Problem of Determining the Density of Sources in a Multidimensional

Heat Equation with the Caputo Time Fractional Derivative

DURDIMUROD DURDIEV

Bukhara Branch of Romanovskii Institute of Mathematics,

Uzbekistan Academy of Sciences,

UZBEKISTAN

also with

obtained by the continuation method used in the theory of partial differential equations with integer

derivatives. The Green’s function of the problem is also constructed in terms of the Fox  function.

Involving the results of solving a direct problem and the overdetermination condition, a uniqueness theorem for

the definition of the spatial part of the multidimensional source function is proved.

Key-Words: - Fractional heat equation, Robin boundary condition, Green’s function, Fox  function,

the solution of which satisfies the initial condition

and the boundary condition

󰇛󰇜

and  is a given finite number.

problem of finding the solution to the initial -

boundary problem (1) - (3) will be called the direct

problem. A regular solution of this problem consists

 for each 

 function 󰇛󰇜 is

is continuously differentiable in  for 

known function, we are interested in finding the

At the present time, fractional PDEs have been

found as useful and applicable tools in applied

sciences. The processes of heat transfer and

diffusion phenomena in media with a fractal

have been studied quite deeply. There are the linear

inverse source and nonlinear inverse coefficient

problems for heat equations with different types of

the initial and initial-boundary problems (direct

problems) and over-determination conditions, [5]. In

this direction we note that the works, [6], [7], [8],

are concerned with inverse memory recovery

problems in parabolic integro-differential equations

of the second order with integral terms of

convolution type. In, [9], [10], there were proven

methods for solving various initial-boundary value

and Caputo using functions of the Mittag-Leffler

type are given in the well-known monograph, [3],

and article, [12].

In recent years, fractional differential equations

have attracted much attention and some analytical

methods for solutions of the initial and initial-

boundary problems for such equations have been

proposed, [13], [14], [15], [16], [17], [18], [19],

[20]. In works, [21], [22], the author obtained the

space with the Robin boundary condition was

considered in, [24]. Using the integral transform

methods, including the Laplace transform and the

Fourier transform it was obtained the exact solution

of the problem.

Among the inverse problems for the fractional

diffusion equation with Riemann-Liouville and

Caputo type derivatives, the very common are

inverse source problems with different over-

determination conditions, [25], [26], [27], [28], [29],

function). For the definition and properties of this

satisfies the condition



are true:





accordance with (10), we obtain:



 





3 Solution of the Direct Problem

Lemma 1 allows us to solve the problem for the

homogeneous heat equation:

󰇛󰇜

with the initial condition:

󰇛󰇜󰇛󰇜󰇛󰇜

satisfies conditions

󰇛󰇜at



 󰇛󰇜

 󰇛󰇜

(16)

According to Lemma 1, it is necessary to continue

obtain the Cauchy problem for the following

in the form of an analog of formula (8), where the



 

after transformations, we get

or, taking into account the formula (9)

above Green’s function 



Thus the fundamental solution  to the heat

equation takes its classical form

Then, as follow from the formula (20) the Green

function of the problem (1)-(3) in the case of 

(i.e. (1) is a multidimensional classical

inhomogeneous heat equation) has the following

5 Investigation of Inverse Problem

following statement about the uniqueness of the

solution of the inverse problem.

every  a finite Fourier transform

󰆻󰇛󰇜 

depending on  in a continuous way. In this case the

estimates (10) satisfies the conditions of

applicability the Fourier-Laplace transform:

After applying this transformation, the equations (1)

- (3) and (5) are reduced to the form

󰇛󰇜󰇛󰇜󰆻󰇛󰇜

At every fixed value of the parameters  and 

(22) is a boundary value problem for an ordinary

 for problem (22). With respect to the

variable  this function is continuous and bounded

generalized sense) the relations:

󰇛󰇜







In view of Green’s function the solution to the

inverse problem (22), (23) is written in the

following form

In this equation  is as a parameter. The

function 󰇛󰇜 being analytical in the domain

Therefore (25) can be rewritten in the following

form 

󰇜󰇛󰇜

every fixed  is a Laplace transform with respect to

the variable  of the function 󰆻󰇛󰇜 But the

function 󰆻󰇛󰇜 is uniquely determined by the

6 Conclusion

In this paper, the technique of the continuation

method of the solution from the infinite axis was

applied to derive an explicit solution to the third

initial-boundary problems for multidimensional

time-fractional heat equation with the Caputo

fractional derivative of the order  ().

This formula for solution contains the Green’s

function Robin boundary condition. The Green’s

theorem for the definition of the spatial part of the

multidimensional source function is proved.

It is of interest both theoretically and practically

to obtain exact formulas for solutions to the

multidimensional time- and space-fractional

diffusion-wave equations (in the case of the

half-space with the Dirichlet, Neumann, Robin

boundary conditions. So far, such problems are

open.

Acknowledgement:

with Bessel operator, Math. Model. Nat.