On the linear algorithm of numerical solution of a boundary value problem for a simple wave equation

The solution of a boundary value problem for a simple wave equation defined on a rectangle can be represented as a sum of two terms. They are solutions of two boundary value problems: in the first case, the boundary functions are constant, while in the second the initial functions have a special form. Such decomposition allows to apply two-dimensional splines for the numerical solution of both problems. The first problem was studied previously, and an economical algorithm of its numerical solution was developed.
To solve the second problem we define a finite-dimensional space of splines of Lagrangian type, and recommend an optimal spline giving the smallest residual as a solution. We obtain exact formulas for the coefficients of this spline and its residual. The formula for the coefficients of this spline is a linear form of initial finite differences defined on the boundary.
The formula for the residual is a sum of two simple terms and two positive definite quadratic forms of new finite differences defined on the boundary. Elements of matrices of forms are expressed through Chebyshev polynomials, both matrices are invertible and have the property that their inverses matrices are of tridiagonal form. This feature allows us to obtain upper and lower bounds for the spectrum of matrices, and to show that the residual tends to zero when the numerical problem dimension increases. This fact ensures the correctness of the proposed algorithm of numerical solution of the second problem which has linear computational complexity.