Abstract

The authors solve the hyperbolic initial-boundary value problem with the integral term in the boundary condition: To find a pair \((u,P)\) of functions satisfying \[ u_{tt}-u_{xx}+Ku +\lambda u_t=f(x,t),\quad 0<x<1, 0<t<T; \] \[ u_x(0,t)=P(t), \] \[ u_x(1,t)+K_1u(1,t)+\lambda_1u_t(1,t)=0, \] \[ u(x,0)=u_0(x), \] \[ u_t(x,0)=u_1(x), \] where \(K,\lambda,K_1,\lambda_1\) are nonnegative constants. The boundary value \(P(t)\) satisfies the Cauchy problem \(P”(t)+\omega^2P(t)=hu_{tt}(0,t), 0<t<T, P(0)=P_0, P’(0)=P_1.\) After evaluating \(P(t)\),df the first boundary condition in an integral form is \(\quad u_x(0,t)=g(t)+hu(0,t)-\int_0^t k(t-s)u(0,s)ds.\) Using the compactness method the existence and the uniqueness of a solution is verified. Numerical results for a particular problem support the theory.