Abstract
The Fujita equation is a special Fredholm integral equation of the first kind with compact integral operator and as such is a typical example of a severely ill-posed problem. It is proved that a class of Fredholm integral equation of the first kind (containing the Fujita equation) has injective integral operators and thus solutions coinciding with the corresponding minimum norm least squares solution. An application of an algorithm for a good choice of the regularization parameter gives satisfactory results for an example of Fujita’s equation. Moreover, the use of Sobolev norms for a partial regularization of the problem may improve these results considerably. The same example shows that collocation methods based on equidistant points are numerically instable and inaccurate.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahlfors, L.A. (1966) Complex analysis (McGraw-Hill, New York).
Edwards, R.E. (1967) Fourier series I (Holt, Rinehart, Winston, New York).
Gehatia, M. (1970) Solution of Fujita’s equation for equilibrium sedimentation by applying Tikhonov’s regularizing functions. J. Polymer Sci., Part A-2 8, 2039–2050.
Golub, G., Reinsch, C. (1970) Singular value decomposition and least squares solutions. Numer. Math. 14, 403–420.
Groetsch, C. (1982) A regularization-Ritz method for Integral equations of the first kind. J. Integral Equations 4, 173–182.
Henrici, P. (1974) Applied and computational complex analysis (Wiley, New York).
Hille, E., Phillips, R.S. (1957) Functional analysis and semi-groups (American Mathematical Society, Providence, RI).
Kammerer, W.J., Nashed, M.Z. (1972) Iterative methods for best approximate solutions of linear integral equations of the first and second kinds. J. Math. Anal. Appl. 40, 547–573.
Marti, J.T. (1980) On the convergence of an algorithm computing minimum norm solutions of ill-posed problems. Math. Comp. 34, 521–527.
(1983) On a regularization method for Fredholm integral equations of the first kind using Sobolev spaces. In “Numerical treatment of integral equations”, Durham-Conference, Eds. Baker, C.T.M. and Miller, G.F. (Academic Press, New York).
Tikhonov, A.N. (1963) Regularization of incorrectly posed problems. Soviet Math. Dokl. 4, 1035–1038.
Tikhonov, A.N. and Arsenin, V.Y. (1977) Solutions of ill-posed problems (Wiley, New York).
Wahba, G. (1978) Smoothing and ill-posed problems. In “Solution methods for integral equations”, Ed. Golberg, M.A. (Plenum Press, New York).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer Basel AG
About this chapter
Cite this chapter
Marti, J.T. (1983). Numerical Solution of Fujita’s Equation. In: Hämmerlin, G., Hoffmann, KH. (eds) Improperly Posed Problems and Their Numerical Treatment. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 63. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5460-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5460-3_13
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5462-7
Online ISBN: 978-3-0348-5460-3
eBook Packages: Springer Book Archive