A composite method for the solution of first-order singular and nonlinear initial value problems

Year 2025, Volume: 74 Issue: 4, 560 - 571, 24.12.2025

Nazile Buğurcan Dişibüyük

https://doi.org/10.31801/cfsuasmas.1551975

Abstract

By using the geometric and algebraic properties of Bernstein polynomials, a composite method for solving the initial value problems (IVP's) with first-order, singular and nonlinear ordinary differential equations (ODE's) has been developed. The Newton’s method is incorporated into the method to solve the resulting system of nonlinear equations. The algorithm of the problem solving is reduced to the calculation of the unknown Bernstein coefficients of the approximate solution. The effectiveness of the proposed method is verified by comparing the present numerical results by other existing ones. The proposed method reduces the computation cost and gives a better approximation to the exact solution even for small degrees of approximation. Another advantage of the present method is the ability to calculate the approximate solution at each point of the solution interval in addition to the grid points.

Keywords

First order singular ODE , first order nonlinear ODE , initial value problem , Bernstein polynomial , composite method

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