The purpose of this paper is to develop the shooting method as a technique for approximating the solution to the two-point boundary value problem on the interval [a,b] with the even order differential equation {i.e. n is even)
u(n)(t) + f(t, u(t), u(i)(t, ),..., u(n-1)(t)) = 0
and boundary conditions
u(a) = A
u(b) = B
and with at most n-2 other boundary conditions specified at either a or b. The basic proceedure will be illustrated by the following example.
Consider the two-point boundary value problem (0.1) (0.2) (0.3) with the additional boundary conditions
u(i)(a) = mi
for i = 1, ... ,k-1,k+l, ... ,n-1. The first step is to find values m1 and m2 such that the solutions or "shots", u1(t) and u2(t), to (0.1) that satisfy the initial conditions
u(a) = A
u(k-1)(a)= mk-1
u(k)(a) = m1
u(k+1)(a) = mk+l
u(n-1)(a) = mn-1
with 1 = 1,2, respectively, with the property that
u1(b) < B < u2(b).
The interval [m1,m2] is then searched by seccussive bisection to find the value, m, such that the solution or "shot", u(t), to the initial value problem with (0.1) and initial conditions
u(a) = A
u(k-1)(a)= mk-1
u(k)(a) = m1
u(k+1)(a) = mk+l
u(n-1)(a) = mn-1
has the property that u(b) = B.
| Identifer | oai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-8048 |
| Date | 01 May 1976 |
| Creators | Baumann, John D. |
| Publisher | DigitalCommons@USU |
| Source Sets | Utah State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | All Graduate Theses and Dissertations |
| Rights | Copyright for this work is held by the author. Transmission or reproduction of materials protected by copyright beyond that allowed by fair use requires the written permission of the copyright owners. Works not in the public domain cannot be commercially exploited without permission of the copyright owner. Responsibility for any use rests exclusively with the user. For more information contact digitalcommons@usu.edu. |
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