An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

Abstract

In this paper, we propose an effective numerical method using two-dimensional Shifted fractional-order Gegenbauer Multi-wavelets to find the approximate solutions of the time-fractional distributed order non-linear partial differential equations. The method is applied to numerically solve the fractional distributed order non-linear Klein–Gordon equation. We derive an exact formula for the Riemann-Liouville fractional integral operator for the Shifted fractional Gegenbauer Multi-wavelets. Applying function approximations obtained by this method turns the considered equation into a system of algebraic equations. Error estimation and convergence analysis of the method are also studied. Some numerical examples are included to show and check the effectiveness of the proposed method.