SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS: AN ADAPTIVE VARIATIONAL ITERATION APPROACH

Abstract

This work presents the study of Adaptive variational iteration method (AVIM), an enhanced semi-analytical technique for solving nonlinear partial differential equations (PDEs). The AVIM is a modification of the classical variational iteration method (VIM), in which an adaptive Lagrange multiplier is dynamically updated at each iteration based on the residual of the preceding approximation. This adaptive mechanism ensures improved convergence behavior and solution accuracy. The method was successfully applied to heat and Burgers’s equation. In each case, iterations were computed and for the first three equations, the results of the iteration solutions were compared against He’s Variational Iteration method and known exact solutions. Numerical comparisons were presented in tabular form, and corresponding graphs were plotted to visualize the convergence and accuracy of the method. Results indicate that AVIM yields highly accurate approximations in fewer iterations than the standard VIM. Furthermore, convergence graphs of successive iterations clearly demonstrate the superior performance of AVIM in capturing the solution behavior of nonlinear equations. The findings validate AVIM as a good and efficient tool for solving nonlinear PDEs.