Collocation Method With Shifted Legendre-Based Approach for Solution of Nonlinear Differential Equations - Ibrahim, Salisu; Izzat, Sharmeen; Abdulla, Chenar (Communications in Mathematics and Applications, 2026)
The technique of solving nonlinear ordinary differential equation (NODE) that we have used in the paper is the Shifted Legendre technique. This is achieved by rewriting the problem in a more stable way with Shifted Legendre polynomials which has been shown to be effective and precise in numerical solutions. Using this method allows solving complicated nonlinear equations and it is not difficult to obtain approximate solutions in a short time. The paper identifies the benefits of the given approach, that is, reduced computing requirements and enhanced precision and is supported by practical exercises. The up-to-date analysis of the solution of nonlinear ordinary differential equations (NODEs) by the Shifted Legendre technique provides a simple and easily applicable answer. Numerical findings of the applied method demonstrate the exact correspondence with the exact solution with the least errors that is better in comparison with the current methods. A number of examples are resolved to demonstrate the success of the suggested method with minimum errors that outperform the existing methods.
Keywords: Shifted Legendre Polynomial; Collocation method; L2-Norm; Nonlinear Ordinary Differential Equations; Numerical Approximation