Fractal diffusion retrospective problems

: In this article we study the retrospective inverse problem. The retrospective inverse problem consists of in the reconstruction of a priori unknown initial condition of the dynamic system from its known final condition. Existence and uniqueness of the solution is proved.


Introduction
In this article we study the retrospective inverse problem. The retrospective Inverse problem consists of in the reconstruction of a priori unknown initial condition of the dynamic system from its known final condition. The direct problem of heat conductivity is well-posed; the inverse problem is not well-posed. In mathematics the vast majority of inverse problems set not wellposed -small perturbations of the initial data (observations) can correspond to arbitrarily large perturbations of the solution. The French mathematician Jacques Hadamard in 1939 defined, the problem is called correct or well-posed problem if a solution exists, the solution is unique, the solution's behavior hardly changes when there's a slight change in the initial condition. If at least one of these three conditions is not fulfilled, problems are termed ill-posed or not well-posed. The most often in the case of ill-posed problems of the third condition are violated the condition of the stability of solutions. In this case, there is a paradoxical situation: the problem is mathematically generated, but the solution cannot be obtained by conventional methods. A classic example of ill-posed problem is retrospective problem for heat equation on the real axis. Mathematically retrospective problem leads to a Fredholm integral equation of the first kind:  (1) in which ˆ( ) fx -is the initial distribution of the temperature field, ˆ( , ) u t x -is the distribution of the fields in the moment of time t. As shown in [1], the solution of equation (1) expressed by the formula:

Problem Statement
In the inverse problem of heat conductivity the initial distribution of sources is unknown. The initial distribution of sources generates the specified temperature distribution in an infinite piecewise-homogeneous rod In. Mathematical statement of the problem consists in finding a solution separatist system (n+1) equations of parabolic type  (5) and the conjugation conditions , 1,..., -given real number, in which the condition of unlimited solvability of the problem considered fulfilled [2]. The solution to problem (3)-(6) is of the form:

Transformation operators
Method of transformation operators is used to solve the problem [2]. Necessary definitions from [2].
is a solution of the system of separate differential equations   is a solution of the system of separate differential equations

Analogues of the system the Hermite functions on piecewise-homogeneous real axis
Let define analogues of the system the Hermite functions on piecewise-homogeneous real axis: Proof. We have the equality: We change the integrals of places, we get:

Main result
The problem of determining the initial distribution of the temperature field () fx mathematically leads to the separate system of integral equations: Method of transformation operators applicable to solving separate system of integral equations (8 Proof . Let's apply the transformation operator 1  J to separate system of integral equations (8). As a result come to a model integral equation (1). Let's apply the operator J in both parts of the obtained equality (9); as a result, taking into account the continuity of the operator J , we find the unknown distribution of temperature: from the definition of the operator conversion of J the equality follows:

Power function with discontinuous coefficients and its application
We consider the Fourier transform of the Delta function We find as a consequence Let's define analog of the power function as follows . 0 We find equality from the definition of the transformation operators The last equality means that the power function with discontinuous coefficients is obtained by the action of the transformation operator to the power function.

Retrospective problem for the system of the diffusion equations
Let's return to the solution of the separate system of integral Let's substitute this decomposition in a formula (11) and let's integrate term by term. We come to the formula (12) in which Let's substitute decomposition of eigenfunction ) , (   x in the generalized power series in a formula (10) we will receive Thus, the solution of the retrospective problem (10) is obtained.

Retrospective problem for fractal systems of diffusion equations
Retrospective problem for fractal system of diffusion equations in the space of generalized functions S  leads to the separatist system of integral equations: -the Mittag-Leffler function [9]. We get the solution of the fractal retrospective problem repeating reasoning's from paragraph VII.
We get the expression, comparing the two views In the end we find the solution of the Cauchy problem for the hyperbolic equation If, as an example, take

The inverse dirichlet problem for a half-plane
Solution of the inverse Dirichlet problem for the right half-plane has the form:

Conclusion
In this article the formal solution of the retrospective problem is provided. The third aspect in determining the well-posed problem is not taken into account. Theorem of existence and uniqueness of solution are given. From the analysis of the formula (9)