On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order . Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler's and Talbot's methods performed better than Stehfest's method.