An Iterative Finite Difference Method for Solving Nonlinear Gordon-Type Problems

Mohamed Ben-Romdhane and Helmi Temimi ()
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Mohamed Ben-Romdhane: College of Integrative Studies, Abdullah Al Salem University, Khaldiya, Kuwait
Helmi Temimi: College of Integrative Studies, Abdullah Al Salem University, Khaldiya, Kuwait

Mathematics, 2025, vol. 13, issue 13, 1-15

Abstract: This paper introduces an enhanced Iterative Finite Difference (IFD) method for efficiently solving strongly nonlinear, time-dependent problems. Extending the original IFD framework for nonlinear ordinary differential equations, we generalize the approach to address nonlinear partial differential equations with time dependence. An improved strategy is developed to achieve high-order accuracy in space and time. A finite difference discretization is applied at each iteration, yielding a flexible and robust iterative scheme suitable for complex nonlinear equations, including the Sine-Gordon, Klein–Gordon, and generalized Sinh-Gordon equations. Numerical experiments confirm the method’s rapid convergence, high accuracy, and low computational cost.

Keywords: Sine-Gordon equation; Klein–Gordon equation; generalized Sinh-Gordon equation; higher-order approximation; finite difference discretization; order of convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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