Solving Nonlinear Partial Differential Equations of Special Kinds of 3rd Order Using Balance Method and Its Models

Most nonlinear partial differential equations have many applications in the physical world. Finding solutions to nonlinear partial differential equations is not easily solvable and hence different modified techniques are applied to get solutions to such nonlinear partial differential equations. Among them, we considered the modified Korteweg–de Vries third order using the balance method and constructing its models using certain parameters. The method is successfully implemented in solving the stated equations. We obtained kind one and two soliton solutions and their graphical models are shown using mathematical software-12. The obtained results lead to shallow wave models. A few illustrative examples were presented to demonstrate the applicability of the models. Furthermore, physical and geometrical interpretations are considered for different parameters to investigate the nature of soliton solutions to their models. Finally, the proposed method is a standard, effective, and easily computable method for solving the modified Korteweg–de Vries equations and determining its perspective models.