Numerical method for restoring the initial condition for the wave equation

The inverse problem of restoring the initial condition for the time derivative for the one-dimensional wave equation is considered. As an additional condition, the solution of the wave equation at a finite time is given. First, the discretization of the derivative with respect to the spatial variable is carried out and the initial problem is reduced to a differential-difference problem with respect to functions depending on the time variable. To solve the resulting differential-difference problem, a special representation is proposed, with the help of which the problem splits into two independent differential-difference problems. As a result, an explicit formula is obtained for determining the approximate value of the desired function for each discrete value of a spatial variable. The finite difference method is used for the numerical solution of the obtained differential-difference problems. The presented results of numerical experiments conducted for model problems demonstrate the effectiveness of the proposed computational algorithm.