Anomalous sub-diffusion equations by the meshless collocation method

Abstract

Recently, many new applications in engineering and science are governed by a series of fractional ordinary differential equation or fractional partial differential equations (FPDEs), in which the differential order is with a fractional order. The anomalous sub-diffusion equation (ASDE) is a typical FPDE. The current dominant numerical method for modelling ASDE is finite difference method, which is based on a pre-defined grid leading to inherited issues or shortcomings. Because of its distinguished advantages, the meshless method has good potential in simulation of ASDE. This paper aims to develop an implicit meshless collocation technique based on the moving least squares (MLS) approximation for numerical simulation of ASDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretisation are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modelling and simulation of ASDEs.

Keywords:

• Fractional differential equation
• anomalous sub-diffusion equation
• meshless method
• moving least squares
• collocation scheme

Additional information

Notes on contributors

Y T Gu

Dr YuanTong Gu is an Associate Professor and an ARC Future Fellow in School of Chemistry, Physics and Mechanical Engineering at Queensland University of Technology (QuT). Before joining QUT, he worked at the University of Sydney and University of California. His current research interests include advanced numerical modelling for mechanical engineering, computational biomechanics, modified finite element modelling and meshless techniques, and nanotechnology.

P Zhuang

Dr Pinghui Zhuang is an Associate Professor in School of Mathematical Sciences at Xiamen University (China). He was a visiting professor at the Queensland University of Technology. His current research interests include computational mathematics, fractional partial differential equations and meshless techniques.