Novel Insights into the Exact Solutions of the Modified (3+1) Dimensional Fractional KS Equation with Variable Coefficients

This study comprehensively explores a unified methodology for deriving exact solutions to the fractional modified (3+1) dimensional Kudryashov–Sinelshchikov (KS) equation featuring variable coefficients. The fractional KS equation, which incorporates fractional local M-derivatives, presents a significant challenge because of its inherent nonlinearity and the introduction of spatially varying coefficients. Of paramount importance are these variable coefficients, as they introduce spatial dependence and intricately capture spatial variations within the equation. This complexity in spatial variation poses a formidable challenge in obtaining exact solutions. Our analytical examination offers profound insights into the intricate interplay between diverse variable coefficients and fractional parameters. This comprehensive analysis greatly enhances our capacity to interpret solutions across various scenarios, enriching our understanding of the nuanced behavior exhibited by the fractional Kudryashov–Sinelshchikov equation. Our investigation encompasses a diverse range of solution forms, including fractional, polynomial, exponential, and others. The outcomes of this study hold profound implications for an extensive array of scientific domains, spanning mathematical physics, fluid dynamics, and nonlinear optics. Furthermore, this research employs advanced data visualization techniques, comprising 3D plots, contour plots, and stream plots, to facilitate a deep comprehension of intricate physical phenomena. These visual aids concurrently illustrate how analytical solutions are influenced by varying conditions.