Application of Adaptive Variational Iteration Method to Telegraph and Schrödinger Equations

Abstract

In this study, an Adaptive Variational Iteration Method (AVIM) is applied to the Telegraph and Schrödinger equations. The method extends He’s Variational Iteration Method by introducing a residual-dependent adaptive Lagrange multiplier into the correction functional to improve convergence and numerical stability. Iterative approximations were constructed for the considered equations and compared with the exact solutions and the classical Variational Iteration Method (VIM). The results obtained showed that the proposed method converges rapidly with smaller absolute errors and improved stability compared to the classical VIM. The adaptive correction mechanism effectively controls the iteration process by dynamically adjusting the multiplier according to the residual magnitude. Numerical results further demonstrated that AVIM provides accurate approximations for both hyperbolic and oscillatory partial differential equations with reduced computational effort. The study establishes that AVIM is an efficient and reliable semi-analytical technique for solving partial differential equations arising in wave propagation and quantum mechanics.