An Advanced Implicit Meshless Approach for the Non-linear Anomalous Subdiffusion Equation
Y. T. Gu; P. Zhuang; and F. Liu;

doi:10.3970/cmes.2010.056.303
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 56, No. 3, pp. 303-334, 2010
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Keywords anomalous subdiffusion equation, fractional differential equation, numerical simulation, meshless method, radial basis functions
Abstract Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation(FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by using the meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formulations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modelling and simulation of the ASDE. Therefore, the meshless technique should have good potential in development of a robust simulation tool for problems in engineering and science which are governed by the various types of fractional differential equations.
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