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Integral identity for a class of ill-posed problems generated by a parabolic equation

  • Sergey A. Vavilov and Kirill V. Svetlov EMAIL logo
Published/Copyright: June 30, 2015

Abstract

The goal of the present study is to derive an integral identity for a class of ill-posed problems generated by a parabolic equation. The obtained result enables us to reduce the original ill-posed problem directly to the first-kind Fredholm equation with translation kernel. Various real world applications to a different fields of knowledge, including ecology and finances, are presented.

MSC 2010: 45B05; 45Q05

Acknowledgements

The authors are grateful to professor V. G. Osmolovskii for his interest to the paper and valuable comments.

References

[1] Björk T., Arbitrage Theory in Continuous Time, 2nd ed., Oxford University Press, Oxford, 2004. 10.1093/0199271267.001.0001Search in Google Scholar

[2] El Badia A., Ha-Duong T. and Hamdi A., Identification of a point source in a linear advection-dispersion-reaction equation: Application to a pollution source problem, Inverse Problems 21 (2005), no. 21, 1121–1136. 10.1088/0266-5611/21/3/020Search in Google Scholar

[3] Jørgensen P. L., Traffic light options, J. Banking Finance 31 (2007), no. 12, 3698–3719. 10.1016/j.jbankfin.2007.01.021Search in Google Scholar

[4] Ladyzhenskaya O. A., Solonnikov V. A. and Ural’tseva N. N., Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. 10.1090/mmono/023Search in Google Scholar

[5] Leonov A. S., Application of function of several variables with limited variations for piecewise uniform regularization of ill-posed problems, J. Inverse Ill-Posed Probl. 6 (1998), no. 1, 67–93. 10.1515/jiip.1998.6.1.67Search in Google Scholar

[6] Leonov A. S., Application of function of several variables with bounded variation to numerical solution of two-dimensional ill-posed problems (in Russian), Sib. Zh. Vychisl. Mat. 2 (1999), no. 3, 257–271. Search in Google Scholar

[7] Lions R. and Lattes J.-L., Methode de Quasi-Reversibilite et Applications, Dunod, Paris, 1967. Search in Google Scholar

[8] Marchuk G. I., Adjoint Equations and Analysis of Complex Systems, Math. Appl. (Dordrecht) 295, Kluwer Academic Publishers, Dordrecht, 1995. 10.1007/978-94-017-0621-6Search in Google Scholar

[9] Øksendal B. K., Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer, Berlin, 1998. 10.1007/978-3-662-03620-4Search in Google Scholar

[10] Tichonov A. N. and Arsenin V. Y., Solutions of Ill-Posed Problems, Halsted Press, New York, 1977. Search in Google Scholar

Received: 2014-11-13
Revised: 2015-5-13
Accepted: 2015-6-2
Published Online: 2015-6-30
Published in Print: 2016-10-1

© 2016 by De Gruyter

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