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dP = P (;c + p)
in the vector form (1.1.1). t>0 p(0) = p0 t>0 P (0) = P0 2. Write the mechanical vibration problem
m d y + c dy + ky = f (t)
dt2 dt t>0 in the form (1.1.1a-c)
8 y(0) = y0 dy(0) = y
0 3. Write the column buckling problem d4y + d2y = 0
dt2 y(0) = dy(0) = 0
dt 0<t<l d2 y(l) = 0
y(l) = dt2 in the form (1.1.1a,b,d) 1.2 Existence, Uniqueness, and Stability of IVPs
Before considering numerical procedures for (1.1.1, 1.1.3), let us review some results from
ODE theory that ensure the existence of unique solutions which depend continuously on
the initial data. For simplicity, let us focus on a scalar IVP having the form (1.1.2). De nition 1.2.1. A function f (t y) satis es a Lipschitz condition in a domain D if there exists a non-negative constant L such that
jf (t y) ; f (t z)j Ljy ; zj 8(t y) (t z) 2 D: (1.2.4) Satisfying a Lipschitz condition guarantees that solutions y(t) of (1.1.2) are unique
as expressed by the following theorem. Theorem 1.2.1. Let f (t y...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14