The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method

Abstract

Introduction: This study describes a novel meshless technique for solving one of common problem within cell biology, computer graphics, image processing and fluid flow. The diffusion mechanism has extremely depended on the properties of the structure. Objectives: The present paper studies why diffusion processes not following integer-order differential equations, and present novel meshless method for solving. diffusion problem on surface numerically. Methods: The variable- order time fractional diffusion equation (VO-TFDE) is developed along with sense of the Caputo derivative for (0<α(t)<1). An efficient and accurate meshfree method based on the singular boundary method (SBM) and dual reciprocity method (DRM) in concomitant with finite difference scheme is proposed on three-dimensional arbitrary geometry. To discrete of the temporal term, the finite diffract method (FDM) is utilized. In the spatial variation domain; the proposal method is constructed two part. To evaluating first part, fundamental solution of (VO-TFDE) is transformed into inhomogeneous Helmholtz-type to implement the SBM approximation and other part the DRM is utilized to compute the particular solution. Results: The stability and convergent of the proposed method is numerically investigated on high dimensional domain. To verified the reliability and the accuracy of the present approach on complex geometry several examples are investigated. Conclusions: The result of study provides a rapid and practical scheme to capture the behavior of diffusion process.

Keywords

• 31B10
• 44A10
• 26A33