Mixed problems for separated variable coefficient wave equations: analytic–numerical solutions with a priori error bounds

Abstract

This paper deals with the construction of accurate analytic-numerical solutions of mixed problems related to the separated variable dependent wave equation u tt=(b(t)/a(x))u xx, 0<x<L, t>0 . Based on the study of the growth of eigenfunctions of the underlying Sturm–Liouville problems, an exact theoretical series solution is firstly obtained. Explicit bounds allow truncation of the series solution so that the error of the truncated approximation is less than ε 1 in a bounded domain Ω(d)={(x,t); 0⩽x⩽L, 0⩽t⩽d } . Since the approximation involves only a finite number of exact eigenvalues λ i 2, 1⩽i⩽n 0 , the admissible error for the approximated eigenvalues λ ̃ i 2, 1⩽i⩽n 0 , is determined in order to construct an analytical numerical solution of the mixed problem, involving only approximated eigenvalues λ ̃ i 2 , so that the total error is less than ε uniformly in Ω(d). Uniqueness of solutions is also treated.