DU HONG AND MU LI-HUA

(Department of Mathematics and Mechanics,Heilongjiang Institute of Science and Technology, Harbin,150027)

Stability of Fredholm Integral Equation of the First Kind in Reproducing Kernel Space?

DU HONG AND MU LI-HUA

(Department of Mathematics and Mechanics,Heilongjiang Institute of Science and Technology, Harbin,150027)

It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in C[a,b]or L2[a,b]. In this paper,the representation of the solution for Fredholm integral equation of the first kind is given if it has a unique solution.The stability of the solution is proved in the reproducing kernel space,namely,the measurement errors of the experimental data cannot result in unbounded errors of the true solution.The computation of approximate solution is also stable with respect toor.A numerical experiment shows that the method given in this paper is stable in the reproducing kernel space.

Freholm integral equation,ill-posed problem,reproducing kernel space

74S30

1 Introduction

The Fredholm integral equation of the first kind is of the form

It is well known that the problem on the stability for Fredholm integral equation of the first kind is an ill-posed problem in C[a,b]or L2[a,b].Some related works can be found in[1–6]. Namely,when given the right-hand side f(x)a perturbation,it could be caused large errors of solution u(y)in L2[0,π].

Many problems in science and engineering lead to seeking for the solution of the first kind of linear integral equations.In[1,7],the 1D heat conduction equation with initial and

boundary conditions

is given.The solution of(1.2)is

In this paper,the representation of the solution is obtained for Fredholm integral equation of the first kind in the reproducing kernel space[a,b].The reproducing kernel space[a,b]was de fi ned in[8].The computation of approximate solution is also stable when a perturbation is convergent to zero in the sense oforin the reproducing kernel space.We illustrate a numerical experiment in the last section of this paper.

2 The Solution of(1.1)

In this section,if the solution of(1.1)is unique,then the representation of the solution is given in the reproducing kernel space for the Fredholm integral equation of the first kind as follows:

Lemma 2.1The operatorAde fi ned in(2.1)is a bounded linear operator from[a,b]to[a,b]under the conditions(2.2)and(2.3).

In order to obtain the representation of the solution of(2.1),set the reproducing kernel Ry(x)in[a,b]as

Therefore,(2.7)is the solution of(2.1).

(ii)If(2.1)has solutions,then any solution could be represented as

3 The Stability of the Solution

It is well known that the problem on the stability of the solution for(2.1)may be an illposed problem in the space C[a,b]or L2[a,b].In this section,we discuss it in the reproducing kernel space[a,b].

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Now,the stability of the solution for(2.1)in[a,b]can be de fi ned.

De fi nition 3.1Letu(x)be a solution of(2.1).We say that the approximate method for the solutionu(x)in relation tou(n)(x),which is the solution of(2.1)with the right-hand sidef(n)(x),is stable in[a,b],if

Proof.Since the spaceΨand ?2are isometric-isomorphism,and ?2is complete,we see thatΨis complete.This completes the proof.

Therefore,the discussion of the stability of any solution for(2.1)is equivalent to that of the stability of the minimal norm solution for(2.1).

4 Numerical Experiments

In this section,we seek for the approximate solution of(1.3)with the right-hand side given a perturbation in the reproducing kernel space W12[a,b].

Take t=1,and

The true solution is u(x)=sinx.We calculate the approximate solution?u(x).All computations are performed by the Mathematica software package.We present the numerical results in Tables 4.1 and 4.2 when the right-hand side of(1.3)is put on perturbations ε=0.05 and ε=0.005,respectively,in the space[0,π].

Table 4.1 The error of solution u(x)with perturbations ε=0.05

Table 4.2 The error of solution u(x)with perturbations ε=0.005

It illustrates that the new method given in the paper is valid.

[1]Groestch C W.Inverse Problems in the Mathematical Sciences.Braunschweig:Vieweg,1993.

[2]Bojarki N N.Inverse black body radiation.IEEE Trans.Antennas and Propagation,1982,30: 778–780.

[3]Hansen J,Maier D,Honerkamp J,Richtering W,Horn M F,Sen ffH.Size distributions out of static light scattering:Inclusions of distortions from the experimental set.J Colloid Interf.Sci., 1999,215:72–84.

[4]Hadamard J.Lectures on the Cauchy Problems in Partial differential Equation.New Haven: Yale Univ.Press,1923.

[5]Tikhonov A N,Arsenin V Y.Solutions of Ill-posed Problems.New York:John Wiley and Sons, 1977.

[6]Yildiz B,Yetiskin H,Sever A.A stability estimate on the regularized solution of the backward heat equation.Appl.Math.Comput.,2003,135:561–567.

[7]Kirsch A.An Introduction to the Mathematical Theory of Inverse Problems.New York: Springer-Verlag New York Incorporated,1996.

[8]Li C L,Cui M.The exact solution for solving a class nonlinear operator equation in reproducing kernel space.Appl.Math.Comput.,2003,143:393–399.

Communicated by Ma Fu-ming

A

1674-5647(2012)02-0121-06

date:Apirl 26,2006.

NSF(A201015)of Heilongjiang Province.