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Rothe’s method for a telegraph equation with integral conditions. (English) Zbl 1171.35306

The paper deals with the telegraph equation \[ \tau\frac{\partial^2 v}{\partial t^2}+a\frac{\partial v}{\partial t}-b\frac{\partial^2 v}{\partial x^2}=f(x,t,v),\quad (x,t)\in (0,1)\times [0,T], \] subject to the initial conditions \[ v(x,0)=v_0(x),\quad \frac{\partial v}{\partial t}(x,0)=v_1(x), \] the Neumann condition \[ \frac{\partial v}{\partial x}(0,t)=G(t) \] and the integral condition \[ \int_0^1v(x,t)\,dx=E(t), \] where \(f,\,v_1,\,v_0,\,G,\,E\) are given functions which verify some assumptions, \(T,\,\tau,\,b\) are positive constants and \(a\geq 1\). By using the Rothe time discretization method, the authors prove the existence and uniqueness of the weak solutions of the above problem.

MSC:

35A35 Theoretical approximation in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35L20 Initial-boundary value problems for second-order hyperbolic equations
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