SHOOTING METHOD INTERACTIVE E-BOOK

Ordinary Differential Equations

MAJOR GENERAL

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Title	An interactive e-book for illustrating Shooting Method of solving ordinary differential equations
Creator	Autar K Kaw
Subject and Keywords	Shooting Method, Ordinary Differential Equations, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	This is an interactive E-book for illustrating the Shooting Method for solving ordinary differential equations. It includes links to examples, simulations in Mathcad, Maple, Mathematica, and Matlab for the algorithm, and a PowerPoint presentation.
Publisher	Holistic Numerical Methods Institute, College of Engineering, University of South Florida, Tampa, FL 33620-5350.
Contributors	Autar Kaw
Format	Text/HTML
Last Revised	May 4, 2006
Identifier	http://numericalmethods.eng.usf.edu/ebooks/Shooting_ode_ebook.pdf
Language	English
Rights	http://numericalmethods.eng.usf.edu/rights.htm

Table of Contents

Background

What are ordinary differential equations?

A Primer on Ordinary Differential Equations

Method

Shooting Method for Ordinary Differential Equations

Presentation

PowerPoint presentation

Example

Cantilever Beam Problem

Simulation of the Shooting Method

[MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

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Background

This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website.

Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose.

Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here.

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Shooting Method for Ordinary Differential Equations

Ordinary differential equations are given either with initial conditions or with boundary conditions. Look at the problem below.

Figure 1. A cantilevered uniformly loaded beam

In this case to find the deflection as a function of location, due to a uniform load q, the ordinary differential equation needed to solve is

(1)

where

L is the length of the beam,

E is the Young’s modulus of the beam,

I is the second moment of area of the cross-section of the beam.

Two conditions are needed to solve the problem, those are

(2a,b)

as it is a cantilevered beam at . These conditions are initial condition as they are given at an initial point, so that we can find the deflection along the length of the beam.

Now see a similar beam problem, where the beam is simply supported on the two ends

Figure 2. A simply supported uniformly loaded beam

In this case to find the deflection, as a function of, due to the uniform load, q the ordinary differential equation needed to solve is

(3)

Two conditions are needed to solve the problem, those are,

(4a,b)

As it is a simply supported beam at and . These conditions are boundary conditions as they are given at the two boundaries, and .

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The Shooting Method

The shooting method uses the methods used in solving initial value problems. This is done by assuming initial values that would have been given if the ordinary differential equation were a initial value problem. The boundary value obtained is compared with the actual boundary value. Using trial and error or some scientific approach, one tries to get as close to the boundary value as possible. This method is best explained by an example.

Figure 3. Cross-sectional geometry of a pressure vessel

Take the case of a pressure vessel that is being tested in the laboratory to check its ability to withstand pressure. For a thick pressure vessel of inner radius a and outer radius b, the differential equation for the radial displacement u of a point along the thickness is given by

(5)

Assume that the inner radius a= 5’’ and the outer radius b= 8’’, and the material of the pressure vessel is ASTM36 steel. The ultimate strength of this type of steel is 36 ksi. Two strain gages that are bonded tangentially at the inner and the outer radius measure normal tangential strain as

(6a, b)

at the maximum needed pressure. Since the radial displacement and tangential strain are related simply by

, (7)

then

(8)

Starting with the ordinary differential equation

Let

(9)

Then

(10)

giving us two first order differential equations as

″

(11a,b)

Let us assume

To set up initial value problem.

(12a,b)

Using Euler’s method,

(13a,b)

Let us consider 4 segments between the two boundaries, ″and ″, then

″

″

″

″

So at we have

While the given value of this boundary condition is

Let us assume a new value for . Based on the first assumed value, may be using twice the value of initial guess.

Using , and Euler’s method, we get

While the given value of this boundary condition is

Can we use the results obtained from the two previous iterations to get a better estimate of the assumed initial condition of . One method is to use linear interpolation on the obtained data for the two assumed values of .

With we obtained, and

with we obtained

so a better starting value of knowing that the actual value at , we get

Using , and repeating the Euler’s method with w(5), we get

while the actual given value of this boundary condition is

.

In this case, this value coincides with the actual value of . If that were not the case, one would continue to use linear interpolation to refine the value of till one gets close to the actual value of . Note that the step size and numerical method used would influence the accuracy for the obtained values. For the last case, the values are as follows

″

″

″

″

″