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

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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)

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.

Suggested Citation

  • Santiago Quinga & Wilson Pavon & Nury Ortiz & Héctor Calvopiña & Gandhy Yépez & Milton Quinga, 2025. "Numerical Solution of the Nonlinear Convection–Diffusion Equation Using the Fifth Order Iterative Method by Newton–Jarratt," Mathematics, MDPI, vol. 13(7), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1164-:d:1625773
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    References listed on IDEAS

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    1. Cordero, Alicia & Rojas-Hiciano, Renso V. & Torregrosa, Juan R. & Vassileva, Maria P., 2025. "Maximally efficient damped composed Newton-type methods to solve nonlinear systems of equations," Applied Mathematics and Computation, Elsevier, vol. 492(C).
    2. Moin-ud-Din Junjua & Saima Akram & Nusrat Yasmin & Fiza Zafar, 2015. "A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications," Journal of Applied Mathematics, Hindawi, vol. 2015, pages 1-14, March.
    3. Moin-ud-Din Junjua & Saima Akram & Nusrat Yasmin & Fiza Zafar, 2015. "A New Jarratt‐Type Fourth‐Order Method for Solving System of Nonlinear Equations and Applications," Journal of Applied Mathematics, John Wiley & Sons, vol. 2015(1).
    4. José J. Padilla & Francisco I. Chicharro & Alicia Cordero & Alejandro M. Hernández-Díaz & Juan R. Torregrosa, 2024. "A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam," Mathematics, MDPI, vol. 12(3), pages 1-16, February.
    5. Chun-Ku Kuo & Sen-Yung Lee, 2015. "A New Exact Solution of Burgers’ Equation with Linearized Solution," Mathematical Problems in Engineering, Hindawi, vol. 2015, pages 1-7, August.
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