New Numerical and Analytical Solutions for Nonlinear Evolution Equations Using Updated Mathematical Methods

Abstract

This study explores adapted mathematical methods to solve the couple-breaking soliton (BS) equations in two-dimensional spatial domains. Using these methods, we obtained analytical soliton solutions for the equations involving free parameters such as the wave number, phase component, nonlinear coefficient, and dispersion coefficient. The solutions are expressed as hyperbolic, rational, and trigonometric functions. We also examined the impact of wave phenomenon on two-dimensional diagrams and used composite two-dimensional and three-dimensional graphs to represent the solutions. We used the finite difference method to transform the proposed system into a numerical system to obtain numerical simulations for the Black–Scholes equations. Additionally, we discuss the stability and error analysis of numerical schemes. We compare the validity and accuracy of the numerical results with the exact solutions through analytical and graphical comparisons. The methodologies presented in this research can be applied to various forms of nonlinear evolutionary systems because they are appropriate and acceptable.