A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coefficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.