP) iterative method accompanied by their computational cost analysis are also given. Numerical experiments are conducted to demonstrate the superiority of the proposed hybrid group iterative methods over the HSP iterative method in terms of the number of iterations, computational cost as well as the CPU times.
In this paper, a hybrid method based on the Laplace transform and implicit finite difference scheme is applied to obtain the numerical solution of the two-dimensional time fractional advection-diffusion equation (2D-TFADE). Some of the major limitations in computing the numerical solution for fractional differential equations (FDEs) in multi-dimensional space are the huge computational cost and storage requirement, which are O(N^2) cost and O(MN) storage, N and M are the total number of time levels and space grid points, respectively. The proposed method reduced the computational complexity efficiently as it requires only O(N) computational cost and O(M) storage with reasonable accuracy when numerically solving the TFADE. The method’s stability and convergence are also investigated. The Results of numerical experiments of the proposed method are obtained and found to compare well with the results of existing standard finite difference scheme.
<abstract><p>In recent years, fractional partial differential equations (FPDEs) have been viewed as powerful mathematical tools for describing ample phenomena in various scientific disciplines and have been extensively researched. In this article, the hybrid explicit group (HEG) method and the modified hybrid explicit group (MHEG) method are proposed to solve the 2D advection-diffusion problem involving fractional-order derivative of Caputo-type in the temporal direction. The considered problem models transport processes occurring in real-world complex systems. The hybrid grouping methods are developed based upon a Laplace transformation technique with a pair of explicit group finite difference approximations constructed on different grid spacings. The proposed methods are beneficial in reducing the computational burden resulting from the nonlocality of fractional-order differential operator. The theoretical investigation of stability and convergence properties is conducted by utilizing the matrix norm analysis. The improved performance of the proposed methods against a recent competitive method in terms of central processing unit (CPU) time, iterations number and computational cost is illustrated by several numerical experiments.</p></abstract>
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