Parabolic Partial Differential Equations with Border Conditions of Dirichlet as Inverse Moments Problem

We considerer parabolic partial differential equations: ( ) ( ) , t x x w w r x t − = under the conditions ( ) ( ) ( ) ( ) 1 1 1 2 , , w a t k t w b t k t = = , ( ) ( ) 2 1 , w x a h t = on a region ( ) ( ) 1 1 2 2 2 , , ; E a b a b b = × = ∞ . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 , , d d b b m x t x t a a e w x t w x t t x m φ − + − = ∫ ∫ . Using the inverse moments problem techniques we obtain an approximate solution ( ) , n p x t of ( ) ( ) , , x t w x t w x t − . Then we find a numerical approximation of ( ) , w x t when solving the integral equation ( )( ) ( )( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 1 1 1 2 , d d , b b m x z t a a e w x t t x t x m z φ − + + − + + − = ∫ ∫ , because solving the previous integral equation is equivalent to solving the equation ( ) ( ) ( ) , , , x t n w x t w x t p x t − = .


Introduction
We considerer parabolic partial differential equation of the form: ( ) ( ) where the unknown function ( ) , w x t is defined in ( ) ( ) This problem was studied under conditions of Cauchy in [1] and under conditions of Neumann in [2].
Parabolic differential equations are commonly used in the fields of engineering and science for simulating physical processes.These equations describe various processes in viscous fluid flow, filtration of liquids, gas dynamics, heat conduction, elasticity, biological species, chemical reactions, environmental pollution, etc. [3] [4].
In a variety of cases, approximations are used to convert parabolic PDEs to ordinary differential equations or even to algebraic equations.The existence and uniqueness properties of this problem are presented in literature.Several numerical methods have been proposed for the solution of this problem [5] [6] [7].
The finite element method for the numerical solution of partial differential equations is a general method covering all the three main types of equations: elliptic, parabolic and hyperbolic equations [12].Some meshless schemes to solve differential partial equations are the diffuse element method [13], the partition of unity method [14], the element-free Galerkin method [15], the reproducing kernel particle method [16], the finite point method [17], the meshless local Petrov-Galerkin method [18], the use of radial basis functions [19] and the general finite difference method [20].
The d-dimensional generalized moment problem [21] [22] [23] [24] [25] can be posed as follows: find a function f on a domain Moment problem is 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  and i λ are coeffi- cients as a function of the i µ .
Solved in the subspace , , , 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.
In this paper we consider a different way to numerically solve the problem given by Equation (1) with conditions ( 2) and ( 3): we first transform it into an integral equation which we then handle as a bidimensional moment problem.This approach was already suggested by Ang [25] in relation with the heat conduction equation.
The work is organized as follows: in Section 2 first we transform the parabolic partial differential equation to the integral equation Using the inverse moments problem techniques we obtain an approximate solution ( ) x t w x t w x t − .Then we find a numerical approximation of ( ) , w x t when solving the integral equation In Section 3 the method is illustrated with examples.

Resolution of the Parabolic Partial Differential Equations
be a partial differential equations such as (1).The solution ( ) , w x t is defined on the region ( ) ( )

, w x a h t =
We apply the technique used in [2].Let be a vectorial field such that w verifies with h * a known function and, reciprocally, if w verifies and we take , , , u m z x t be the auxiliary function , , , where ( ) Then of ( 6) and ( 7) On the other hand it can be proved that, after several calculations, ( 8) is written as , L a b and then the above equation can be transformed into a generalized moment problem We can apply the truncated expansion method detailed in [24] and generalized in [25] [26] to find an approximation ( ) x t w x t w x t − for the corresponding finite problem with 0,1, , where n is the number of moments i µ .We consider the base ( ) We approach the solution ( ) ( ) , , x t w x t w x t − with [25] [26]: ( ) ( ) 0 0 , , where 0,1, 2, , The terms of the diagonal are ( ) The proof of the following theorem is in [27] [28].In [28] he proof is done for b = ∞ instead of taking polynomials the Legendre are taken po- lynomials of Laguerre.In [2] the demonstration is done for the one-dimensional case.
Theorem.Let { } 0 n i i µ = be a set of real numbers and suppose that ( ) , f x t verify for some ε and E (two positive numbers) ( ) ( ) where C is the triangular matrix with elements ij C ( ) If we apply the truncated expansion method to solve Equation ( 9) we obtain an approximation ( ) . Then we have an equation in first order partial derivatives of the form It is solved as in [28], we can prove that solving this equation is equivalent to solving the integral equation Again we take a base 2 L E and then the above equation can be transformed into a generalized moment problem ( ) ( )

Example 1
We consider the equation The solution is ( ) ( ) We take the base ( )

Example 2
We consider the equation The solution is ( ) We take the base ( ) We take the base ( )

Conclusions
An equation in parabolic partial derivatives of the form We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution.First we transform the parabolic partial differential equation to the integral equation the sequence of real numbers { } i i N µ ∈ is the known data.The moments problem of Hausdorff is a classic example of moments problem, and is to find a function ( ) f x in ( ) we have the problem of moments of Stieltjes; if the interval of integration is ( ) , −∞ ∞ we have the problem of moments of Hamburger.
u m z x a x w b t u m z b t w a t u m z a t t m z w x t u t x w x t u t x m w x t u m m x t t x w x a u m m x a x w b t u m m b t w a t u m m a t t m

, 0, 1
, 2, i x t i φ =  obtained by applying the Gram-Schmidt orthonormalization process on ( ) , 0,1, 2, i H x t i =  and add-ing to the resulting set the necessary functions until reaching an orthonormal basis.
m z b t w b t u m n a t w a t t u m z x b w x b u m n x a w x a x up t x Applying again the techniques of generalized moments problem we found an approximate solution for ( ) , w x t .
a) the exact solution and the approximate solution are compared Second step: approximates ( ) , w x t b) the exact solution and the approximate solution are compared.
Figure 1.(a) a) the exact solution and the approximate solution are compared Second step: approximates ( ) , w x t b) the exact solution and the approximate solution are compared.
Figure 2. (a) a) the exact solution and the approximate solution are compared Second step: approximates ( ) b) the exact solution and the approximate solution are compared.
the unknown function ( ) , w x t is defined in