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On a special function arising in the time fractional diffusion-wave equation. (English) Zbl 0921.33010
Rusev, P. (ed.) et al., Transform methods and special functions. Proceedings of the 1st international workshop, Bankya, Bulgaria, August 12–17, 1994. Sofia: SCT Publishing, 171-183 (1995).
Summary: The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order $$\alpha$$ with $$0< \alpha<2$$. Using the method of the Laplace transform, it is shown that the fundamental solution of the basic Cauchy problem can be expressed in terms of an auxiliary function $$M(z;\beta)$$, where $$z$$ is the similarity variable and $$\beta= \alpha/2$$. This function, which reduces to the Gaussian function in the case of classical diffusion $$(\beta=1/2)$$, is shown to be a Wright-type function, and therefore it turns out to be an entire function of the complex variable $$z$$ for any value of the parameter $$\beta$$ in its range $$0< \beta<1$$. For if we provide series and integral representations and the differential equation of fractional order to be satisfied in the complex plane; the above properties allow us to consider $$M(z;\beta)$$ as a sort of generalized hyper-Airy function. The problem of its asymptotic representation for large $$| z|$$ is preliminarily discussed using the saddle point method.
For the entire collection see [Zbl 0914.00064].

##### MSC:
 33E20 Other functions defined by series and integrals 26A33 Fractional derivatives and integrals 35L05 Wave equation