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.
Similar content being viewed by others
References
Nakhushev, A.M., Loaded equations and their applications, Differ. Equations, 1983, vol. 19, no. 1, pp. 74–81.
Dzhenaliev, M.T., Loaded equations with periodic boundary conditions, Differ. Equations, 2001, vol. 37, no. 1, pp. 51–57.
Kozhanov, A.I., A nonlinear loaded parabolic equation and a related inverse problem, Math. Notes, 2004, vol. 76, no. 6, pp. 784–795.
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.
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.
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.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.N. Baranovskaya, E.N. Novikov, N.I. Yurchuk, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 9, pp. 1176–1183.
Rights and permissions
About this article
Cite this article
Baranovskaya, S.N., Novikov, E.N. & Yurchuk, N.I. Directional Derivative Problem for the Telegraph Equation with a Dirac Potential. Diff Equat 54, 1147–1155 (2018). https://doi.org/10.1134/S0012266118090033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266118090033