Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Alternative Variational Iteration Method for Solving Time-Fractal-Fractional Partial Differential Equations
Isah, I. O., Senu, N., and Ahmadian, A.
Corresponding Email: norazak@upm.edu.my, ahmadian.hosseini@unirc.it
Received date: 29 July 2024
Accepted date: 3 July 2025
Abstract:
The alternative variational iteration method (AVIM) is applied to solve time fractal-fractional partial differential equations. We investigate the one-dimensional time fractional diffusion-wave equation and a generalized time-fractional partial differential equation (PDE) with variable coefficients, modelling wave propagation in mathematical physics under the new generalized Caputo-type fractal-fractional derivative. The validity and efficiency of the method, as well as the effect of the fractal dimension, are demonstrated through numerical test examples. Graphical representations of numerical results for different parameter values are also provided. The acquired numerical results employing the proposed method are compared equally with those gotten using the traditional variational iteration method and AVIM with fractional derivatives only, as found in the literature. The results indicate that the proposed approach has the advantage of being able to solve the problem with fewer iterations due to the extra parameter and fractal dimension utilized in the fractal-fractional derivative.
Keywords: fractal-fractional operator; new generalized Caputo derivative; alternative variational iterative method; Fornberg–Whitham equation with variable coefficients; diffusion equation
