Variational Iteration Method with an Auxiliary Parameter for Some Kinds of Partial Differential Equations

In this paper, we present the Modified Variational Iteration Algorithm-I (MVIA-I) for obtaining numerical solutions of the modified equal width (MEW) wave equation and the regularized long wave (RLW) equation. This recently developed technique introduces an auxiliary parameter that significantly accelerates the convergence of the series solutions. The method provides both approximate and exact solutions with easily computable terms for linear and nonlinear PDEs, without requiring Adomian polynomials, small perturbations, discretization, or linearization. To evaluate its accuracy, reliability, and efficiency, the results obtained by the proposed algorithm are compared with those of the standard Variational Iteration Method (VIM) and the Reduced Differential Transform Method (RDTM). The comparison clearly shows that MVIA-I is computationally efficient, highly accurate, and superior to existing methods. Two numerical test problems are included to assess the performance of the modified algorithm, and the accuracy is measured by calculating the absolute errors for different parameter values. The results demonstrate that MVIA-I converges rapidly and provides solutions of excellent precision, confirming its effectiveness as a robust tool for solving nonlinear PDEs.