Divergent series and the mixed problem for the wave equation with free endpoints

For a mixed problem for homogeneous wave equation with a summable potential, free endpoints, and zero initial velocity, we obtain necessary and sufficient conditions of the existence of the classic solution and a generalized solution for a summable initial function. The resolvent approach, the Fourier method, A. N. Krylov's ideas on the acceleration of convergence of Fourier series, and important Euler's ideas on application of divergent series allow us to obtain a generalization of d'Alembert formula for the classic solution in the form of uniformly converging series whose terms are solutions of corresponding mixed problems for a inhomogeneous wave equation with zero potential, free endpoints, and zero initial data. This series also converges if the initial function is summable and, therefore, it is a generalized solution of the mixed problem in this case.