Parabolic Partial Differential Equations as Inverse Moments Problem

Maria B. Pintarelli^{1,2}

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Received 10 December 2015; accepted 22 January 2016; published 25 January 2016

1. Introduction

We considerer parabolic partial differential equation of the form:

(1)

where the unknown function is defined in. is known function. We consider conditions

This problem was studied under conditions of Cauchy in [1] .

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. [2] [3] .

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 [4] -[6] .

Next section is devoted to showing how the differential equation (1) is transformed into integral equation of first kind that can be seen as generalized moments problem. In Section 3 there we present 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 parabolic PDEs of the form (1). Finally in Section 5 the method is applied to solve the one-dimensional one-phase inverse Stefan problem.

The Stefan problem consists of finding w y s such that

The classical Stefan problem is a nonlinear initial value problem with a moving boundary whose position is unknown a priori and it must be determined as part of the solution. The differential equations of parabolic type governing heat diffusion with phase change are an important class of Stefan problems.

The direct Stefan problem requires determining both the temperature and the moving boundary interface when the initial and boundary conditions, and the thermal properties of the heat conducting body are known. Conversely, inverse Stefan problems require determining the initial and/or boundary conditions, and/or thermal properties from additional information which may involve the partial knowledge or measurement of the moving boundary interface position, its velocity in a normal direction, or the temperature at selected interior thermo- couples of the domain.

In this paper we solve the inverse Stefan problem: find with known such that the above con- ditions are met.

The d-dimensional generalized moment problem [7] -[10] can be posed as follows: find a function u on a domain satisfying the sequence of equations

(2)

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 considerer the multidimensional moment problems

Moment problems are usually ill-posed. There are various methods of constructing regularized solutions, that is, a approximate solution stable with respect to the given data. One of them is the method of truncated expan- sion.

The method of truncated expansion consists in approximating (2) by finite moment problems

(3)

Solved in the subspace generated by (3) is stable. Considering the case where the data are inexact, we apply some convergence theorems and error estimates for the regularized solutions [9] [11] .

2. Parabolic Partial Differential Equations as Integral Equations of First Kind

Let be a partial differential equations such as (1). The solution is defined on the region and verifies conditiones on the boundary:

We apply the technique used in [1] . Let be a vectorial field such that w verifies

with a known function and, reciprocally, if w verifies then

Specifically in this case and we take

(4)

(5)

Let be the auxiliary function

(6)

Since

we have

Moreover, as

and

we obtain

(7)

where

We consider the integral

(8)

Integrating by parts:

(9)

Note that in (9) if x is a natural number then

(10)

and if then

(11)

Thus if and:

(12)

Also if we write (see Figure 1) then

Figure 1. Domain E and contour.

We write

finally, if y we get:

(13)

If then you can take

(14)

and we must have when

3. Solution of Generalized Moment Problems

Equation (13) is of the form:

We assign natural values to x and t: and and we consider the corre- sponding generalized finite moment problem bi-dimensional [12] [13]

To obtain a numerical approximation of the solution the truncated expansion method is applied [9] [11] .

We considerer the basis obtained from the sequence with by Gram-Schmidt method and addition of the necessary functions in order to have an orthonormal basis.

To facilitate the calculations we write and with

We then approximate the solution with

and

where the coefficients verifies

(15)

(16)

The proof of the following theorem is in [14] .

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

(17)

(18)

then

(19)

where C is the triangular matriz with elements .

And

(20)

If, then (18) it is replaced by

and we must have

and

4. Numerical Examples

4.1. Example 1

Let considerer the equation

in the domain

and boundary condition on

given by

The exact solution is

In Figure 2 the approximate numerical solution and the exact one are compared.Were taken moments.

Figure 2. and.

The accuracy is, in this case

4.2. Example 2

Let considerer the equation

in the domain and boundary condition on given by

The exact solution is

In Figure 3 the approximate numerical solution and the exact one are compared.Were taken moments.

To apply Gram Schmidt to we consider the inner product

The accuracy is, with this inner product

5. The One-Dimensional One-Phase Inverse Stefan Problem

5.1. The Inverse One-Phase Stefan Problem as Integral Equation

The Stefan problem consists of finding w y s such that

(21)

(22)

(23)

Figure 3. and.

(24)

(25)

(26)

We want to solve the inverse Stefan problem: to find with known such that the above con- ditions are met.

We write

(27)

We take the auxiliary function

Therefore

We consider the vector field with. In this manner. In consequence if w it is solution of the Equation (27), then. Reci-

procally, if w satisfies, then w it is solution of the Equation (27).

We write. We use that

if it is the scalar product and is the gradient operator we get

By the divergence theorem

with, in consequence

We calculate:

First we write

and, (Figure 4) then we take.

If then

As by (23) and then

(28)

Now we developed. Observe the Figure 5 and:

Figure 4. Domain E_{T}.

Figure 5. Domain E_{T}.

(29)

(30)

(31)

(32)

Then

(33)

To solve the inverse problem, where is known and is unknown we do

In this manner:

(34)

We assign values to t:

(35)

We can interpret (35) as a one-dimensional generalized moments problem.

We solve the problem numerically considering the finite generalized moments problem

(36)

and we apply the truncated expansion method.

5.2. Numerical Approximation to the Solution of the Inverse Stefan Problem

To obtain a numerical approximation of the solution the procedure is analogous to that presented in Section 3. To approximate is taken a base of obtained from the sequence by Gram-Schmidt method and necessary functions are added in order to have an orthonormal basis. We then approximate the solution with:

where

and the coefficients verifies

(37)

(38)

The following theorem is the one-dimensional version of Theorem 1. In [15] is the demonstration when the domain is bounded.

We present here the demonstration when the domain is the interval

Theorem 2. Let be a set of real numbers and let and M be two positive numbers such that

(39)

(40)

then if

(41)

and

(42)

where C is the triangular matriz with elements .

Proof. Since the problem is linear we can assume.

We applied Gram-Scmidt method on in and we get then add the resulting set of necessary functions to obtain an orthonormal basis.

We write as

where it is the orthogonal projection of on the linear space generated by the set and it is the orthogonal projection of on the orthogonal complement. Here the underlying structure is the space. We can write

(43)

where are the Fourier coefficients in the expansion of.

To estimate we consider the relationship between the Fourier coefficients and the moments

:

(44)

where they are given in (37) y (38).

In matrix notation

(45)

Then

(46)

By (43) until ( 46) we can write

(47)

To estimate we see that each element of the orthonormal set can be expanded in terms of the

elements other orthonormal basis, in particular the base, where it represents the Laguerre polynomial of degree i.

These polynomials satisfy

(48)

or also

(49)

Then

(50)

then using (50)

(51)

After several calculations

(52)

and

(53)

where

(54)

Now multiplying by and integrating both sides of the differential Equation (49) and assuming that we get:

(55)

then by (53) and (55):

(56)

from (47) and (56):

(57)

This inequality remains valid if we replace any integer i between 0 and n for n. Then the result (41) it demon- strated. An analogous demonstration proves inequality (42). □

6. Numerical Example

Find such that

The exact solution is

In Figure 6 the approximate numerical solution and the exact one are compared.Were taken moments.

To apply Gram Schmidt to en we considerer the inner product

The accuracy is, with this inner product

7. Conclusions

The parabolic partial differential equations

on a region can be written as an Fredholm integral equation

Figure 6. and.

This equation is of the form:

with

If, then this Fredholm integral equation of first kind can be transformed into a bi-dimen- sional generalized moment problem assigning integer values greater than or equal to zero to variables x and t

(58)

As the functions are linearly independent then the generalized moment problem defined by (58) can be solved numerically considering the correspondent finite problem.

The inverse Stefan problem which it is to find being unknown and such that the follow- ing conditions are met

is equivalent to solve the integral equation

which is equivalent to the generalized moments problem

and can be solved numerically considering the correspondent finite problem.

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