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Regularization for \(n\)th-order linear boundary value problems using \(m\)th-order differential operators. (English) Zbl 0594.47008

Let \(X\) and \(Y\) denote real Hilbert spaces, and let \(L: X\to Y\) be a closed densely-defined linear operator having closed range. Given an element \(y\in Y\), we determine least squares solutions of the linear equation \(Lx=y\) by using the method of regularization. Let \(Z\) be a third Hilbert space, and let \(T: X\to Z\) be a linear operator with \({\mathcal D}(L)\subseteq {\mathcal D}(T)\). Under suitable conditions on \(L\) and \(T\) and for each \(\alpha\) \(\neq 0\), we show that there exists a unique element \(x_{\alpha}\in {\mathcal D}(L)\) which minimizes the functional \(G_{\alpha}(x)=\| Lx-y\|^ 2+\alpha^ 2\| Tx\|^ 2\), and the \(x_{\alpha}\) converge to a least squares solution \(x_ 0\) of \(Lx=y\) as \(\alpha\) \(\to 0.\)
We apply our results to the special case where \(L\) is an \(n\)th-order differential operator in \(X=L^ 2[a,b]\), and we regularize using for \(T\) an mth-order differential operator in \(L^ 2[a,b]\) with \(m\leq n\). Using an approximating space of Hermite splines, we construct numerical solutions to \(Lx=y\) by the method of continuous least squares and the method of discrete least squares.

MSC:

47A50 Equations and inequalities involving linear operators, with vector unknowns
41A15 Spline approximation
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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