The method of least squares for boundary value problems

Author:
John Locker

Journal:
Trans. Amer. Math. Soc. **154** (1971), 57-68

MSC:
Primary 65.62

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281359-1

MathSciNet review:
0281359

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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 *n*th 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.

- P. G. Ciarlet, M. H. Schultz, and R. S. Varga,
*Numerical methods of high-order accuracy for nonlinear boundary value problems. I. One dimensional problem*, Numer. Math.**9**(1966/67), 394–430. MR**221761**, DOI https://doi.org/10.1007/BF02162155 - P. G. Ciarlet, M. H. Schultz, and R. S. Varga,
*Numerical methods of high-order accuracy for nonlinear boundary value problems. II. Nonlinear boundary conditions*, Numer. Math.**11**(1968), 331–345. MR**229391**, DOI https://doi.org/10.1007/BF02166686 - Nelson Dunford and Jacob T. Schwartz,
*Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space*, Interscience Publishers John Wiley & Sons New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR**0188745** - John Locker,
*An existence analysis for nonlinear boundary value problems*, SIAM J. Appl. Math.**19**(1970), 199–207. MR**265669**, DOI https://doi.org/10.1137/0119018 - John Locker,
*An existence analysis for nonlinear equations in Hilbert space*, Trans. Amer. Math. Soc.**128**(1967), 403–413. MR**215142**, DOI https://doi.org/10.1090/S0002-9947-1967-0215142-9 - S. G. Mikhlin,
*Variational methods in mathematical physics*, The Macmillan Co., New York, 1964. Translated by T. Boddington; editorial introduction by L. I. G. Chambers; A Pergamon Press Book. MR**0172493** - S. G. Mihlin,
*Variational methods of solving linear and non-linear boundary value problems*, Differential Equations and Their Applications (Proc. Conf., Prague, 1962), Publ. House Czechoslovak Acad. Sci., Prague; Academic Press, New York, 1963, pp. 77–92. MR**0170132** - S. G. Mihlin and H. L. Smolickiĭ,
*Priblizhennye metody resheniya differentsial′nykh i integral′nykh uravneniĭ*, Izdat. “Nauka”, Moscow, 1965 (Russian). MR**0192630** - W. V. Petryshyn,
*Direct and iterative methods for the solution of linear operator equations in Hilbert space*, Trans. Amer. Math. Soc.**105**(1962), 136–175. MR**145651**, DOI https://doi.org/10.1090/S0002-9947-1962-0145651-8 - J. Schwartz,
*Perturbations of spectral operators, and applications. I. Bounded perturbations*, Pacific J. Math.**4**(1954), 415–458. MR**63568** - Richard S. Varga,
*Hermite interpolation-type Ritz methods for two-point boundary value problems*, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 365–373. MR**0205475**

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Keywords:
Method of least squares,
boundary value problems,
formal differential operator,
boundary values,
differential operator in Hilbert space,
equivalent norms,
complete sequences,
Gram-Schmidt process,
nonhomogeneous boundary conditions

Article copyright:
© Copyright 1971
American Mathematical Society