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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.
Quinga et al. (Tue,) studied this question.