A Review on Some Numerical Methods on Solving the Fractional Initial Value Problems of Fractional Differential Equations

Abstract

Over the past few years, there has been an in creasing interest in fractional differential equations (FDEs), owing to the enhanced effectiveness of fractional calculus compared to traditional calculus. Fractional calculus, with its non-integer order derivatives, provides more accurate models for complex phenomena in science and engineering, capturing memory effects and hereditary properties in processes like viscoelasticity, anomalous diffusion, and signal processing. This has made FDEs a valuable tool for advancing the modeling of intricate systems. This paper aims to examine various numerical methods for solving fractional initial value problems associated with FDEs. A comparative analysis of the Fractional Explicit Adams Method of Order 3, the Fractional Adams Method of Explicit Order 2, Implicit Order 2, the Fourth-order 2 point Fractional Block Backward Differentiation Formula, the Fractional Explicit Method and the PECE method of Adams Bashforth-Moulton type are presented with respect to their performance against the exact solution. The comparison focuses on key metrics, including convergence and accuracy of the methods. Three numerical problems were successfully solved to evaluate the methods. The results showed that all of the five methods are reliable when solving FDEs.