Coefficient Determination in Parabolic Equations Solved as a Moment Problem Two-Dimensional in a Rectangular Domain

The problem is to considerer a parabolic equation depending on a coefficient ( ) a t , and find the solution of the equation and the coefficient. The objective is to solve the problem as an application of the inverse moment problem. An approximate solution and limits will be found for the error of the estimated solution using the techniques of inverse problem moments. In addition, the method is illustrated with several examples.


Introduction
We want to find ( ) a t and ( ) , w x t such that ( )( ) ( ) where ( ) x ϕ , ( ) , r x t and ( ) E t are known functions and α is an arbitrary real number other than zero.
We also assume that the underlying space is ( ) This problem is studied in [1].Citing the abstract of this work: "this paper investigates the inverse problem of simultaneously determining the time-dependent thermal diffusivity and the temperature distribution in a parabolic equation in the case of nonlocal boundary conditions containing a real parameter and integral overdetermination conditions, and under some consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the classical solution are shown by using the generalized Fourier method".
In general the methods applied to solve the problem are varied.Other works that solve the parabolic equation but under different conditions are [2] [3] [4].
There is a great variety of inverse problems in which a parabolic equation must be solved and additionally we must determine an unknown parameter, under various conditions [5] [6] [7] and [8] [9] [10] [11], to name some examples.
I have considered one of these problems and my objective in this work is to show that we can solve this problem using the techniques of inverse moments problem two-dimensional as an alternative and different technique.We focus the study on the numerical approximation.
The problem has already been solved as a moment problem two-dimensional in [12] for a domain But if you want to apply this work for 0 t T < < it would be necessary to know the value of the function ( ) , w x t in t T = and this data is not considered in the boundary conditions.For this reason we must make a change in the way of solving the problem, and this implies significant differences with the work done in [12].
As was done in [12], first we find an exact expression for ( ) ( ) We resolve a first step in numerical form ( ) ( ) ψ is written in terms of known expressions, and it is the function to be determined.
In a second step the following integral equation is solved in numerical form Then we find an approximation ( ) , aAp x t for ( ) a t using the solution found in the second step and condition (3).
Finally we find an approximation for ( ) , w x t using ( ) aAp t and the solution found in the second step.

Inverse Generalized Moment Problem
The d-dimensional generalized moment problem [13] [14] [15] and [16] [17] can be posed as follows: find a function f on a domain where ( ) i g is a given sequence of functions lying in ( ) 2 Ω L linearly independent, and the sequence of real numbers { } i i N µ ∈ are the known data.N is the set of natural numbers.
The moments problem of Hausdorff is a classic example of moments problem, is to find a function have the problem of moments of Hamburger.
It can be proved that [17] a necessary and sufficient condition for the existence of a solution of (4) is that ( ) where ij C are given by (11) and (12).
Moment problem are usually ill-posed in the sense that there may be no solution and if there is no continuous dependence on the given data.There are various methods of constructing regularized solutions, that is, approximate solutions stable with respect to the given data.One of them is the method of truncated expansion.
The method of truncated expansion consists in approximating (4) by finite moment problems and consider as an approximate solution of ( ) , , , n g g g  generated by 1 2 , , , n g g g  (5) is stable.Considering the case where the data ( ) , , , n µ µ µ µ =  are inexact, convergence theorems and error estimates for the regularized solutions they are applied.

Resolution of the Parabolic Partial Differential Equation
We consider the equation w a t w r x t = + .If we integrate with respect to x between 0 and 1 we obtain On the other hand we consider the vector field , , , u i z x t be the auxiliary function

div uF ua t w ua t w u a t w u t w u a t w ua t w ua t w
where ( ) Then of ( 7) and ( 8) Can be proven that, after several calculations, ( 9) is written as In the deduction of the previous formula it is used that ( ) At work [8] the auxiliary function is ( ) Then ( ) ( ) We wrote We solve the integral equation numerically , 1 and we will obtain an approximate solution for ( ) , .

G x t
We can apply the truncated expansion method detailed in [16] and generalized in [17] [18] [19] to find an approximation ( ) We approach the solution ( ) , , where , 1, 2, , The terms of the diagonal are The proof of the following theorem is in [19] [20].In [20] he proof is done for t in a finite interval.In [21] the demonstration is done for the one-dimensional case.We consider a more general notation: Theorem Let { } 0 n i i µ = be a set of real numbers and suppose that ( ) ( ) ( ) ( ) verify for some ε and M (two positive numbers) where C is the triangular matrix with elements ( ) Dem.) The demonstration is similar to that we have done for the unidimensional generalized moment problem [18], which is based in results of Talenti [16] for the Hausdorff moment problem.Here we simply introduce the necessary modification for the bi-dimensional case.
In matricial notation: To estimate the norm of ( ) , n d x t we observe that each element of the or- thonormal basis can be written as a function of the elements of another orthonormal basis, in particular the set , , a b , ( ) The Legendre polynomials and analogous property for the polynomials From these equations we deduce that Adding the expressions for the two standards ( ) is reached.An analogous demonstration proves inequality (15).If we apply the truncated expansion method to solve Equation ( 10) we obtain an approximation ( ) Then we have an equation in first order partial derivatives

A x t w x t A x t w x t p x t + =
where ( ) i.e., we can prove that solving this equation is equivalent to solving the integral equation ( ) ( ) ( ) In the deduction of the expression ( ) ϕ it is also used that ( ) Again we consider the base ( ) We can measure the accuracy of the approximation ( 16) using the previous theorem, where i µ would be the ith generalized moment of ( ) , wAp x t , that is, we consider the moments of ( ) , w x t measured with error.
An analogous argument is used to measure the accuracy of the approximation ( ) aAp t .

Numerical Examples
To obtain an approximation ( ) obtained by applying the Gram-Schmidt orthonormalization process on , In other words, it applies the Gram-Schmidt orthonormalization process on


We will obtain, by applying the truncated expansion method, ( ) To apply the method must be ( ) It may happen that ( 16) or ( 17) have discontinuities because the denominator is overridden for certain values of t.In this case we can vary the number of moments that are taken so that the denominator does not have real roots that cancel it.
It is observed that the greater is M, the more moments are needed to achieve precision in approximate solution, which is related to the length of the interval ( ) 0,T .

Example 1
We consider the equation

Example 2
We consider the equation

Example 4
We consider the equation ( )( ) The following conditions are met:

.
In Figure1and Figure 2 the exact solution and the approximate solution are compared.
Figure 1. ( ) a t and

Figure 5
Figure 5. ( ) a t and