DOI: https://doi.org/10.26089/NumMet.v19r429

On some statements of nonlinear parabolic problems with boundary conditions of the first kind and on methods of their approximate solution

Authors

  • N.L. Gol’dman

Keywords:

parabolic equations
boundary value problem of the first kind
Hölder spaces
Rothe method
inverse problems
final observation
stability estimates
quasisolutions

Abstract

We study two statements of nonlinear problems in H"{o}lder spaces for a parabolic equation with an unknown coefficient at the time derivative and with boundary conditions of the first kind. One statement is a system containing a boundary value problem and an equation for the time dependence of the sought coefficient. In the other statement, in addition it is necessary to determine a boundary function in one of the boundary conditions by using an additional information on this coefficient at a final time. For these statements we justify a construction of approximate solutions on the basis of the Rothe method and the method of quasisolutions.

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Published

2018-12-24

Issue

Vol. 19 (2018): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biography

N.L. Gol’dman

Lomonosov Moscow State University,
Research Computing Center,
Ленинские горы, 119991, Москва
• Leading Researcher

References

  1. N. L. Gol’dman, “A Nonlinear Problem for a Parabolic Equation with an Unknown Coefficient at the Time Derivative and Its Applications in Mathematical Models of Physico-Chemical Processes,” Vychisl. Metody Programm. 18, 247-266 (2017).
  2. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; SIAM, Providence, 1968).
  3. N. L. Gol’dman, Inverse Stefan Problems (Kluwer, Dordrecht, 1997).
  4. N. L. Gol’dman, Inverse Stefan Problems. Theory and Methods of Solution (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
  5. V. K. Ivanov, “The Solution of Operator Equations That Are Not Well-Posed,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 112, 232-240 (1971) [Proc. Steklov Inst. Math. 112, 241-250 (1971)].
  6. S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Nauka, Moscow, 1969; Springer, New York, 1975).
  7. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon, New York, 1982).
  8. A. N. Tikhonov, “Solution of Incorrectly Formulated Problems and the Regularization Method,” Dokl. Akad. Nauk SSSR 151 (3), 501-504 (1963) [Sov. Math. Dokl. 5 (4), 1035-1038 (1963)].
  9. O. A. Liskovets, “Incorrectly Posed Problems and Stability of Quasisolutions,” Sib. Mat. Zh. 10 (2), 373-385 (1969) [Sib. Math. J. 10 (2), 266-274 (1969)].
  10. V. K. Ivanov, “On Ill-Posed Problems,” Mat. Sb. 61 (2), 211-223 (1963).
  11. B. M. Budak, A. Vin’oli, and Yu. L. Gaponenko, “A Regularization Method for a Continuous Convex Functional,” Zh. Vychisl. Mat. Mat. Fiz. 9 (5), 1046-1056 (1969) [USSR Comput. Math. Math. Phys. 9 (5), 82-95 (1969)].

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