Solution of Partial Derivative Equations of Poisson and Klein-Gordon with Neumann Conditions as a Generalized Problem of Two-Dimensional Moments

It will be shown that ﬁnding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solv-ing an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation 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 with R(x, t) known about a D domain where The underlying space is L 2 (D). Under the conditions w(x, a 2 ) = s 1 (t) w(x, b 2 ) = s 2 (t) The problem has been largely studied and solved with different methods such as the method of finite differences [1][2][3][4] to name a few.
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.
We want to present a an alternative method to solve a Poisson equation under Neumann conditions using techniques of generalized inverse moment problem, independently of other commonly used existing methods: finite difference method, Galerkin method, among many others. The interest is not to compare with the existing methods, but to present a different method to my novel criteria, and the one that I have already applied in other cases of partial differential equations under other conditions, for example the Poisson equation under Cauchy conditions or from Dirichlet. It turns out that a change in conditions implies a different approach. This is a significant change in the problem statement for its resolution.
The generalized moments problem [5][6][7] is to find a function f (x) about a domain Ω ⊂ R d that satisfies the sequence of equations 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 } i N is known data. The problem of Hausdorff moments [5][6][7], is to find a function f (x) en (a, b) such that 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 [5][6][7]. 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 [5][6][7]. There are several methods to build regularized solutions. One of them is the truncated expansion method [5].Ṫhis method is to approximate (2) with the finite moments problem 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 n N is chosen in an appropriate way then the solution of (6) it approaches the solution of the original problem (2).
In the case where the data µ i are inaccurate the convergence theorems should be applied and error estimates for the regularized solution (p. 19 a 30 de [5]).

Resolution of the Poisson Equation
We consider We take as an auxiliary function If the D domain is bounded the conditions are: If the region D is not bounded the conditions are: We define the vector field In addition, as udiv( Integrating by parts: On the other hand, We interpret (8) as a moments problem of two-dimensional generalized. p n (x, t) is the numerical approximation with the truncated expansion method for w(x, t) with n = n 1 .n 2 H mr (x, t) = u(m, r, x, t) m = 0, 1, 2, ...n 1 − 1; r = 1, 2, ..., n 2

Solution of the Generalized Moments Problem
We can apply the detailed truncated expansion method in [7] and generalized in [8] and [9] to find an approximation p n (x, 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 w(x, t) with [7] and generalized in [8] y [9]: And the coefficients C ij verify The terms of the diagonal are The proof of the following theorem is in [9,10]. In [10] the demonstration is made for b 2 finite. If b 2 = ∞ instead of taking the Legendre polynomials we take the Laguerre polynomials. En [11] 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 C ij (1 < i ≤ n; 1 ≤ j < i) and b2 a2 b1 a1 If b 2 it is not finite then (9) change by b2 a2 b1 a1 And it must be fulfilled that

Example 1
We consider the equation We take n = 9 moments and is approaching w(x, t) where the accuracy is

Conclusions
An equation in partial Poisson derivatives of the form w xx + w tt = R(x, t) or from Klein-Gordon w xx − w tt = R(x, t) where the unknown function w(x, t) is defined in D = (0, b 1 ) × (a 2 , b 2 ) or D = (0, b 1 ) × (a 2 , ∞) under Neumann's conditions can be solved numerically by applying inverse moment problem techniques.