The fundamental solutions to time-fractional advection diffusion equation in a plane and a half-plane are obtained using the Laplace integral transform with respect to time

The classical advection diffusion equation

In the case of one spatial coordinate

Investigation of different physical phenomena in media with complex internal structure has led to considering differential equations with derivatives of fractional order. The space-fractional [

In [

In this paper, we study the fundamental solutions to time-fractional advection diffusion equation

In (

Consider the time-fractional advection diffusion equation

The zero conditions at infinity are also imposed:

Introducing the new sought function

Inversion of the integral transforms gives

Solution (

Due to periodic properties of the integrand

Changing variable

The particular case of solution (

The results of numerical computations for

Dependence of the fundamental solution to the Cauchy problem on distance (the classical advection diffusion equation,

Dependence of the fundamental solution to the Cauchy problem on distance (the time-fractional advection diffusion equation,

The following nondimensional quantities:

To calculate the Mittag-Leffler function

Consider the time-fractional advection diffusion equation with the source term

The integral transform technique leads to

The particular case of solution (

The results of numerical computations for

Dependence of the fundamental solution to the source problem on distance (the time-fractional advection diffusion equation,

In this case the time-fractional advection diffusion equation,

As above, the new sought function

Introducing the polar coordinates in the

Changing variables

The particular case of solution (

The results of numerical computations according to solution (

Dependence of the fundamental solution to the Dirichlet problem on distance (the classical advection diffusion equation,

Dependence of the fundamental solution to the Dirichlet problem on distance (the time-fractional advection diffusion equation,

We have considered the time-fractional advection diffusion equation in a plane and in a half-plane. The fundamental solutions to the Cauchy problem and to the source problem in a plane have been obtained as well as to the Dirichlet problem in a half-plane. It should be emphasized that the fundamental solution to the Cauchy problem in the case

The author declares that there is no conflict of interests regarding the publication of this paper.