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 $\tau f = {g_0},{B_i}(f) = 0$ for $i = 1, \ldots ,k$, on the interval [a, b], where $\tau$ is an nth order formal differential operator, ${g_0}(t)$ is a given function in ${L^2}[a,b]$, and ${B_1}, \ldots ,{B_k}$ are linearly independent boundary values. Letting ${H^n}[a,b]$ denote the space of all functions $f(t)$ in ${C^{n - 1}}[a,b]$ with ${f^{(n - 1)}}$ absolutely continuous on [a, b] and ${f^{(n)}}$ in ${L^2}[a,b]$, a sequence of functions ${\xi _i}(t)\;(i = 1,2, \ldots )$ in ${H^n}[a,b]$ is constructed satisfying the boundary conditions and a completeness condition. Assuming the boundary value problem has a solution, the approximate solutions ${f_i}(t) = \Sigma _{j = 1}^ia_j^i{\xi _j}(t)\;(i = 1,2, \ldots )$ are constructed; the coefficients $a_j^i$ are determined uniquely from the system of equations \[ \sum \limits _{j = 1}^i {(\tau {\xi _j},\tau {\xi _l})a_j^i = ({g_0},\tau {\xi _l}),\quad l = 1, \ldots ,i,} \] where (f, g) denotes the inner product in ${L^2}[a,b]$. The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.