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Numerical Solution of the Retrospective Inverse Problem of Heat Conduction with the Help of the Poisson Integral

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Abstract

We consider the retrospective inverse problem that consists in determining the initial solution of the one-dimensional heat conduction equation with a given condition at the final instant of time. The solution of the problem is given in the form of the Poisson integral and is numerically realized by means of a quadrature formula leading to a system of linear algebraic equations with dense matrix. The results of numerical experiments are presented and show the efficiency of the numerical method including the case of the final condition with random errors.

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References

  1. R. Lattés and J.-L. Lions, Méthode de quasi-réversibilitéet applications (Dunod, Paris, 1967; Mir, Moscow, 1970).

    MATH  Google Scholar 

  2. A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Ill-Posed Problems (Nauka, Moscow, 1979) [in Russian].

    MATH  Google Scholar 

  3. V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978) [in Russian]. Теория линейных некорректных задач и ее приложения. М.: Наука, 1978.

    MATH  Google Scholar 

  4. M. M. Lavrent’ev, V. G. Romanov, and S. T. Shishatskii, Ill-Posed Problems ofMathematical Physics and Analysis (Nauka, Moscow, 1980; Amer.Math. Soc., Providence, 1986).

    Book  Google Scholar 

  5. V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984; VNU Science Press, Utrecht, 1987).

    Google Scholar 

  6. A. M. Denisov, Introduction to Theory of Inverse Problems (Izd. Moskov. Gos. Univ., Moscow, 1994) [in Russian].

    Google Scholar 

  7. A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Dekker, Marcel, 2000).

    MATH  Google Scholar 

  8. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods of Solving the Inverse Problems of Mathematical Physics (Editorial URSS, Moscow, 2004) [in Russian].

    MATH  Google Scholar 

  9. V. Isakov, Inverse Problems for Partial Differential Equations (Springer, New York, 2006).

    MATH  Google Scholar 

  10. S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sibir. Nauchn. Izd., Novosibirsk, 2009) [in Russian].

    Google Scholar 

  11. A. A. Samarskii, P. N. Vabishchevich, and V. I. Vasil’ev, “IterativeModeling of Retrospective Inverse Problem of Heat Conduction,” Mat. Model. 9 (5), 119–127 (1997).

    MathSciNet  MATH  Google Scholar 

  12. D. Lesnic, L. Elliott, and D. B. Ingham, “An Iterative Boundary Element Method for Solving the Backward Heat Conduction Problem Using an Elliptic Approximation,” Inverse Probl. Eng. 6 (2), 255–279 (1998).

    Article  Google Scholar 

  13. B. T. Johansson and D. Lesnic, “A Procedure for Determining a Spacewise Dependent Heat Source and the Initial Temperature,” Appl. Anal. 87 (3), 265–276 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Ismail-Zadeh, A. Korotkii, G. Schubert, and I. Tsepelev, “Numerical Techniques for Solving the Inverse Retrospective Problem of Thermal Evolution of the Earth Interior,” Comput. Struct. 87 (11–12), 802–811 (2009).

    Article  Google Scholar 

  15. I. A. Tsepelev, “Iterative Algorithmfor Solving the Retrospective Problem of Thermal Convection in aViscous Fluid,” Fluid Dynam. 46 (5), 835–842 (2011).

    Article  MathSciNet  Google Scholar 

  16. Zh. Zhao and Z. Meng, “A Modified Tikhonov Regularization Method for a Backward Heat Equation,” Inverse Probl. Sci. Eng. 19 (8), 1175–1182 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. N. H. Tuan, Ph. H. Quan, D. D. Trong, and L. M. Triet, “On a BackwardHeat Problem with Time-Dependent Coefficient: Regularization and Error Estimates,” Appl.Math. Comput. 219, 6066–6073 (2013).

    MathSciNet  MATH  Google Scholar 

  18. V. I. Vasil’ev, A. M. Kardashevsky, and P. V. Sivtsev, “Computational Experiment on the Numerical Solution of Some Inverse Problems of Mathematical Physics,” in 11th International Conference “Mesh Methods for Boundary-Value Problems and Applications,” IOP Conference Series Material Science and Engineering, Vol. 158, p. 012093 (2016).

    Article  Google Scholar 

  19. V. I. Vasil’ev and A. M. Kardashevsky, “Iterative Solution of the Retrospective Inverse problem for a Parabolic Equation Using the Conjugate Gradient Method,” in The 6th Conference on Numerical Analysis and Applications, June 15–22, 2016. Lozenetz, Bulgaria (LNCS, 10187, 2017), pp. 698–705.

    Google Scholar 

  20. A. A. Samarskii, P. N. Vabishchevich, and E. A. SamarskAYA, Tasks and Exercises for Numerical Methods (Editorial URSS, Moscow, 2000) [in Russian].

    Google Scholar 

  21. A. B. Vasil’ev and A. N. Tikhonov, Integral Equations (Izd. Moskov. Gos. Univ., Moscow, 1989) [in Russian].

    Google Scholar 

  22. O. Yaremko and E. Mogileva, “The Solution of Fractal Diffusion Retrospective Problem,” Appl.Math. Phys. No. 3, 60–66 (2013).

    Google Scholar 

  23. I. I. Bavrin, V. L. Matrosov, and O. E. Yaremko, Transform Operators for Boundary Value Problems, Integral Representations, and Recovery of Dependences (Prometei, Moscow, 2016) [in Russian].

    Google Scholar 

  24. A. L. Karchevsky, “On a Solution of the Convolution Type Volterra Equation of the 1-st Kind,” Adv. Math. Models & Applications. 2 (1), 1–5 (2017).

    Article  Google Scholar 

  25. P. N. Vabishchevich, NumericalMethods. ComputationalWorkshop (URSS, Moscow, 2016) [inRussian].

    Google Scholar 

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Authors and Affiliations

  1. Ammosov North-Eastern Federal University, ul. Belinskogo 58, Yakutsk, 677000, Russia

    V. I. Vasil’ev & A. M. Kardashevskii

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  1. V. I. Vasil’ev
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  2. A. M. Kardashevskii
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Correspondence to V. I. Vasil’ev.

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Original Russian Text © V.I. Vasil’ev, A.M. Kardashevskii, 2018, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2018, Vol. XXI, No. 3, pp. 26–36.

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Vasil’ev, V.I., Kardashevskii, A.M. Numerical Solution of the Retrospective Inverse Problem of Heat Conduction with the Help of the Poisson Integral. J. Appl. Ind. Math. 12, 577–586 (2018). https://doi.org/10.1134/S1990478918030158

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  • DOI: https://doi.org/10.1134/S1990478918030158

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