## All Graduate Theses and Dissertations

5-1976

Thesis

#### Degree Name:

Master of Science (MS)

#### Department:

Mathematics and Statistics

Mathematics

Robert Gunderson

Robert Gunderson

#### Abstract

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.

#### Checksum

b0c84b68d359b5760b942cbbe0a3a811

COinS

#### DOI

https://doi.org/10.26076/c8b1-81c7