The Modified Homogeneous Balance Method for solving fractional Cahn–Allen and equal width equations

This paper presents exact solutions to the Fractional Cahn–Allen (FC–A) and the Fractional Equal Width (FEW) equations using the Modified Homogeneous Balance Method (MHBM). The MHBM transforms the FC–A and FEW equations into fractional ordinary differential equations via a wave transformation. By balancing the highest-order derivative with the leading nonlinear term, the method determines the appropriate polynomial degree. A fractional Riccati equation with a quadratic nonlinearity facilitates the construction of exact solutions without resorting to infinite series expansions. Compared to existing methods, the MHBM offers a finite and well-defined solution structure, avoiding the rigidity of the $tan\left(\frac{\xi \left(\eta \right)}{2}\right)$-expansion method and the complexities associated with the Riemann–Hilbert and algebro–geometric methods. It also provides clearer criteria for convergence analysis than the ${\varphi }^6$-expansion method. The MHBM accommodates various solution types, including trigonometric, hyperbolic, rational, and elliptic functions, with fewer parameter restrictions and potential for multi-wave structures. Numerical simulations shows that as the spatial variable $x$ increases, the solitons tend to stabilize, and the plots for different values of the fractional order $\alpha$ closely aligned, indicating minor sensitivity to $\alpha$. Furthermore, the FEW soliton exhibits a dense tiling structure along the time axis in its surface plot, while the FC–A soliton demonstrates a smooth kink-like transition along $\xi$, characteristic of solutions connecting two stable equilibrium states. These findings underscore the robustness and versatility of the MHBM in analyzing fractional nonlinear evolution equations.