Work

Numerical Solution of the Nonlinear Convection–Diffusion Equation Using the Fifth Order Iterative Method by Newton–Jarratt

Year: 2025

Type: article

Abstract: This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solutions of the nonlinear convection–diffusion equation. A MATLAB script function was developed to implement the approach in two stages: first, discretizing the equation using the Crank–Nicolson Method, and second, solving the resulting nonlinear systems using Newton’s iterative method enhanced by a three-step Jarratt variant. A comprehensive analysis of the results highlights the method’s convergence and accuracy, comparing the numerical solution with the exact solution derived from linear parabolic partial differential transformations. This innovative fifth-order method provides an efficient numerical solution to the nonlinear convection–diffusion equation, addressing the problem through a systematic methodology that combines discretization and nonlinear equation solving. The study underscores the importance of advanced numerical techniques in tackling complex problems in physics and mathematics.

Source: Mathematics

DOI: 10.3390/math13071164

Authors: Santiago Quinga, Wilson Pavon, Nury Ortiz, Héctor Calvopiña, Gandhy Yépez, Milton Quinga

Institutions: Universidad de las Fuerzas Armadas ESPE, Universidad UTE

Language: English

Topic: Iterative Methods for Nonlinear Equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: Universidad de las Fuerzas Armadas ESPE