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

摘要

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.