Analytical solution of the generalized space-time fractional ultra-hyperbolic differential equation

Abstract

A generalization of the ultra-hyperbolic time-fractional diffusion-wave equation introduced by the author in Dorrego [The Mittag–Leffler function and its application to the ultra-hyperbolic time-fractional diffusion-wave equation. Integral Transforms Spec. Funct. 2016;27(5):392–404] is presented and studied. The results obtained here extend the previous to the whole space ${\mathbb{R}}^n$. The problem is solved by applying the integral transforms method and the analytical solution is given by the Fox H-function. Some interesting particular cases are analysed.

Keywords:

• Fractional power
• ultra-hyperbolic operator
• integrals transforms
• Mittag–Leffler type function
• Fox's H-function
• Hilfer fractional derivative

AMS Classifications:

• 26A33
• 33E12
• 33E20
• 35R11

Acknowledgments

The author wishes to thank Dr R. Cerutti for the suggestions made on the possibility of defining the Mittag–Leffler type function (3.5(3.5) $E_{\alpha ,\beta }(P\pm i0{)}^{\lambda }=\sum _{n=0}^{\mathrm{\infty }}\frac{(P\pm i0{)}^{\lambda n}}{\mathrm{\Gamma }(\alpha n+\beta )},$(3.5) ).

Disclosure statement

No potential conflict of interest was reported by the author(s).