In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is $(4-\alpha)$, where $\alpha ~(0<\alpha<1)$ is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of generalized fractional advection-diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss about the stability and convergence of the difference scheme. Numerical examples are presented to examine the theoretical claims. The convergence order of the difference scheme is analyzed numerically, which is $(4-\alpha)$ in time and second-order in space.
翻译:在本条中,基于立方内插公式的高阶时间跨步法被认为是接近通用的卡普托分数衍生物(GCFD)的近似方法,这一方法的趋同顺序为$(4-\alpha),其中美元=(0 ⁇ alpha < 1美元)是GFD的顺序。还提供当地截断错误。然后,我们采用发达的办法,建立一个差别办法,解决与Drichlet边界条件的普遍分数对流-扩散方程式。此外,我们讨论了差异办法的稳定性和趋同性。提供了数字例子来审查理论索赔。对差异办法的趋同顺序进行了数字分析,在时间和空间的第二顺序上是$(4-alpha)。