Numerical method for generalized time fractional KdV‐type equation

In this article, an efficient numerical method for linearized and nonlinear generalized time‐fractional KdV‐type equations is proposed by combining the finite difference scheme and Petrov–Galerkin spectral method. The scale and weight functions involved in generalized fractional derivative cause too much difficulty in discretization and numerical analysis. Fortunately, motivated by finite difference method for fractional differential equation on graded mesh, the stability and convergence of the constructed method are established rigorously. It is proved that the full discretization schemes of generalized time‐fractional KdV‐type equation is unconditionally stable in linear case. While for nonlinear case, it is stable under a CFL condition and for not small ϵ, coefficient of the high‐order spatial differential term. In addition, the full discretization schemes with respect to linear and nonlinear cases respectively converge to the associated exact solutions with orders Oτ2−α+N−m and Oτ+N−m , where τ, α, N and m accordingly indicate the time step size, the order of the fractional derivative, polynomial degree, and regularity of the exact solution. Numerical experiments are carried out to support the theoretical results.

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