Abstract
Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.
Highlights
Fractional differential equations have been widely used in medicine, mechanics, control theory, environmental science, and finance [1,2,3,4]
This paper considers multi-term time fractional diffusion equation [2,5]
For multi-term time fractional diffusion-wave equation, Dehghan et al (2015) [19] combined with finite difference method and Galerkin spectral method to give a numerical algorithm with fourth-order precision
Summary
Fractional differential equations have been widely used in medicine, mechanics, control theory, environmental science, and finance [1,2,3,4]. For multi-term time fractional diffusion-wave equation, Dehghan et al (2015) [19] combined with finite difference method and Galerkin spectral method to give a numerical algorithm with fourth-order precision. For multi-term time fractional diffusion equation, Gao et al (2017) [20] firstly constructed a numerical difference formula with second-order accuracy to approximate multiple Caputo type time fractional derivatives, and applied fourth-order difference format in space and proposed the high precision difference scheme for time fractional diffusion equations. Wang et al (2016) [33] proposed an efficient parallel algorithm for Caputo fractional reaction–diffusion equation with implicit difference scheme They developed a new tridiagonal reduced system with elimination method. We try to construct a class of alternating segment pure explicit–implicit (PASE-I) and pure implicit–explicit (PASI-E) parallel nature difference methods for solving multi-term time fractional diffusion equations. Numerical experiments are used to verify the correctness of theoretical analysis
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