The method of least squares for boundary value problems

Abstract

The method of least squares is used to construct approximate solutions to the boundary value problem if=go, B,(f) = 0 for i=1, ...,k, on the interval [a, b], where r is an nth order formal differential operator, go(t) is a given function in L2[a, b], and B1, ..., Bk are linearly independent boundary values. Letting Hn[a, b] denote the space of all functions f(t) in C -'[a, b] with f(n -1) absolutely continuous on [a, b] andftn) in L2[a, b], a sequence of functions e1(t) (i= 1, 2, .. .) in Hn[a, b] is constructed satisfying the boundary conditions and a completeness condition. Assuming the boundary value problem has a solution, the approximate solutions 1 a'e(t) (i = 1, 2, ...) are constructed; the coefficients a' are determined uniquely from the system of equations 2 (Te, rel)a' = (go, e), I = 1.1, where (f, g) denotes the inner product in L2[a, b]. The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.