Bounds for Initial Value Problems

[+] Author and Article Information
C. A. Bell

Department of Mechanical Engineering, Texas Tech University, Lubbock, Texas

F. C. Appl

Department of Mechanical Engineering, Kansas State University, Manhattan, Kans.

J. Appl. Mech 40(4), 1097-1102 (Dec 01, 1973) (6 pages) doi:10.1115/1.3423132 History: Received April 01, 1972; Revised April 01, 1973; Online July 12, 2010


A new method has been developed for finding rigorous upper and lower bounds to the solution of a wide class of initial value problems. The method is applicable to initial value problems of the following type:

ẍ(t) + f(t, x, ẋ) = 0,
 x(0) = X0,
 ẋ(0) = V0,
where f is continuous with continuous first derivatives, Lipschitzian, and ∂f/∂x ≥ 0. An original bounding theorem has been formulated and proven and a numerical technique has been developed for finding the bounding functions in analytic form as linear combinations of Tchebyshev polynomials. The method has been applied to several problems of engineering interest.

Copyright © 1973 by ASME
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