An efficient numerical technique for variable order time fractional nonlinear Klein-Gordon equation

This paper studies the variable order time fractional nonlinear Klein-Gordon equation (V-TFNKGE). An optimization method using the Caputo type definition and a set of basis functions, namely the generalized polynomials (GP), is proposed. The solution is expanded in terms of the GP with unknown free coefficients and control parameters. First, a new variable order time fractional operational matrix of the Caputo type for the GP is derived. Then, based on a operational matrix and the Lagrange multipliers, an optimization process achieves the approximate solution. Additionally, the convergence of the proposed method is analyzed and two numerical examples illustrate its good performance. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.