Application of natural and Aboodh transforms to solve Caputo Fractional Differential Equations

Abstract

Fractional calculus, which generalizes differentiation and integration to non-integer orders, is widely applied in modeling processes with memory and hereditary properties. However, solving Fractional Differential Equations (FDEs) remains analytically challenging. This study investigates the effectiveness of the Natural and Aboodh transforms for solving frac tional differential equations defined in the Caputo sense. Using the operational properties of the transforms, fractional derivatives are simplified, the equations are reformulated in the transform domain, and explicit solutions are obtained through inverse transforms. Both homogeneous and nonhomogeneous cases are addressed, supported by illustrative exam ples. The transforms yield exact analytical solutions with reduced computational effort, offering accuracy and efficiency in handling fractional models compared to conventional approaches. The Natural and Aboodh transforms are shown to be reliable and elegant tools for solving complex fractional differential equations, making them valuable alternatives for ap plied sciences and engineering problems.