Approximate solution of fractional differential equations using transformation and perturbation method

Fractional calculus extends basic calculus, specifically differentiation and integration to non-integer orders which include real and complex numbers, enabling more accurate modelling of actual systems. The fractional differential equation (FDEs) is one kind of fractional calculus that has been widely implemented in a variety of fields, including biology, engineering, and physics. FDE extends ordinary and partial differential equations of non-integer order. This makes them more difficult to solve, since there are very few well-developed analytical methods available to solve FDE that have a complicated solution. This study explores the numerical methods, the Homotopy Perturbation Transform Method (HPTM) in solving FDE, specifically time-fractional diffusion equations, using the Riemann-Liouville definition of fractional derivatives. HPTM combines the Laplace transform as well as Homotopy Perturbation Method (HPM) that simplifies the equation and approaches the solution efficiently. Thus, the findings indicate that the suggested method is highly efficient and is able to simplify the solution of FDE. The solutions were also analyzed using Maple software and have been plotted for various fractional derivative orders, α and several numerical illustrations are provided.