Parabolic Partial Differential Equations as Inverse Moments Problem

We considerer parabolic partial differential equations ( ) ( ) t x x w w r x t , − = under the conditions ( ) ( ) ( ) ( ) 1 1 1 2 , , x x w a t k t w b t k t = = ( ) ( ) ( ) ( ) 2 1 2 2 , , w x a h t w x b h t = = on a region ( ) ( ) E a b a b 1 1 2 2 , , = × . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the onedimensional one-phase inverse Stefan problem.


Introduction
We considerer parabolic partial differential equation of the form: ( ) ( ) where the unknown function ( ) , w x t is defined in ( ) ( ) , , E a b a b = × . ( ) , r x t 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 thermocouples of the domain.
In this paper we solve the inverse Stefan problem: find ( ) f t with ( ) s t 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 where ( ) n g is a given sequence of functions lying in ( ) 2 Ω L linearly independent.Many inverse problems can be formulated as an integral equation of the first kind, namely, 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 expansion.
The method of truncated expansion consists in approximating (2) by finite moment problems Solved in the subspace , , , n µ µ µ µ =  are inexact, we apply some convergence theorems and error estimates for the regularized solutions [9] [11].

Parabolic Partial Differential Equations as Integral Equations of First Kind
, , F w x t r x t = be a partial differential equations such as (1).The solution ( ) , w x t is defined on the region ( ) ( )

, . w x a h t w x b h t = =
We apply the technique used in [1].Let ( ) ( ) ( ) where ( ) Integrating by parts: Note that in (9) if x is a natural number then ( ) and if 1 0 a = then ( ) Thus if x N ∈ and 1 0 a = : ( ) Also if we write We write and we must have ( )

Solution of Generalized Moment Problems
Equation ( 13) is of the form: We assign natural values to x and t: , 0,1, ,  and we consider the corresponding generalized finite moment problem bi-dimensional [12] [13] To obtain a numerical approximation of the solution ( ) , w τ ξ the truncated expansion method is applied [9] [11].
We considerer the basis  by Gram-Schmidt method and addition of the necessary functions in order to have an orthonormal basis.
To facilitate the calculations we write We then approximate the solution ( ) ( ) The proof of the following theorem is in [14].
Theorem 1.Let { } 0 r m m µ = be a set of real numbers and let ε and E be two positive numbers such that ( ) ( ) where C is the triangular matriz with elements ij C ( ) and we must have ( )

Numerical Examples
The accuracy is, with this inner product , , 0.0250523.p x t w x t − =

The Inverse One-Phase Stefan Problem as Integral Equation
The Stefan problem consists of finding w y s such that  ( ) We want to solve the inverse Stefan problem: to find ( ) f t with ( ) s t known such that the above con- ditions are met.
We write We take the auxiliary function We write if ⋅ it is the scalar product and ∇ is the gradient operator we get ( ) ( )

T T s T s p E F u A u F w F w u uw x t
To solve the inverse problem, where ( ) We assign values to t: We can interpret (35) as a one-dimensional generalized moments problem.We solve the problem numerically considering the finite generalized moments problem and we apply the truncated expansion method.

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 by Gram-Schmidt and necessary functions are added in order to have an orthonormal basis.We then approximate the solution ( ) 0, w τ ξ with: ( ) ( ) 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 ( ) 0, .∞ Theorem 2. Let { } 0 n i i µ = be a set of real numbers and let ε and M be two positive numbers such that ( ) ( ) ( ) where C is the triangular matriz with elements ij C ( ) Since the problem is linear we can assume 0, 0,1, , ( ) ( ) ( ) it is the orthogonal projection of ( ) x ξ on the orthogonal complement.Here the underlying structure is the space where ( ) ( )  are the Fourier coefficients in the expansion of ( ) x ξ .
To estimate ( ) n h ξ we consider the relationship between the Fourier coefficients i λ and the moments ( ) ( ) where ij C they are given in (37) y (38).In matrix notation 0 0 1 1 .
By ( 43) until ( 46) we can write ( ) To estimate n f we see that each element of the orthonormal set or also ( ) ( ) ( ) .
After several calculations ( ) ( ) Now multiplying by ( ) x ξ and integrating both sides of the differential Equation (49) and assuming that ( ) then by ( 53) and (55): from ( 47) and ( 56): ( ) ( ) This inequality remains valid if we replace any integer i between 0 and n for n.Then the result (41) it demonstrated.An analogous demonstration proves inequality (42).□

Numerical Example
Find ( ) The exact solution is ( ) 6 the approximate numerical solution and the exact one are compared.Were taken 6 r = moments.
To apply Gram Schmidt to The accuracy is, with this inner product ( ) ( )

Figure 2
the approximate numerical solution and the exact one are compared.Were taken 9 r = moments.

2 ,=
consequence if w it is solution of the Equation (27), then , then w it is solution of the Equation (27).

.
Observe the Figure5 and:
it is the orthogonal projection of ( )x ξ on the linear space generated by the set written as an Fredholm integral equation

,(
then this Fredholm integral equation of first kind can be transformed into a bi-dimensional generalized moment problem assigning integer values greater than or to zero to variables x and t 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 and can be solved numerically considering the correspondent finite problem.
x 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 , K x y and ( ) f x are given functions and ( ) u y is a solution to be determined; ( ) f