The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain

In this paper, for a class of fractional sub-diffusion equations on a space unbounded domain, firstly, exact artificial boundary conditions, which involve the time-fractional derivatives, are derived using the Laplace transform technique. Then the original problem on the space unbounded domain is reduced to the initial-boundary value problem on a space bounded domain. Secondly, an efficient finite difference approximation for the reduced initial-boundary problem on the space bounded domain is constructed. Different from the method of order reduction used in [37] for the fractional sub-diffusion equations on a space half-infinite domain, the presented difference scheme, which is more simple than that in the previous work, is developed using the direct discretization method, i.e. the approximate method of considering the governing equations at mesh points directly. The stability and convergence of the scheme with numerical accuracy O ( τ 2 - γ + h 2 ) are proved by means of discrete energy method and Sobolev imbedding inequality, where γ is the order of time-fractional derivative in the governing equation, τ and h are the temporal stepsize and spatial stepsize, respectively. Thirdly, a compact difference scheme for the case of γ ⩽ 2 / 3 is derived with the truncation errors of fourth-order accuracy for interior points and third-order accuracy for boundary points, respectively. Then the global convergence order O ( τ 2 - γ + h 4 ) of the compact difference scheme is proved. Finally, numerical experiments are used to verify the numerical accuracy and the efficiency of the obtained schemes.