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Fractional-order diffusion-wave equation

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The fractional-order diffusion-wave equation is an evolution equation of order α ε (0, 2] which continues to the diffusion equation when α → 1 and to the wave equation when α → 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.

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El-Sayed, A.M.A. Fractional-order diffusion-wave equation. Int J Theor Phys 35, 311–322 (1996).

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