A detailed analysis of the improved modified Korteweg-de Vries equation via the Jacobi elliptic function expansion method and the application of truncated M-fractional derivatives

Abstract

Using the Jacobi elliptic function expansion method, which is improved by the novel use of truncated M-fractional derivatives, we thoroughly analyze the improved modified Korteweg-de Vries problem in this paper. Not only does this method provide new insight into the improved modified Korteweg-de Vries Equation, but it also represents a giant leap forward for nonlinear differential equations. Periodic, solitary, and trigonometric waves are among the answers found by us; all of them have important consequences for mathematical physics and engineering.The MATLAB-generated visual representations of our results are crucial because they provide aconcrete shape to the once intangible mathematical ideas. In addition to showing the solutions, these images highlight the waves' dynamic behaviors and stability under various conditions, providing a better understanding of the physical processes that the improved modified Korteweg-de Vries equation attempts to explain. The graphic representations in this study demonstrate how the equation can be used to simulate complicated systems in many scientific disciplines. The relevance of our findings goes beyond only theoretical studies. Our study lays the groundwork for more accurate predictive models in fields including quantum physics, optical fibers, and fluid dynamics by giving exact and concrete solutions to the improved modified Korteweg-de Vries equation. The combination of the Jacobi Elliptic Function Expansion Method with truncated M-fractional derivatives provides a robust framework for solving comparable complicated differential equations, which in turn opens up new research and development opportunities. Overall, this research both deepens our understanding of the Improved Modified Korteweg-de Vries equation and provides more evidence that theoretical mathematics can be useful in solving practical problems. Mathematical modeling and computational visualization have the potential to make substantial contributions to engineering and the sciences, and our results support a multidisciplinary approach to study.