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UDC 517.9 + 519.6 DOI: 10.14529/ctcr150206 Numerical Method for Solving Some Inverse Heat Conduction Problems with Unknown Initial Conditions N.M. Yaparova, South Ural State University, Chelyabinsk, Russian Federation, ddjy@math.susu.ac.ruAbstractThe article deals with some inverse Cauchy problems for parabolic equations with unknown initial conditions. The principal possibility of construction the numerical solution of these problems in this domain is showed. The computational scheme is proposed which allows to construct the numerical solution of the inverse problem both on the whole domain and on the boundary under unknown initial conditions.To evaluate the efficiency of the proposed method the computational experiment was carried out. The results of the experiments show the sufficient stability of numerical solutions and the advantages of the proposed method. Keywordsparabolic equations, inverse boundary problem, regularization method, numerical method, computational scheme References1. Alifanov O.M. Inverse Heat Transfer Problems International Series in Heat and Mass Transfer. New York, Springer, 2011, 324 p. 2. Vasin V.V. Modified Newton-type Processes Generating Feje'r Approximations of Regularized Solutions to Nonlinear Equations. Proceedings of the Steklov Institute of Mathematics, 2014, 284 (1), pp. 145–158. 3. Tanana V.P. Oder-Optimal Method for Solving an Inverse Problem for a Parabolic Equation. Mathematics Reports, 2006, 407 (3), pp. 316–318. 4. Tanana V.P., Yaparova N.M. [The Optimum in Order Method of Solving Conditionally-correct Problems]. Siberian Journal of Numerical Mathematics, Sib. Branch of Russ. Acad. of Sci., 2006, Vol. 9, no 4, pp. 353–368. (in Russ.) 5. Berman P., Levi O., Parmet Y., Saunders M. and Wiesman Z. Laplace Inversion of Lowresolution NMR Relaxometry Data Using Sparse Representation Methods. Concepts in Magnetic Resonance Part A, 2013, vol. 42, no 3, pp. 72–88. 6. Yaparova N.M. Numerical Methods for Solving a Boundary Value Inverse Heat Conduction Problem. Inverse Problems in Science and Engineering, 2014, vol. 22, no 5, pp. 832–847. 7. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya parabolicheskogo tipa [Linear and Quasilinear Equations of Parabolic Type]. Moscow, Nauka, 1980, 736 p. 8. Lavrentiev M.M., Romanov V.G., Shishatskii S.P. Nekorrektnye zadachi matematicheskoy fiziki i analiza [Ill-Posed Problems of Mathematical Physics and Analysis], Moscow, Nauka, 1980, 286 p. 9. Samarskii A.A. Teoriya raznostnykh skhem [The Theory of Difference Schemes], Moscow, Nauka, 1977, 656 p. 10. Yaparova N.M. Programma modelirovaniya raspredeleniya odnomernogo teplovogo rezhima na granites pri neizvestnykh nachal’nykh usloviyakh [The Program of Modeling of Distribution of the One-dimensional Thermal Mode on the Border under Unknown Initial Conditions]. Certificate on the state registration of the computer programs № 2014614775 (demand № 2014612053, 12.03.2014). SourceBulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control, Radio Electronics, 2015, vol. 15, no. 2, pp. 55-65. (in Russ.) (Modeling and Computer Technologies) |