A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations

2018/08/07 by Fanhai Zeng, Zeng, Fanhai, Ian Turner +5
Mathematics · #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1808.02170

openalex publication_date 2018/08/07 · openalex created_date 2018/08/22 · openalex updated_date 2026/08/01

Abstract

We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters. Subsequently, we develop an efficient strategy to calculate the discrete convolution for the approximation of the fractional operator in the semi-implicit method and we derive an error bound of the fast convolution. The memory requirement and computational cost of the present semi-implicit methods with a fast convolution are about O(Nlog n_T) and O(Nn_Tlog n_T), respectively, where N is a suitable positive integer and n_T is the final number of time steps. Numerical simulations, including the solution of a system of two nonlinear fractional diffusion equations with different fractional orders in two-dimensions, are presented to verify the effectiveness of the semi-implicit methods.

Citations

• A Fast Time Stepping Method for Evaluating Fractional Integrals
• High-order splitting methods for the incompressible Navier-Stokes equations
• Fast Summation by Interval Clustering for an Evolution Equation with Memory

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