Fractional-order diffusion-wave equation

The fractional-order diffusion-wave equation is an evolution equation of order alpha is an element of (0, 2] Which continues to the diffusion equation when a --> 1 and to the wave equation when alpha --> 2. We prove some properties of its solution and give some examples. We define a new fractional calculus (negative-direction fractional calculus) and study some of its properties. We study the existence, uniqueness, and properties of the solution of the negative-direction fractional diffusion-wave problem.