Abstract
In this work we obtain the first and second fundamental solutions (FS) of the multidimensional time-fractional equation with Laplace or Dirac operators, where the two time-fractional derivatives of orders $$\alpha \in ]0,1]$$ and $$\beta \in ]1,2]$$ are in the Caputo sense. We obtain representations of the FS in terms of Hankel transform, double Mellin-Barnes integrals, and H-functions of two variables. As an application, the FS are used to solve Cauchy problems of Laplace and Dirac type.
Highlights
In the last years, the interest in the study of fractional differential equations has increased considerably due essentially to the wide range of applications
The telegraph equation is used as an alternative to the diffusion equation, since it has the potential to describe both diffusive and wave-like phenomena, due to the simultaneous presence of first and second order time derivatives
In the case of the transport of energetic charged particle in turbulent magnetic fields such as low-energy cosmic rays in the solar wind, the diffusion equation can not be used to describe the transport for early times because it leads to a non-zero probability density everywhere, which would correspond to an infinite propagation speed
Summary
The interest in the study of fractional differential equations has increased considerably due essentially to the wide range of applications These type of equations are used mostly to model processes in the fields of engineering, viscoelastic materials, hydrology, system control, just to mention some. The aim of this paper is to obtain explicit integral representations for the first and second FS of the timefractional telegraph equation with Laplace or Dirac operators. In both cases, we deduce several representations of the FS in the form of Hankel transform, double Mellin-Barnes integrals, and Hfunctions of two variables.
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