一类分数阶导数微分方程的隐式差分解法 An Implicit Finite Difference Scheme for Space-Time Fractional Partial Differential Equation

分数阶导数微分方程作为通常微分方程的推广，被广泛地应用于工程，物理，信息处理，金融，水文等领域。本文给出了数值求解一类时间空间分数阶导数的双边空间微分方程的一种隐式差分格式，并对其稳定性和收敛性进行了理论分析，证明了格式的无条件稳定性并给出了收敛阶估计。 Fractional order differential equations are generalizations of classical differential equations. They are widely used in the fields of diffusive transport, finance, nonlinear dynamics, signal processing and others. In this paper, an implicit finite difference method for a class of initial-boundary value space-time fractional two-sided space partial differential equations with variable coefficients on a finite domain is established. The stability and convergence order are analyzed for the resulted implicit scheme. With mathematical induction skills, the scheme is proved to be unconditionally stable and convergent.