Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ( x ), where ψ( x ) is absolutely continuous, ψ'( x ) ∈ L 2 [0,1], and ψ(0) = ψ(1) = 0. In the case ψ( x ) ∈ L [0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.