Existence, Stability, and Numerical Analysis of a Cubic Nonlinear Differential Equation via Picard Iteration

Abstract

This paper presents the investigation of the presence, stability, and convergence of the numerical solution for a third-order differential equation with cubic nonlinearity using the integral iterative method and numerical computation. The nonlinear initial-value problem is converted into an integral equation, and the iterative procedure in the spirit of Picard iteration is introduced to generate the solution. Analysis of stability is accomplished by obtaining the equilibrium points of the nonlinear differential system and then using the derivative test to determine their nature. Numerical simulations are done in MATLAB utilizing the built-in ODE45 Runge–Kutta algorithm as the reference solution. The comparison of the results of the integral iterative solution and the numerical reference solution indicates that the equilibrium point u = 0 is asymptotically stable, while the other equilibrium points are unstable. The findings illustrate that the iteration scheme can produce the exact solution only in the stable domain; however, the solution diverges numerically due to the influence of the cubic nonlinearity in the unstable domain. The error analysis by calculating the error with maximum norm and L2 norm proves the instability of the numerical method for long time intervals.