SOLUTION OF AN EQUATION IN POISSON PARTIAL DERIVATIVES WITH CONDITIONS OF DIRICHLET USING TECHNIQUES OF THE INVERSE MOMENTS PROBLEM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, it will be shown that finding solutions from the Helmholtz equation and the non-linear Poisson equation under Dirichlet conditions is equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular in principle. We will see that an approximate solution of the equation in partial derivatives can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution. The method consists of two steps.In each one an integral equation is solved numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.


INTRODUCTION
You want to find w(x,t) such that using the problem generalized moments techniques.
The problem has been solved using other techniques such as the Galerkin method [1], or the method of finite differences in regions of irregular shape [2].
The objective of this work is to show that we can solve the problem using the techniques of inverse moments problem.We focus the study on the numerical approximation.
The generalized moments problem [3,4]is to find a function f (x) about a domain Ω ⊂ R d that satisfies the sequence of equations ( 1) where N is the set of the natural numbers,(g i (x)) is a given sequence of functions in L 2 (Ω) linearly independent known and the succession of real numbers {µ i } iεN are known data.The problem of Hausdorff moments [3,4], is to find a function f (x) en (a, b) such that (2) In this case g i (x) = x i with i belonging to the set N.
If the integration interval is (0, ∞) we have the problem of Stieltjes moments; if the integration interval is (−∞, ∞) we have the problem of Hamburger moments [3,4].
The moments problem is an ill-conditioned problem in the sense that there may be no solution and if there is no continuous dependence on the given data [3,4].There are several methods to build regularized solutions.One of them is the truncated expansion method [3].Ṫhis method is to approximate(1) with the finite moments problem (3) where it is considered as approximate solution of f (x) to p n (x) = ∑ n i=0 λ i φ i (x) , and the functions {φ i (x)} i=1,...,n result of orthonormalize g 1 , g 2 , ..., g n being λ i the coefficients based on the data µ i .In the subspace generated by g 1 , g 2 , ..., g n the solution is stable.If nεN is chosen in an appropriate way then the solution of (3) it approaches the solution of the original problem (1).
In the case where the data µ i are inaccurate the convergence theorems should be applied and error estimates for the regularized solution (pg.19 a 30 de [3]).

RESOLUTION OF THE POISSON EQUATION
You want to find w(x,t) such that (4) We consider (5) If w x = w t we can take k = 1.
We consider as auxiliary function If the region E is bounded the conditions are: If the region E it is not bounded the conditions are: (7) w(a We define the vector field As udiv(F * ) = uG(x,t) we have to: Moreover, as udiv(F * ) = div(uF * ) − F * .∇u,then where ∇u = (u x , u t ).
Besides that Then from (8) y ( 9): On the other hand, it can be proven, after several calculations that, integrating by parts: To solve this integral equation we take a base ψ i (r) = r i e −r i = 0, 1, 2, ..., n of L 2 (E).
Then we multiply both members of (12) by ψ i (r) = r i e −r and we integrate with respect to r, we obtain ( 13) where We can interpret (13) as a generalized two-dimensional moment problem.We solve it numerically with the truncated expansion method and we found an approximation p n (x,t) for

SOLUTION OF THE GENERALIZED MOMENTS PROBLEM
We can apply the detailed truncated expansion method in [4] and generalized in [1] and [5] to find an approximation p n (x,t) of − √ kw x + kw t for the corresponding finite problem with i = 0, 1, 2, ..., n, where n is the number of moments µ i .We consider the basis φ i (x,t) i = 0, 1, 2, ..., n obtained by applying the Gram-Schmidt orthonormalization process on H i (x,t) i = 0, 1, 2, ..., n.
We approximate the solution − √ kw x + kw t with [4] and generalized in [3] y [5]: And the coefficients C i j verify The terms of the diagonal are The proof of the following theorem is in [6,7].In [7] the demonstration is made for b 2 finite.If b 2 = ∞ instead of taking the Legendre polynomials we take the Laguerre polynomials.En [8] the demonstration is made for the one-dimensional case.
This Theorem gives a measure about the accuracy of the approximation.
Theorem.Let {µ i } n i=0 be a set of real numbers and suppose that f (x,t) ∈ L 2 ((a 1 , b 1 ) × (a 2 , b 2 )) for two positive numbers ε and M verify: where C it is a triangular matrix with elements If b 2 it is not finite then (14) change by And it must be fulfilled that So we have an equation in first order partial derivatives of the form that is, it can be written as where It is resolved as in [9], that is, we can prove that solving this equation is equivalent to solving the integral equation where now it is taken as an auxiliary function The values of m 1 and m 2 are chosen in a convenient way to avoid discontinuities.and Again we take a base: ψ i j (m, r) = m i r j e −(m+r) i = 0, 1, ..., n 1 j = 0, 1, 2, ..., n 2 and we multiply both members of (17) by ψ i j (m, r) and we integrate with respect to m and r We have then the generalized moments problem where We apply the truncated expansion method and find a numerical approximation for w(x,t).

Numerical examples.
Example 1.We consider the equation in (0, 1) × (0, 1) Conditions: The solution is : For the first step we take n = 5 moments and we approximate −w x (x,t) + w t (x,t) = G(x,t) In the Figure 1 we show p 5 (x,t) and G(x,t) overlapping.For the second step we take m 1 = 1 and m 2 = 2.We also consider n 1 = 3 and n 2 = 2 , that is 6 moments.
We approximate w(x,t) with accuracy  Ejemplo 2. We consider the equation The solution is : w(x,t) = e −1−x−t .
For the first step we take n = 5 moments and we approximate − In this example we take k = 2,since otherwise G(x,t) = 0.
In the Figure 3 we show p 5 (x,t) and G(x,t) overlapping.
For the second step we take m 1 = 1 and m 2 = 2.We also consider n 1 = 3 y n 2 = 2 , that is 6 moments.
We approximate w(x,t) with accuracy 2 0 ∞ 0 (p 6 (x,t) − w(x,t)) 2 dtdx = 0.0427058 In the Figure 4 we show p 6 (x,t) and w(x,t) overlapping.Ejemplo 3. In general, the method could be applied to any region that can be written simultaneously as We can apply the to a circular region: We consider the equation Conditions:we must know w(x,t) on the edge of E The solution is : w(x,t) = e −1−x−2t .
The integrals are solved in numerical form without making change to polar coordinates using the Mathematica software.
For the first step we take n = 5 moments and we approximate −w x (x,t) + w t (x,t) = G(x,t) with accuracy In this example we take k = 1.
In the Figure 5 we show p 5 (x,t) and G(x,t) overlapping.For the second step we take m 1 = 1 and m 2 = 2.We also consider n 1 = 3 y n 2 = 2 , that is 6 moments.
Ejemplo 4. We consider the equation or The solution is : w(x,t) = e −1−x−2t .
For the first step we take n = 5 moments and we approximate −w x (x,t) + w t (x,t) = G(x,t) with accuracy In this example we take k = 1.
In the Figure 7 we show p 5 (x,t) and G(x,t) overlapping.
For the second step we take m 1 = 1 and m 2 = 2.We also consider n 1 = 3 y n 2 = 2 , that is 6 moments.
We approximate w(x,t) with accuracy E (p 6 (x,t) − w(x,t)) 2 dtdx = 0.00137451 In the Figure 8 we show p 6 (x,t) and w(x,t) overlapping.we can solve it numerically as a inverse moments problem, and we get an approximate solution for − √ kw x (x,t) + kw t (x,t).
(2) as a second step we consider the integral equation E K(m, r, x,t)w(x,t)dtdx = ϕ 2 (m, r) and again it can be solved numerically by applying inverse moments problem techniques, and we get an approximate solution for w(x,t).
The function f (x,t) it is not used in calculations, but it is implicitly considered in the boundary conditions.
An equation in partial Poisson derivatives of the form w xx + w tt = f (x,t) where the unknownfunction w(x,t) is defined in E = (a 1 , b 1 ) × (a 2 , b 2 ) or E = (a 1 , b 1 ) × (a 2 , ∞) or E a region that can be written simultaneously asE = {(x,t); a 1 < x < b 1 ; g 1 (x) < t < g 2 (x)} and E = {(x,t); h 1 (t) < x < h 2 (t); a 2 < t < b 2 }under the conditions of Dirichlet can be solved numerically by applying inverse problem techniques of moments in two steps:(1) first we consider the integral equation E u(− √ kw x + kw t )dA = ϕ 1 (r)