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

Santiago Quinga, Wilson Pavon (), Nury Ortiz, Héctor Calvopiña (), Gandhy Yépez and Milton Quinga
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Santiago Quinga: Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE, Sangolqui 171103, Ecuador
Wilson Pavon: Facultad de Ciencias de la Ingeniería e Industrias, Universidad UTE, Av. Mariscal Sucre, Quito 170129, Ecuador
Nury Ortiz: Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE, Sangolqui 171103, Ecuador
Héctor Calvopiña: Departamento de Ciencias de la Energía y Mecánica, Universidad de las Fuerzas Armadas ESPE, Sangolqui 171103, Ecuador
Gandhy Yépez: Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE, Sangolqui 171103, Ecuador
Milton Quinga: Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE, Sangolqui 171103, Ecuador

Mathematics, 2025, vol. 13, issue 7, 1-20

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.

Keywords: convection–diffusion; Crank–Nicolson; Newton–Jarrat; iterative method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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