# 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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