Journal of Applied Nonlinear Dynamics
Solution of New Generalized DiffusionWave Equation Defined in a Bounded Domain
Journal of Applied Nonlinear Dynamics 3(2) (2014) 159171  DOI:10.5890/JAND.2014.06.006
Yufeng Xu$^{1}$; Om P. Agrawal$^{2}$; Nimisha Pathak$^{3}$
1. School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China
2. Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA
3. Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA
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Abstract
This paper concerns with solutionof a Generalized DiffusionWave 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 diffusionwave 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 diffusionwave 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.
Acknowledgments
The first author would like to thank Department of Mechanical Engineering and Energy Processes at SIUC for hosting him and providing him research facilities.
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