Application of the Shehu Variational Iteration Method to Fractional-Order Newell–Whitehead–Segel Equations

Abstract

In this paper, the Shehu Variational Iteration Method (SVIM) employed to obtain approximate analytical solutions for time-fractional Newell–Whitehead–Segel (NWS) equations. The fractional derivative is considered in the Caputo sense, with fractional order 0 < α ≤ 1. The proposed approach combines the Shehu transform with the variational iteration method and provides a convenient iterative procedure for constructing the solution in the form of a rapidly convergent series. The applicability and effectiveness of the method demonstrated through several linear and nonlinear time-fractional NWS examples. For the linear problem, the resulting series solution is expressed in terms of the Mittag–Leffler function and reduces to the classical exact solution when α = 1. For the nonlinear problems, SVIM produces approximate analytical series solutions whose classical limits agree with the corresponding known solutions. The obtained results are consistent with solutions reported by other established methods, particularly the Adomian decomposition method. Therefore, the Shehu Variational Iteration Method provides an effective and straightforward semi-analytical technique for solving linear and nonlinear time-fractional NWS equations.