Comparison of Caputo fractional and integer order derivatives forthird order partial differential equation by finite differencemethod

Fractional partial differential equations (FPDEs) have been developed in various disciplines including physics, engineering, biology, finance, viscosity, and fluid mechanics to describe anomalous diffusion phenomena. Consequently, numerical techniques are indispensable for solving these equations. This thesis is structured into two main parts: the first investigates the approximation of the Caputo fractional derivative, while the second examines differential equations with integer order derivatives. This study explores a third-order partial differential equation (PDE) using both CFDs and IODs for the first time, extending beyond the typical scope of third-order time-varying systems models. At β = 1, this fractional PDE is transformed into an abstract form of an ordinary differential equation. A first-order finite difference scheme is developed for the transformed problem, and the stability of the abstract model problem is demonstrated using Cardano's relation method in a Hilbert space. The explicit finite difference method (EFDM) is utilized to approximate the solution of the third-order PDE, employing both CFD and IOD of order β ∈ (0, 1]. The obtained approximate solutions are compared with analytical solutions using the proposed method. Additionally, the DLT method is employed to derive the analytical solution. Finally, simulations for the third-order PDE based on both CFDs and IODs are presented for analytical and approximate solutions.