Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Extended Laplace Power Series Method for Solving Non-linear Caputo Fractional Volterra Integro-differential Equations

Version 1 : Received: 23 May 2023 / Approved: 25 May 2023 / Online: 25 May 2023 (08:06:04 CEST)

A peer-reviewed article of this Preprint also exists.

Alomari, A.-K.; Alaroud, M.; Tahat, N.; Almalki, A. Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations. Symmetry 2023, 15, 1296. Alomari, A.-K.; Alaroud, M.; Tahat, N.; Almalki, A. Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations. Symmetry 2023, 15, 1296.

## Abstract

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm's accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of problem. Under Caputo meaning the obtained results are illustrated numerically, and graphically. Geometrically, the behavior of the solution declares that the changing of the fractional derivative parameter values' in their domain alters the style of the attained solution in a symmetrical meaning and fully coinciding to the ordinary derivative value'. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for Integral, and integro-differental equations of fractional order.

## Keywords

Fractional Volterra integro-differential equation; Laplace fractional power series; Caputo fractional derivative, Laplace transform.

## Subject

Computer Science and Mathematics, Applied Mathematics