Analytic Solutions of the Time-Fractional Date-Jimbo-Kashiwara-Miwa Equation via New Function Transformations (European Journal of Pure and Applied Mathematics, 2025)
In this paper, a new analytical framework for solving the fundamental nonlinear model, the time-fractional Date-Jimbo-Kashiwara-Miwa equation, is proposed. The time-fractional Date-Jimbo-Kashiwara-Miwa equation is made simpler by reducing it to an ordinary differential equation through the Power Index Method’s multiplication of variables x and t. Because it permits a change in variables, which can uncover a hidden pattern in the equation, multiplying x and t is significant.The original equation’s terms may be removed or their complexity decreased with the aid of this transformation. The benefits of various variable transformations vary depending on the particular issue, but this transformation has the advantage of simplifying the equation, which makes it simpler to solve, analyze, and produce precise and explicit solutions. Both rational and logarithm functions are present in the solutions that were obtained. Through 3D visualizations of the general solutions, our method offers a deeper comprehension of the dynamics of the equation. The behavior of the solution to the time-fractional Date-Jimbo-Kashiwara-Miwa equation is depicted in the paper’s 3D visualization. The solitons and nonlinear wave solutions are depicted in each plot. Our findings show the effectiveness of the Power Index Method in this situation and highlight its capacity to address challenging issues.
Keywords: Nonlinear Date Jimbo Kashiwara-Miwa equation, Analytic solution, Euler's second order ODE, Time-fractional derivative, Fractional calculus, Traveling wave transformation