Advertisement

Differential Equations

, Volume 54, Issue 9, pp 1147–1155 | Cite as

Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

  • S. N. Baranovskaya
  • E. N. Novikov
  • N. I. Yurchuk
Partial Differential Equations
  • 18 Downloads

Abstract

In the domain Q = [0,∞)×[0,∞) of the variables (x, t), for the telegraph equation with a Dirac potential concentrated at a point (x0, t0) ∈ Q, we consider a mixed problem with initial (at t = 0) conditions on the solution and its derivative with respect to t and a condition on the boundary x = 0 which is a linear combination with coefficients depending on t of the solution and its first derivatives with respect to x and t (a directional derivative). We obtain formulas for the classical solution of this problem under certain conditions on the point (x0, t0), the coefficient of the Dirac potential, and the conditions of consistency of the initial and boundary data and the right-hand side of the equation at the point (0, 0). We study the behavior of the solution as the direction of the directional derivative in the boundary condition tends to a characteristic of the equation and obtain estimates of the difference between the corresponding solutions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nakhushev, A.M., Loaded equations and their applications, Differ. Equations, 1983, vol. 19, no. 1, pp. 74–81.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dzhenaliev, M.T., Loaded equations with periodic boundary conditions, Differ. Equations, 2001, vol. 37, no. 1, pp. 51–57.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kozhanov, A.I., A nonlinear loaded parabolic equation and a related inverse problem, Math. Notes, 2004, vol. 76, no. 6, pp. 784–795.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dzhenaliev, M.T. and Ramazanov, M.I., On a boundary value problem for a spectrally loaded heat operator. I, Differ. Equations, 2007, vol. 43, no. 4, pp. 513–524.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dzhenaliev, M.T. and Ramazanov, M.I., On a boundary value problem for a spectrally loaded heat operator. II, Differ. Equations, 2007, vol. 43, no. 6, pp. 806–812.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baranovskaya, S.N. and Yurchuk, N.I., Cauchy problem and the second mixed problem for parabolic equations with the Dirac potential, Differ. Equations, 2015, vol. 51, no. 6, pp. 819–821.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lomovtsev, F.E. and Novikov, E.N., Necessary and sufficient conditions for the vibrations of a bounded string with directional derivatives in the boundary conditions, Differ. Equations, 2014, vol. 50, no. 1, pp. 128–131.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baranovskaya, S.N. and Yurchuk, N.I., Mixed problem for the string vibration equation with a timedependent oblique derivative in the boundary condition, Differ. Equations, 2009, vol. 45, no. 8, pp. 1212–1215.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Moiseev, E.I. and Yurchuk, N.I., Classical and generalized solutions of problems for the telegraph equation with a Dirac potential, Differ. Equations, 2015, vol. 51, no. 10, pp. 1330–1337.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • S. N. Baranovskaya
    • 1
  • E. N. Novikov
    • 1
  • N. I. Yurchuk
    • 1
  1. 1.Belarusian State UniversityMinskBelarus

Personalised recommendations