Mixed problem for a homogeneous wave equation with a nonzero initial velocity (Q1757645)

The study refers to an IBVP for the wave equation \(u_{tt}=u_{xx}-q(x)u(x,t)\), \(x\in[0,1]\), \(t\geq0\), \(u(0,t)=u(l,t)=0\), \(u(x,0)=0\), \(u_t(x,0)=\Psi(x)\), the complex function \(q(x)\) being summable on \([0,1]\). The classical solution is computed by means of the well known Fourier method, and to accelerate convergence of the Fourier series the Krylov method is applied. The solution \(u(x ,t)\) is represented as a sum of six functions \(u_j(x,t)\), \(j=1,\dots,6\), representing the sum of absolute and uniform convergent series on \([0,1]\times [0,T]\). These functions are solutions for specific IBV problems. The second section of the paper is devoted to the computation of a generalized solution.