Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space. (English) Zbl 1145.65113

Du, Hong; Cui, Minggen

Appl. Math. Lett. 21, No. 6, 617-623 (2008).

The authors present a new method for solving Fredholm integral equations of the first kind in a reproducing kernel Hilbert space (RKHS). Following the introduction, emphasising the practical application of, and hence the need for, the research reported in the paper, a RKHS is introduced in section 2. Theorems providing the representation of the solution of the Fredholm integral equation (when one exists) in a RKHS are presented and proved in section 3. In section 4 the stability of the solution is discussed and a theorem, stating that the approximate method for obtaining the minimal norm solution is stable, is presented and proved. The paper concludes with an illustrative numerical example to support the validity of the new method.

Reviewer: Pat Lumb (Chester)

MSC:

65R20	Numerical methods for integral equations
65R30	Numerical methods for ill-posed problems for integral equations
45B05	Fredholm integral equations
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
45M10	Stability theory for integral equations

Keywords:

Fredholm integral equations of the first kind; ill-posed problem; reproducing kernel Hilbert space; stability; minimal norm solution; numerical example

Software:

GENEREG; NLREG

References:

[1]	Groestch, C.W., Inverse problems in the mathematical sciences, (1993), Vieweg Braunschweig
[2]	Bojarki, N.N., Inverse black body radiation, IEEE trans. antennas and propagation, 30, 778-780, (1982)
[3]	Hadamard, J., Lectures on the Cauchy problems in partial differential equation, (1923), Yale University Press New Haven · JFM 49.0725.04
[4]	Tikhonov, A.N., Solutions of ill-posed problems, (1977), John Wiley and Sons New York
[5]	Roths, T.; Marth, M.; Weese, J.; Honerkamp, J., A generalization regularization method for nonlinear ill-posed problems enhanced for nonlinear regularization terms, Comput. phys. commun., 139, 279-296, (2001) · Zbl 0986.65140
[6]	Yildiz, Bunyamin, A stability estimate on the regularized solution of the backward heat equation, Appl. math. comput., 135, 2-3, 561-567, (2003) · Zbl 1135.35368
[7]	Xiong, Xiang-Tuan, Central difference schemes in time and error estimate on a non-standard inverse heat conduction problem, Appl. math. comput., 157, 1, 77-99, (2004) · Zbl 1068.65117
[8]	Li, Chun-Li; Cui, Ming-Gen, The exact solution for solving a class nonlinear operator equation in reproducing kernel space, Appl. math. comput., 143, 2-3, 393-399, (2003) · Zbl 1034.47030
[9]	Kirsch, A., An introduction to the mathematical theory of inverse problems, (1996), Springer-Verlag New York Inc. New York · Zbl 0865.35004