We consider linear partial differential equation of first order of the general form:

where the unknown function is defined in. We will consider Dirichlet conditions on the boundary and, , and are known functions.

Equation (1) is a particular case of the quasi-linear equation

The conventional method to solve this equation is reduced to find all surfaces that satisfy the above equation. This equation expresses that the tangent to a curve on the surface is proportional to. The solution of the quasi-linear equation can therefore be expressed by

where is a parametric curve belonging to the solution surface. Then we must solve a system of three simultaneous differential equations of the first order.

The general solution of this system of three equations consists of families of curves which are described by a system of three parametric equations with three arbitrary constants determined by initial conditions. This system is generally not linear and it is known that a system of non linear ordinary differential equations is difficult to solve explicitly. In general, geometrically in, the curves are determined by at least two intersecting surfaces transversely. This can be accomplished, for example, eliminating the parameter s and obtain

where and are arbitrary constants. The general solution will be

where is an arbitrary function of. For a particular solution you can find the function de modo que so to satisfy y.

We will show that, the partial differential Equation (1) can be transformed into a integral equation and that this one can be numerically solved using techniques normally employed with generalized moment problems [1] -[3] . This approach was already suggested by Ang [4] in relation with the heat conduction equation and we have applied to the non linear Klein-Gordon equation [5] .

Next section is devoted to show how the differential Equation (1) is transformed into integral equation of first kind that can be seen as generalized moments problem as is shown in Section 3. There we also proof a theorem that guarantees under certain conditions the stability and convergence of the finite generalized moment problem. In Section 4, we exemplify the general method by applying it to some linear PDEs which are particular cases of Equation (1). Finally in Section 5, the method is applied to solve an equation of Klein Gordon with boundary conditions in a rectangular region.

The d-dimensional generalized moment problem [1] [2] can be posed as follows: find a function u on a domain satisfying the sequence of equations

where is a given sequence of functions lying in linearly independent.

Many inverse problems can be formulated as an integral equation of the first kind, namely,

and are given functions and is a solution to be determined, is a result of experimental measurements and hence is given only at finite set of points. It follows that the above integral equation is equivalent to the following moment problem

Also we consider the multidimensional moment problems

Moment problem are usually ill-posed [6] [7] . There are various methods of constructing regularized solutions, that is, stable appoximate solutions with respect to the given data μ_n. One of them is the method of truncated expansion [4] .

The method of truncated expansion consists in approximating (5) by finite moment problems

Solved in the subspace generated by (6) is stable. Considering the case where the data are inexact, we apply some convergence theorems and error estimates for the regularized solutions.

Let be a partial differential equations such as (1). The solution is defined on the re-

gion and verifies Dirichlet conditiones on the boundary:

Let be a vectorial field such that w verifies with a known function and, reciprocally, if w verifies then

Let be the auxiliary function such that

Since

we have

Moreover, as

and

we obtain

where

Then (7) gives:

and

Then

where

We apply this to the Equation (1). For this we write:

We take as vector field

and

where y are arbitrary constants. Then

Therefore, Equation (8) yields

If (9) can be written in the form:

with, then taking a basis of this Fredholm integral equation of first kind can be transformed into a bi-dimensional generalized moment problem

where

and the moments are

If the functions are linearly independent then the generalized moment problem defined by Equations (10), (11) and (12) can be solved considering the correspondent finite problem

whose solution we denote

If has continuous inverse, then is an estimation of.

To reach this result let consider the basis obtained from the sequence by

Gram-Schmidt method and addition of the necessary functions in order to have an orthonormal basis.

We then approximate the solution de (13) with

with

where the coefficients verifies

We extend to the bi-dimensional case the arguments of reference [8] [9] and we have the following.

Theorem 1. Let be a set of real numbers and let and E be two positive numbers such that

y

then

where C is the triangular matriz with elements

And

Si is Lipschitz in, ie if for some and then

Proof. The demonstration is similar to that we have done for the unidimensional generalized moment problem [8] , which is based in results of Talenti [10] for the Hausdorff moment problem. Here we simply introduce the necessary modification for the bi-dimensional case.

Without loss of generality we take in (16).

We write

where is the orthogonal projection of on the linear space that the set gene- rates and is the orthogonal projection of on the orthogonal complement.

In terms of the basis the functions and reads

with

and the matrix elements given by (14) and (15).

In matricial notation:

Besides

Therefore

To estimate the norm of we observe that each element of the orthonormal basis can

be written as a function of the elements of another orthonormal basis, in particular the set con

with Legendre polynomial in, Legendre polynomial in

The Legendre polynomials verify

and analogous property for the polynomials

Defining we can demonstrate that

and

From these equations we deduce that

Adding the expressions for the two standards y result (17) is reached. An analogous demonstration proves inequality (18). □

Let consider the equation

in the domain and boundary condition on given by

The exact solution is

In Figure 1(a) the approximate numerical solution (dark gray) and the exact one (light gray) are compared.

Was taken with and

And in u.

Thus were taken moments.

The accuracy is, in this case

Let consider the equation

in the domain and boundary condition on given by

The exact solution is

In Figure 1(b) the approximate numerical solution (dark gray) and the exact one (light gray) are compared.

Was taken with and

And in u.

Thus were taken moments.

The accuracy is, in this case

We want to find with such that satisfies the Klein-Gordon equation

where h y r are known functions.

And boundary conditions

we write

We take as vector field

and

Then

where

Since

we have

Moreover, as

Therefore

in addition

then (22) and (23) we obtain:

Also doing integration by parts is reached:

with

From (23), (24) and (25) and after several calculations:

If then

We write (26) as:

We can see that (27) is an integral equation of the form

where the unknown function is, the kernel is and

To solve (27) as a problem of two-dimensional moments we apply seen in Section 3 and we obtain an approximation to.

Now we solve the partial differential equation of the first order

where, , y.

To find the solution of the Equation (28) algorithm of Section 3 applies.

We want to find with such that satisfies the Klein-Gordon equation

with boundary conditions

The exact solution is.

In Figure 2(a) the approximate numerical solution (dark grey) and the exact one (light grey) are compared.

For the first step was taken the base with and as an auxiliary function . For the second step was taken the base with and

And in order to avoid discontinuities.

Thus were taken moments.

The accuracy is, in this case

We want to find with such that satisfies the Klein-Gordon equation

with boundary conditions

The exact solution is.

In Figure 2(b) the approximate numerical solution (dark grey) and the exact one (light grey) are compared.

For the first step was taken the base with and as an auxiliary function . For the second step was taken the base with and

And in order to avoid discontinuities.

Thus were taken moments.

The accuracy is, in this case

The linear partial differential equations of first order

on a region can be written as an Fredholm integral equation

If (31) can be written in the form:

with, then taking a basis of this Fredholm integral equation of the first kind can be transformed into a bi-dimensional generalized moment problem

where

and the moments are

If the functions are linearly independent then the generalized moment problem defined by Equations (32), (33) and (34) can be solved considering the correspondent finite problem.