Parabolic Partial Differential Equations with Border Conditions of Dirichlet as Inverse Moments Problem ()
1. Introduction
We considerer parabolic partial differential equation of the form:
(1)
where the unknown function
is defined in
and
is known function. Under the conditions
(2)
(3)
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] .
Parabolic partial differential equations have been numerically solved by using a variety of techniques [8] [9] [10] [11] .
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
on a domain
satisfying the sequence of equations
(4)
where
is a given sequence of functions lying in
linearly independent, and the sequence of real numbers
is the known data.
The moments problem of Hausdorff is a classic example of moments problem, and is to find a function
in
such that
In this case the functions
. If the interval of integration is
we have the problem of moments of Stieltjes; if the interval of integration is
we have the problem of moments of Hamburger.
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
(5)
and consider as an approximate solution of
to
. The
result from orthonormalize
and
are coefficients as a function of the
.
Solved in the subspace
generated by
(5) is stable. Considering the case where the data
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
of
. Then we find a numerical approximation of
when solving the integral equation
In Section 3 the method is illustrated with examples.
2. Resolution of the Parabolic Partial Differential Equations
Let
be a partial differential equations such as (1). The solution
is defined on the region
and verifies on the boundary
:
We apply the technique used in [2] . Let
be a vectorial field such that
verifies
with
a known function and,
reciprocally, if
verifies
then
Specifically in this case
and we take
Let
be the auxiliary function
Since
we have
Moreover, as
(6)
where
besides
(7)
Then of (6) and (7)
(8)
On the other hand it can be proved that, after several calculations, (8) is written as
and if
then
We take a base
of
and then the above equation can be transformed into a generalized moment problem
(9)
where
and
We can apply the truncated expansion method detailed in [24] and generalized in [25] [26] to find an approximation
for
for the corresponding finite problem with
where
is the number of moments
. We consider the base
obtained by applying the Gram-Schmidt orthonormalization process on
and adding to the resulting set the necessary functions until reaching an orthonormal basis.
We approach the solution
with [25] [26] :
And the coefficients
verifies
The terms of the diagonal are
The proof of the following theorem is in [27] [28] . In [28] he proof is done for
finite. If
instead of taking polynomials the Legendre are taken polynomials of Laguerre. In [2] the demonstration is done for the one-dimen- sional case.
Theorem. Let
be a set of real numbers and suppose that
verify for some
and
(two positive numbers)
then
where
is the triangular matrix with elements
and
It must be fulfilled that
If we apply the truncated expansion method to solve Equation (9) we obtain an approximation
for
. 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
where
and
. Again we take a base
of
and then the above equation can be transformed into a generalized moment problem
where
and
Applying again the techniques of generalized moments problem we found an approximate solution for
.
3. Numerical Examples
3.1. Example 1
We consider the equation
and conditions
The solution is
First step: approximates
We take the base
Accuracy is
.
In Figure 1(a) the exact solution and the approximate solution are compared
Second step: approximates
We take the base
Accuracy is
.
In Figure 1(b) the exact solution and the approximate solution are compared.
3.2. Example 2
We consider the equation
and conditions
(a)(b)
Figure 1. (a)
and
; (b)
and
.
The solution is
First step: approximates
We take the base
Accuracy is
.
In Figure 2(a) the exact solution and the approximate solution are compared
Second step: approximates
We take the base
Accuracy is
.
In Figure 2(b) the exact solution and the approximate solution are compared.
3.3. Example 3
We consider the equation
where
is unknown. And conditions
(a)(b)
Figure 2. (a)
and
; (b)
and
.
The solution is
if
First step: approximates
We take the base
Accuracy is
.
In Figure 3(a) the exact solution and the approximate solution are compared
Second step: approximates
We take the base
Accuracy is
.
In Figure 3(b) the exact solution and the approximate solution are compared.
4. Conclusions
An equation in parabolic partial derivatives of the form
where the unknown function
is defined in
under the conditions
(a)(b)
Figure 3. (a)
and
; (b)
and
.
can be solved numerically by applying inverse moments problem techniques in two steps: first consider the integral equation
and we can solve it numerically as an inverse moments problem, and get an approximate solution for
. Then in a second step we consider the integral equation
and again we can solve it numerically as an inverse moments problem, and get an approximate solution for
. It is observed that the function
is not used in calculations, but it is implicitly considered in the boundary conditions.
In this way it would be possible to solve, for example, the problem of finding
that satisfies
under the conditions
with unknown
and known
.
Acknowledgements
We thank the Editor and the referee for their comments.