Mixed problems for separate variable coefficient wave equations: The non-dirichlet case. Continuous numerical solutions with a priori error bounds

This paper deals with the construction of continuous numerical solutions of non-Dirichlet mixed wave systems of the form utt = (b(t))(a(x))uxx, 0 < x < L, t > 0, a1u(0, t) + a2ux(0, t) = 0, b1u(L, t) + b2ux(L, t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 ≤ x ≤ L. Uniqueness and existence of an exact series solution are studied. Given an admissible error ϵ> 0 and a bounded domain D(T) = [0, L] × [0, T], T > 0, an approximate continuous numerical solution involving only a finite number of eigenvalues and eigenfunctions is given so that the error with respect to the exact solution is less than ϵ uniformly in D(T). The admissible error for the finite number of approximated eigenvalues, eigenfunctions, and Sturm-Liouville coefficients is determined in order to grantee the required accuracy.