Numerical Simulation and Convergence of Variational Iteration Method on Kuramoto-Sivashinsky Equation

Partial differential equations (PDEs) are a useful tool for modelling a variety of mathematical problems, including those in the field of mathematical physics. The Kuramoto–Sivashinsky (K-S) equation is one type of partial differential equation (PDE). A numerical approach is necessary to achieve a solution since not all partial differential equations (PDEs) can be solved analytically in reality. Similar to the K-S equation, this equation cannot be solved without the use of a numerical technique. Variational Iteration Method (VIM) is one of the techniques used to solve the K-S problem. Three core ideas form the basis of the Variational Iteration Method (VIM): the generalized Lagrange multiplier, finite variation and correction function. The aforementioned three basic concepts can be employed in the formulation of the iteration formula. The objective of this research is to implement the VIM numerical scheme on the K-S equation. A wave-shaped graph is obtained based on the K-S equation, which has one valley and two hills, starting with the solution at when As increases, the solution value decreases and reaches a minimum at when . Subsequently, the curve ascends once more, crossing the -axis and reaching a maximum valueEven though just a small number of iterations are carried out, the Variational Iteration approach is successful in obtaining correct answers, according to the convergence analysis of the method on the K-S problem. It can be concluded that VIM is an appropriate and efficacious instrument for the resolution of equations of the K-S variety. It may be employed as a method of solution exploration and as a verification tool for the exact solution.