Numerical Solution of the Gardner Equation Using the Method of Lines and an Improved Fifth-Order Runge-Kutta Scheme
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Gardner Equation, Method of Lines, Improved Runge-Kutta Method, Solitary Pulse, Kink-LikeWave, Numerical Simulation
Abstract
The Gardner equation is a nonlinear partial differential equation that arises in the modeling of nonlinear dispersive wave phenomena. Analytical solutions of this equation are limited to specific cases, which motivates the development of reliable numerical approaches. This study presents a numerical scheme based on the Method of Lines combined with an Improved Runge-Kutta method of order five to solve the Gardner equation. Spatial discretization is performed using second-order central finite difference schemes, which transform the governing equation into a system of ordinary differential equations. The resulting system is integrated in time using the fifth-order Improved Runge-Kutta method to achieve high accuracy and computational efficiency. Simulations targeting solitary pulse and kink-like wave solutions reveal that the MOL-IRK5 scheme consistently outperforms the classical MOL-RK4 method. The numerical results show strong agreement with the exact solutions at all observed time levels. Wave profiles are preserved during propagation, and no spurious oscillations are observed. Global error measures, including the maximum error, mean absolute error, and root mean square error, remain small throughout the simulation interval, indicating stable numerical performance. The results demonstrate that the proposed MOL-IRK5 scheme provides accurate and efficient approximations for different types of Gardner equation solutions. The numerical approach presented in this study offers a reliable framework for solving nonlinear dispersive equations and can be extended to more complex mathematical models in applied science and engineering.
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