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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A uniqueness theorem for a boundary value problem


Author: Riaz A. Usmani
Journal: Proc. Amer. Math. Soc. 77 (1979), 329-335
MSC: Primary 34B05; Secondary 65L10
DOI: https://doi.org/10.1090/S0002-9939-1979-0545591-4
MathSciNet review: 545591
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Abstract: In this paper it is proved that the two-point boundary value problem, namely $ ({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y''(0) - {B_1} = y''(1) - {B_2} = 0$, has a unique solution provided $ {\inf _x}f(x) = - \eta > - {\pi ^4}$. The given boundary value problem is discretized by a finite difference scheme. This numerical approximation is proved to be a second order convergent process by establishing an error bound using the $ {L_2}$-norm of a vector.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0545591-4
Keywords: Brouwer's fixed point theorem, Euclidean and $ {L_2}$-norm of a vector, finite difference scheme, local truncation error, plate deflection theory, posteriori error bound, spectral norm of a matrix, two-point boundary value problem
Article copyright: © Copyright 1979 American Mathematical Society