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

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.

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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