Fundamental Solutions to Time-Fractional Advection Diffusion Equation in a Case of Two Space Variables

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 t and the Fourier transforms with respect to the space coordinates x and y. The Cauchy, source, and Dirichlet problems are investigated.The solutions are expressed in terms of integrals of Bessel functions combined with Mittag-Leffler functions. Numerical results are illustrated graphically.


Introduction
The classical advection diffusion equation where  is the diffusivity coefficient, k is the velocity vector, has several physical interpretations in terms of Brownian motion, diffusion or heat transport with external force or with additional velocity field, diffusion of charge in the electrical field on comb structure, transport processes in porous media, groundwater hydrology, and so forth [1][2][3][4][5][6][7].
In [24,25], the analytical solution to one-dimensional time-fractional advection diffusion equation was obtained in terms of integrals of the -function.
In this paper, we study the fundamental solutions to timefractional advection diffusion equation in a plane and a half-plane.The Laplace transform with respect to time and the Fourier transform with respect to the space coordinates are used.The Cauchy and the source problems in a plane and the Dirichlet problem for a half-plane are solved.The analytical solutions are expressed in terms of integrals of the Mittag-Leffler functions.Numerical results are illustrated graphically.In (3) we use the Caputo fractional derivative [40][41][42]: where Γ() is the gamma function.For its Laplace transform rule, the Caputo fractional derivative requires the knowledge of the initial values of the function () and its integer derivatives of order  = 1, 2, . . .,  − 1: where  is the transform variable.

The Fundamental Solution to the Cauchy Problem
Consider the time-fractional advection diffusion equation under initial condition In (7) we have introduced the constant multiplier  0 to obtain the nondimensional quantity  (see (23)) displayed in Figures.
The zero conditions at infinity are also imposed: Introducing the new sought function and taking into account that for the Dirac delta function, ()() = (0)(), the initial-value problem ( 6)-( 8) is reduced to the following ones: lim Next, we use the Laplace transform with respect to time  (designated by the asterisk) and the double exponential Fourier transform with respect to the space coordinates  and  (marked by the tilde).In the transform domain, we get Here,  is the Laplace transform variable and  and  are the Fourier transform variables.Inversion of the integral transforms gives where the formula [40-42] has been used with   () being the Mittag-Leffler function in one parameter : ,  > 0,  ∈ .
Solution ( 14) is not convenient for numerical calculations.To obtain the solution amenable to numerical treatment, we introduce the polar coordinates in the (, )-plane: Hence, Due to periodic properties of the integrand Changing variable  = sin  and taking into account the following integral [43]: where   () is the Bessel function of the order , we arrive at and, returning to the quantity (, , ) according to (9), we get The particular case of solution (22) corresponding to the time-fractional diffusion equation (V = 0) was considered in [44,45].
The results of numerical computations for  = 0 are presented in Figure 1 for  = 1 and in Figure 2 for  = 0.5.
The following nondimensional quantities: and the nondimensional coordinates (the similarity variables) have been introduced.

The Fundamental Solution to the Source Problem
Consider the time-fractional advection diffusion equation with the source term under zero initial condition,  = 0:  = 0 (26) and conditions (8) at infinity.The integral transform technique leads to Here,  , () is the generalized Mittag-Leffler function in two parameters  and : ,  > 0,  > 0,  ∈ , (29) and the formula [40-42] for the inverse Laplace transform has been used.The particular case of solution (28) corresponding to the time-fractional diffusion equation with V = 0 was considered in [45,47].Solutions ( 22) and ( 28) coincide for  = 1.
The results of numerical computations for  = 0 are presented in Figure 3 for  = 0.5 with

The Fundamental Solution to the Dirichlet Problem
In this case the time-fractional advection diffusion equation, is considered under zero initial condition and the Dirichlet boundary condition The zero conditions at infinity are imposed as follows: As above, the new sought function  is introduced (see ( 9)), and, for (10) in the half-plane  > 0, the Laplace transform with respect to time , the exponential Fourier transform with respect to the spatial coordinate , and the sin-Fourier transform with respect to the spatial coordinate  are used.In the transform domain, we get and, after inversion of the integral transforms, Introducing the polar coordinates in the (, )-plane gives Changing variables  = sin  and taking into account the following integral [43]: we obtain The particular case of solution (41) corresponding to the time-fractional diffusion equation (V = 0) was considered in [48].The results of numerical computations according to solution (41) for  = 0 are presented in Figure 4 for  = 1 and in Figure 5 for  = 0.5 with Other nondimensional quantities are the same as in ( 23) and (24).

Conclusions
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 0 <  < 1 has the logarithmic singularity at the origin: This result is similar to the case of the time-fractional diffusion equation when V = 0 (see [44,49]).Such a singularity disappears only for the classical advection diffusion equation ( = 1).Due to singularity of the solution at the origin, in the case of 0 <  < 1, drift caused by the quantity V is less noticeable than in the case of  = 1 (compare Figures 1 and  2).

2 Figure 1 :
Figure 1: Dependence of the fundamental solution to the Cauchy problem on distance (the classical advection diffusion equation,  = 1).

1 Figure 2 :
Figure 2: Dependence of the fundamental solution to the Cauchy problem on distance (the time-fractional advection diffusion equation,  = 0.5).

)Figure 3 :
Figure 3: Dependence of the fundamental solution to the source problem on distance (the time-fractional advection diffusion equation,  = 0.5).

Figure 4 :
Figure 4: Dependence of the fundamental solution to the Dirichlet problem on distance (the classical advection diffusion equation,  = 1).

3 Figure 5 :
Figure 5: Dependence of the fundamental solution to the Dirichlet problem on distance (the time-fractional advection diffusion equation,  = 0.5).