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) 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. KeywordsFractional advection-diffusion equation, Laplace transform, Finite difference scheme, Stability, Convergence.

中文翻译：

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二维时间分数阶对流扩散方程数值解的计算有效混合方法

本文采用基于Laplace变换和隐式有限差分格式的混合方法，获得了二维时间分数对流扩散方程（2D-TFADE）的数值解。在多维空间中计算分数阶微分方程（FDE）数值解的一些主要限制是巨大的计算成本和存储要求，这是O（N）成本和O（MN）的存储量，N和M是时间级别总数和空间网格点总数。该方法有效地降低了计算复杂度，因为在数值求解TFADE时只需要O（N）的计算成本和O（M）的存储，并具有合理的精度。还研究了该方法的稳定性和收敛性。获得了该方法的数值实验结果，并与现有标准有限差分方案的结果进行了比较。分数维对流扩散方程拉普拉斯变换有限差分格式稳定性收敛性