EXACT AND NUMERICAL SOLUTIONS OF THE SPACE-TIME FRACTIONAL FW EQUATION VIA IMETF AND MVI METHODS

Abstract

This paper investigates the space-time fractional nonlinear Fornberg–Whitham (FW) equation, a model that captures complex wave phenomena with memory effects and non-local interactions through fractional derivatives. By applying the Improved Modified Extended Tanh-Function (imETF) method, we derive several new exact solutions, including soliton and lump waves, which reveal how fractional-order parameters influence wave shape, dispersion, and stability. We also use the Modified Variational Iteration (MVI) method to generate approximate numerical solutions, comparing them to the analytical results and analyzing absolute errors. Our study demonstrates that fractional characteristics introduce fractal scaling and multiscale structures into wave dynamics, leading to broader, oscillatory profiles compared to classical solutions. These findings underscore the importance of fractional calculus in modeling real-world systems with complex geometries, such as turbulent flows, plasma waves, and optical media. The methodologies and results presented here offer a robust framework for analyzing nonlinear fractional partial differential equations and have potential applications in fluid dynamics, optics, plasma physics, and other fields where understanding wave propagation is critical.