A wavelet method to solve nonlinear variable-order time fractional 2D Klein–Gordon equation

In this study, we introduce the nonlinear variable-order time fractional two-dimensional (2D) Klein–Gordon equation by using the concept of variable-order fractional derivatives. The variable-order fractional derivative operator is defined in the Caputo type. Due to the useful properties of wavelets, we propose an efficient semi-discrete method based on the 2D Legendre wavelets (LWs) to numerically solve this equation. In fact, according to the proposed method, the variable-order time fractional derivative should be discretized in the first stage, and then the solution of the problem expanded in terms of the 2D LWs. However, the main objective of this study is to illustrate that the 2D LWs can be a useful tool for solving the nonlinear variable-order time fractional 2D Klein–Gordon equation. Stability analysis of the presented method is investigated theoretically and numerically. Moreover, the applicability and accuracy of the method are investigated by solving some numerical examples. Numerical results confirm the spectral accuracy of the established method.