# Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

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## Abstract

In the domain *Q* = [0,∞)×[0,∞) of the variables (*x*, *t*), for the telegraph equation with a Dirac potential concentrated at a point (*x*_{0}, *t*_{0}) ∈ *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 (*x*_{0}, *t*_{0}), 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.

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## References

- 1.Nakhushev, A.M., Loaded equations and their applications,
*Differ. Equations*, 1983, vol. 19, no. 1, pp. 74–81.MathSciNetzbMATHGoogle Scholar - 2.Dzhenaliev, M.T., Loaded equations with periodic boundary conditions,
*Differ. Equations*, 2001, vol. 37, no. 1, pp. 51–57.MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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.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.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.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.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