^{1, 2}

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The time-fractional heat conduction equation with the Caputo derivative of the order

The generalized Fourier law, the time-nonlocal dependence between the heat flux vector

Recall that the Riemann-Liouville fractional integral

A detailed explanation of derivation of time-fractional heat conduction equation (

Similarly, for

If the heat conduction equation is investigated in a bounded domain, the boundary conditions should be prescribed. The mathematical Robin boundary condition is a specification of a linear combination of the values of temperature and the values of its normal derivative at the boundary of the considered domain

The literature on mathematical aspects concerning correctness of initial-boundary-value problems for time-fractional diffusion equation and form and properties of its solutions is quite extensive (see, e.g., [

There are only a few papers [

The Laplace transform is defined as

The inverse Laplace transfrom is carried out according to the Fourier-Mellin formula:

The Laplace transform of the Riemann-Liouville fractional integral of the order

The Caputo derivative for its Laplace transform requires the knowledge of the initial values of the function

The Hankel transform is used to solve problem in cylindrical coordinates in the domain

The following formula is fulfilled:

In the case of boundary condition of the third kind with the prescribed boundary value of linear combination of a function and its normal derivative

Application of the sin-cos-Fourier transform to the second derivative of a function gives

Consider the axisymmetric time-fractional heat conduction equation in cylindrical coordinates

The zero conditions at infinity are also assumed

The solution to the initial-boundary-value problem (

The Laplace transform with respect to time

Invertion of the integral transforms leads to

The case

In the case of classical heat conduction (

For the wave equation (

Of particular interest is also the case

Consider the following axisymmetric time-fractional heat conduction equation:

The solution to the initial-boundary-value problem (

The Laplace transform with respect to time

In this case, the kernel of the sin-cos-Fourier transform with respect to the spatial coordinate

Inversion of the Laplace transform in (

Formulae (

We have derived the analytical solutions to time-fractional heat conduction equation in a half-space under mathematical and physical Robin boundary conditions. The integral transform technique has been used. It should be emphasized that in the case of physical Robin boundary condition, the order of integral transforms is important as the kernel of the sin-cos-Fourier transform depends on the Laplace transform variable. The limiting case