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Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. (English) Zbl 1481.65153

Summary: In this work, two fully novel finite difference schemes for two-dimensional time-fractional mixed diffusion and diffusion-wave equation (TFMDDWEs) are presented. Firstly, a Hermite and Newton quadratic interpolation polynomial have been used for time discretization and central quotient has used in spatial direction. The H2N2 finite difference is constructed. Secondly, in order to increase computational efficiency, the sum-of-exponential is used to approximate the kernel function in the fractional-order operator. The fast H2N2 finite difference is obtained. Thirdly, the stability and convergence of two schemes are studied by energy method. When the tolerance error \( \epsilon \) of fast algorithm is sufficiently small, it proves that both of difference schemes are of \( 3-\beta\) \((1<\beta<2) \) order convergence in time and of second order convergence in space. Finally, numerical results demonstrate the theoretical convergence and effectiveness of the fast algorithm.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65T50 Numerical methods for discrete and fast Fourier transforms
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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