On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives

• Published in: Axioms 2023-09, Vol.12 (9), p.901
• Main Authors: Alsidrani, Fahad, Kılıçman, Adem, Senu, Norazak
• Format: Article
• Language: English
• Subjects: Calculus; Control engineering; Decomposition; Differential equations; Finite volume method; fractional derivatives; Homotopy theory; integral transforms; Investigations; Iterative methods; Lagrange multiplier; Laplace transformation; Laplace transforms; Mathematicians; Methods; nonlinear equation; Numerical analysis; Numerical methods; Operators (mathematics); partial differential equation; Partial differential equations
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms12090901

This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable ν. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs.