ИСТИНА

In the present paper we study initial boundary value problem for telegraph equation utt(x, t)−uxx(x, t)+c 2u(x, t) = 0 with zero initial conditions and boundary conditions u(0, t) = µ(t) ∈ Lp[0, T], u(l, t) = 0. We found in an explicit analytical form of solution on reqtangle QT = [0 6 x 6 l] × [0 6 t 6 T]. We will prove theorems of existence and uniqueness solution u(x, t) from class Lp(QT ), where p ≥ 1 . Also, we show that the function u(x, t) is solution not only from the class Lp(QT ), but belongs to class Lp[0 6 t 6 T] for all x ∈ [0, l], and belongs to Lp[0 6 x 6 l] for all t ∈ [0, T].