The Difference Methods Of Time Fractional Wave Equation

Fractional order calculus is an expansion of traditional integral calculus theory.It first appeared in the letter written by German mathematician G.W.Leibniz to L'Hospital in 1695.Fractional differential operator is very suitable for describing various kinds of complex mechanics,physical behavior and the materials with memory and heredity due to its non-locality.While the analytical solution of fractional differential equation is difficult to solve,we have to pay more attention to the efficient numerical solutions.This paper is devoted to constructing several difference methods that based on rectangular formula with weighted fractional order for the two-dimensional time fractional wave equations.Firstly,we construct a two order center difference scheme for the two-dimensional fractional wave equation with a time fractional derivative of order a(1<?<2).Based on the equivalence between Caputo fractional derivative and Riemann-Liouville fractional derivative under certain conditions,we transform the two-dimensional fractional wave equation into an integro-differential equation.Then the central difference schemes and the weighted fractional rectangular formula are used to approximate the space derivatives and the time derivative,respectively.A difference scheme with the truncation error order of ??+h12+h22 is derived for the two-dimensional fractional wave equation,where ? is the time step and h1,h2 are steps in the x and y directions,respectively.The numerical stability and convergence of the two order center difference scheme are presented by Gronwall's inequality.Secondly,we construct a high order compact difference scheme for the two-dimensional fractional wave equation.The average operators H1 and H2 are introduced into the spatial discretization,and the weighted fractional rectangular formula is used to the time stepping.A high order compact difference scheme with the truncation error order of ??+h14+h24 is derived.The numerical stability and convergence of the difference scheme are also proved.Thirdly,we construct a high order compact alternating direction implicit(ADI)scheme based on the high order compact difference scheme.Meanwhile,by adding a higher order item,we apply the ADI to the time stepping to improve the calculation speed of numerical algorithm.It is proved that when the weight is ??1/2,the convergent order of time direction is ??,and the convergence order of spatial direction is h14+h24.Finally,numerical examples are provided to verify the effectiveness of three differential methods proposed in this paper.