Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature

Abstract

This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.