A robust numerical method for a fractional differential equation
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摘要
This paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order α ∈ (0, 1). First the fractional differential equation is transformed into an equivalent Volterra integral equation of the second kind with a weakly singular kernel. Based on the a priori information about the exact solution, an integral discretization scheme on an a priori chosen adapted mesh is proposed. By applying the truncation error estimate techniques and a discrete analogue of Gronwall’s inequality, it is proved that the numerical method is first-order convergent in the discrete maximum norm. Numerical results indicate that this method is more accurate and robust than finite difference methods when α is close to 0.
论文关键词:Fractional differential equation,Caputo fractional derivative,Volterra integral equation,Adapted mesh,Convergence analysis
论文评审过程:Received 6 March 2017, Revised 3 August 2017, Accepted 6 August 2017, Available online 16 August 2017, Version of Record 16 August 2017.
论文官网地址:https://doi.org/10.1016/j.amc.2017.08.011