Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

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.