Solution of New Generalized Diffusion-Wave Equation Defined in a Bounded Domain - Xu, Yufeng; Agrawal, Om P.; Pathak, Nimisha (Journal of Applied Nonlinear Dynamics, 2026)
This paper concerns with solutionof a Generalized Diffusion-Wave Equation (GDWE) defined in a bounded space domain. In the model proposed, the GDWE is defined using operator BαP introduced recently. In contrast to fractional derivatives which employ fractional power kernels in their definitions, operator BαP allows the kernels to be arbitrary. Therefore, it offers more generality to a diffusion-wave equation than a fractional derivativedoes. Intheschemeproposed, the method of separation of variables is used to separate the space and time domains. The space equation in conjunction with boundary conditions are used to identify the eigenfunctions. The time dependent equation is solved analytically for a specific kernel. For a general kernel, a closed form solution for time equation may not be available. For this reason, we present a numerical scheme to solve this equation. The analytical solution for an exponential kernel is used to verify the numerical scheme. Two examples are presented to show applications of these models. It is hoped that operator BαP will allow us to model diffusion-wave behaviors of a system for which fractional derivatives may not be suitable, and the analytical and numerical schemes will allow us to solve these equations.