A numerical solution to nonlinear ordinary differential equations based on Bell polynomials

Abstract

This article examines the solutions of high-order nonlinear ordinary differential equations with cubic terms under initial conditions using Bell polynomials, their derivatives, and collocation points. The nonlinear differential equation and the corresponding conditions are transformed into matrix form by means of Bell polynomials and reduced to an algebraic system. From the solution of this system, the unknown Bell coefficients are determined. By substituting these coefficients, the approximate solution of the problem is expressed in terms of Bell polynomials. To illustrate the method, some numerical examples are presented. For these examples, the Bell solutions and the absolute error functions are calculated, and the results are shown in tables and figures for comparison with the exact solutions.