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 for , on the interval [*a, b*], where is an *n*th order formal differential operator, is a given function in , and are linearly independent boundary values. Letting denote the space of all functions in with absolutely continuous on [*a, b*] and in , a sequence of functions in is constructed satisfying the boundary conditions and a completeness condition. Assuming the boundary value problem has a solution, the approximate solutions are constructed; the coefficients are determined uniquely from the system of equations

*f, g*) denotes the inner product in . The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.

**[1]**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**(1966/1967), 394–430. MR**0221761**, https://doi.org/10.1007/BF02162155**[2]**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**0229391**, https://doi.org/10.1007/BF02166686**[3]**Nelson Dunford and Jacob T. Schwartz,*Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space*, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR**0188745****[4]**John Locker,*An existence analysis for nonlinear boundary value problems*, SIAM J. Appl. Math.**19**(1970), 199–207. MR**0265669**, https://doi.org/10.1137/0119018**[5]**John Locker,*An existence analysis for nonlinear equations in Hilbert space*, Trans. Amer. Math. Soc.**128**(1967), 403–413. MR**0215142**, https://doi.org/10.1090/S0002-9947-1967-0215142-9**[6]**S. G. Mikhlin,*Variational methods in mathematical physics*, Translated by T. Boddington; editorial introduction by L. I. G. Chambers. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0172493****[7]**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****[8]**S. G. Mihlin and H. L. Smolickiĭ,*\cyr Priblizhennye metody resheniya differentsial′nykh i integral′nykh uravneniĭ*, Izdat. “Nauka”, Moscow, 1965 (Russian). MR**0192630****[9]**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**0145651**, https://doi.org/10.1090/S0002-9947-1962-0145651-8**[10]**J. Schwartz,*Perturbations of spectral operators, and applications. I. Bounded perturbations*, Pacific J. Math.**4**(1954), 415–458. MR**0063568****[11]**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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Additional Information

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

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