Advertisement

Differential Equations

, Volume 55, Issue 7, pp 940–948 | Cite as

Iterative Method for Solving an Inverse Problem for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative

  • A. M. DenisovEmail author
Numerical Methods
  • 2 Downloads

Abstract

We consider the Cauchy problem for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problem of finding an unknown function that is a coefficient of the equation and also occurs in the initial condition is posed. The values of the solution of the Cauchy problem and its derivative at x = 0 are given as additional information for solving the inverse problem. An iterative method for determining the unknown function is constructed, and its convergence is proved. Existence theorems are proved for the solution of the inverse problem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lavrent’ev, M.M., Romanov, V.G., and Shishatskii, S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza (Ill-Posed Problems in Mathematical Physics and Calculus), Novosibirsk: Nauka, 1980.Google Scholar
  2. 2.
    Romanov, V.G., Obratnye zadachi matematicheskoi fiziki (Inverse Problems of Mathematical Physics), Moscow: Nauka, 1984.zbMATHGoogle Scholar
  3. 3.
    Kabanikhin, S.I. and Lorenzi, A., Identification Problems of Wave Phenomena, Utrecht: VSP, 1999.zbMATHGoogle Scholar
  4. 4.
    Belishev, M.I. and Gotlib, V.Yu., Dynamical variant of the BC-method: Theory and numerical testing, J. Inverse Ill-Posed Probl., 1999, vol. 7, no. 3, pp. 221–240.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Prilepko, A.I., Orlovsky, D.G., and Vasin, I.V., Methods for Solving Inverse Problems in Mathematical Physics, New York: Marcel Dekker, 2000.zbMATHGoogle Scholar
  6. 6.
    Blagoveshchenskii, A.S., Inverse Problems of Wave Processes, Utrecht: VSP, 2001.CrossRefzbMATHGoogle Scholar
  7. 7.
    Isakov, V., Inverse Problems for Partial Differential Equations, New York: Springer, 2006.zbMATHGoogle Scholar
  8. 8.
    Kabanikhin, S.I., Obratnye i nekorrektnye zadachi (Inverse and Ill-Posed Problems), Novosibisk: Sibirskoe Nauchn. Izd., 2008.zbMATHGoogle Scholar
  9. 9.
    Denisov, A.M. and Solov’eva, S.I., Numerical determination of the initial condition in Cauchy problem for hyperbolic equation with a small parameter, Comput. Math. Model., 2018, vol. 29, no. 1, pp. 1–9.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Denisov, A.M. and Solov’eva, S.I., Numerical solution of inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative, Differ. Equations, 2018, vol. 54, no. 7, pp. 900–910.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lattes, R. and Lions, J.-L., The Method of Quasi-Reversibility: Applications to Partial Differential Equations, Moscow: Mir, 1970.zbMATHGoogle Scholar
  12. 12.
    Samarskii, A.A. and Vabishchevich, P.N., Chislennye metody resheniya zadach konvektsii-diffuzii (Numerical Methods for Solving Convection-Diffusion Problems), Moscow: Editorial URSS, 1999.Google Scholar
  13. 13.
    Korotkii, A.I., Tsepelev, I.A., and Ismail-zade, A.E., Numerical modeling of inverse retrospective heat convection problems with applications to geodynamic problems, Izv. Ural. Gos. Univ., 2008, no. 58, pp. 78–87.Google Scholar
  14. 14.
    Denisov, A.M., Asymptotic expansions of solutions to inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative, Comput. Math. Math. Phys., 2013, vol. 53, no. 5, pp. 580–587.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Belov, Yu.Ya. and Kopylova, V.G., Determination of the source function in a composite system of equations, Zh. Sib. Fed. Univ. Ser. Mat. Fiz., 2014, vol. 7, no. 3, pp. 275–288.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations