A Meshless Solution for the Variable-Order Time Fractional Nonlinear Klein–Gordon Equation

In this paper, we study the variable-order (V-O) time fractional Klein–Gordon equation which widely appears in the various fields of engineering and mathematical physics. The numerical method which we have used for solving this equation is based on a combination of the radial basis functions (RBFs) method and finite difference scheme. In the first stage the V-O time-dependent derivative is discreticized, and then we approximate the solution by the RBFs. The main goal is to show that the collocation method based on RBFs is suitable for solving V-O fractional differential equations. The applicability of the proposed method is investigated by solving some numerical examples. The obtained results show that the proposed approach is very efficient and accurate. Also, the effect of replacing V-O fractional derivative of order $$\alpha (x,t)$$ with its approximations on the behavior of approximate solutions relative to the exact solution is investigated numerically.