Solution of mixed problems with boundary elastic-force control for the telegraph equation
From the abstract: We consider the problem on the boundary elastic-force control \(u_{ x }(0, t) = \mu (t)\) at the endpoint \(x = 0\) for the process described by the Klein-Gordon-Fock equation under the condition that the other endpoint \(x = l\) is either fixed, or free, or is controlled by an elastic force. For any time interval \(T\), we obtain the solution \(u(x,t)\) in terms of a finite-energy generalized solution of the telegraph equation. Three solutions are given, each belonging to the class \(\hat{W}_2 ^1(Q_T)\). One example is \[ \begin{split} &u(x,t)=-\sum_ {k=0}^{2n+1}(-1)^k \int_{0}^{t-x-2kl}H(\tau)J_{0}(c\sqrt{(t-\tau-2kl)^2-x^2})d\tau \\& -\sum_ {k=1}^{2n+2}(-1)^k\int_{0}^{t+x-2kl}H(\tau)J_{0}(c\sqrt{(t-\tau-2kl)^2-x^2})d\tau, \\ \end{split} \] where \(H(\tau)\) is the Heaviside step function and \(J_{0}\) is a Bessel function.
scientific article; zbMATH DE number 5963571