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We consider linear partial differential equations of first order <br/> on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of the first kind and will solve this latter with the techniques of inverse problem moments. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on problem of moments.

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

where the unknown function

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

The conventional method to solve this equation is reduced to find all surfaces

where

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

where

where

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 [

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 [

where

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

Also we consider the multidimensional moment problems

Moment problem are usually ill-posed [_{n}. One of them is the method of truncated expansion [

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

Solved in the subspace

Let

gion

Let

Let

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

Therefore, Equation (8) yields

If (9) can be written in the form:

with

where

and the moments

If the functions

whose solution we denote

If

To reach this result let consider the basis

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

We then approximate the solution

with

where the coefficients

We extend to the bi-dimensional case the arguments of reference [

Theorem 1. Let

y

then

where C is the triangular matriz with elements

And

Si

Proof. The demonstration is similar to that we have done for the unidimensional generalized moment problem [

Without loss of generality we take

We write

where

In terms of the basis

with

and the matrix elements

In matricial notation:

Besides

Therefore

To estimate the norm of

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

The Legendre polynomials

and analogous property for the polynomials

Defining

and

From these equations we deduce that

Adding the expressions for the two standards

Let consider the equation

in the domain

The exact solution is

In

Was taken

And

Thus were taken

The accuracy is, in this case

Let consider the equation

in the domain

The exact solution is

In

Was taken

And

Thus were taken

The accuracy is, in this case

We want to find

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

We write (26) as:

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

where the unknown function is

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

Now we solve the partial differential equation of the first order

where

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

Numerical ExamplesWe want to find

with boundary conditions

The exact solution is

In

For the first step was taken the base

And

Thus were taken

The accuracy is, in this case

We want to find

with boundary conditions

The exact solution is

In

For the first step was taken the base

And

Thus were taken

The accuracy is, in this case

The linear partial differential equations of first order

on a region

If (31) can be written in the form:

with

where

and the moments

If the functions