Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

Abstract

In this paper, a class of integrodifferential equations with the Caputo fractal-fractional derivative is considered. We study the exact and numerical solutions of the said problem with a fractal-fractional differential operator. The abovementioned operator is arising widely in the mathematical modeling of various physical and biological problems. In our scheme, first, the integrodifferential equation with the fractal-fractional differential operator is converted to an integrodifferential equation with the Riemann–Liouville differential operator. Additionally, the obtained integrodifferential equation is then converted to the equivalent integrodifferential equation involving the Caputo differential operator. Then, we convert the integrodifferential equation under the Caputo differential operator using the Laplace transform to an algebraic equation in the Laplace space. Finally, we convert the obtained solution from the Laplace space into the real domain. Moreover, we utilize two techniques which include analytic inversion and numerical inversion. For numerical inversion of the Laplace transforms, we have to evaluate five methods. Extensive results are presented. Furthermore, for numerical illustration of the abovementioned methods, we consider three problems to demonstrate our results.