Abstract
In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(tau ^{3-alpha } + h^2), 0<alpha <1 are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where tau and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.
Highlights
We will consider the numerical methods for solving the following time fractional partial differential equation
Yan et al [33] introduced a numerical method for solving linear fractional differential equation with convergence order O(τ 3−α), 0 < α < 1 by approximating the Hadamard finite-part integral with the quadratic interpolation polynomials following Diethelm’s idea in [4]
We choose Uh0 = Rhu0 and we assume that Uh1 ∈ Sh is a suitable approximation of u(t1) which can be obtained by using some special numerical methods and satisfies the condition (3.3) below
Summary
We will consider the numerical methods for solving the following time fractional partial differential equation (1.1). Yan et al [33] introduced a numerical method for solving linear fractional differential equation with convergence order O(τ 3−α), 0 < α < 1 by approximating the Hadamard finite-part integral with the quadratic interpolation polynomials following Diethelm’s idea in [4] They obtained an asymptotic expansion of the error, but there are no error estimates proved in [33]. By c0, c1, c2 we denote some particular positive constants independent of the functions and parameters concerned
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