St. John's University of Tanzania

MAT210 NUMERICAL ANALYSIS2013/14 Semester II

DIFFERENTIAL EQUATIONSBoundary Value Problems

Kaw, Chapter 8.06-8.07Some parts of this presentation are based on resources at

http://nm.MathForCollege.com, primarily http://http://mathforcollege.com/nm/mtl/gen/08ode/index.html

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Ordinary Differential Equations● Topics

● 1st order ODE– Euler's Method – Runge-Kutta Methods

● Higher order Initial Value● Higher order Boundary Value

– Shooting Method– Finite Differences

Today's discussion

….Read Kaw 8.05

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Beyond First Order● 2nd order ODE's

● Require 2 conditions– Two initial: y(x0) and y'(x0) – still an IVP

– Two boundaries: y(x0) and y(x1) – now a BVP

● Kaw 8.05 describes how to decompose the 2nd order IVP into two 1st order IVPs and use Euler or Runge-Kutta to solve

● Boundary Value Problems require something more– The Shooting Method extension of the IVP techniques– The Finite Difference Method

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IVP versus BVP: The IVP

● To find the deflection υ as a function of location x, due to a uniform load q, the ordinary differential equation that needs to be solved is

subject to two initial conditions:

d2νd x2

=q

2EI(L−x)2

ν(0)=0, ν '(0)=0

Beam with● Young's elastic modulus E● 2nd moment of inertia of the

cross-section of the beam I● Support on one end only

Solve as a pair of 1st order IVPs

No problem

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The BVP

● To find the deflection υ as a function of location x, due to a uniform load q, the ordinary differential equation becomes

but subject to boundary conditions:

d2νd x2

=q x2EI

(x−L)

ν(0)=0, ν(L)=0

Beam is now supported on both ends, making it a boundary value problem.

Not as simple as just progressing from 0 to L

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The Shooting Method● Why not approach the BVP using the IVP

techniques, replacing one boundary condition with a “guess” of another initial condition that causes the IVP solution to “hit” the other boundary condition?● Progressively refine the guess until it hits● Refinement could be through interpolation

– Aha, a combination of techniques to reach the desired objective

● This is the Shooting Method

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Summary of the Method● Use y(x0) and a reasonable guess for y'(x0)

● Usually y'(x0)≈(y(x1)-y(x0))/(x1-x0) … FDA

● Use Runge-Kutta to “shoot” to an approximation for y(x1)● If close enough, then stop, solution is done● If not, pick another y(x0) and repeat Runge-Kutta

to get a second approximation for y(x1)

● Now use interpolation for 3rd guess● Repeat the process until solution “hits” y(x1)

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More on the Method● Read the example

in Kaw Ch 8.06● Summarize the

steps to produce an algorithm

● Then it will make more sense

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The Finite Difference Method● Why guess and correct?

Why not feed the information from the other boundary condition back through the intervals?

● That is the heart of the Finite Difference (FD) Method● Create a set of problems where the

subintervals share a boundary condition● End up with a linear algebra problem

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An Exampled2 νd x2

−T yEI

=q x2EI

(x−L)

ν(0)=0, ν(L)=0

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Approximating the derivatives

● Now apply the equation to each of the interior nodes (2 & 3) and the boundary conditions to end nodes (1 & 4)

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Node equations

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Solve in matrix form

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Examining the errorThe exact solution:

The error calculation:

All in one step● Use a fast matrix

inversion method

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Finite Difference Errors● Truncation errors (aka Discretization Errors)

● The approximation of the derivatives has an error, e.g. O(h) ● Applying Fast Fourier Transforms can help, eg. spectral methods

Understand the problem, do some analysis, improve the discretization

● Rounding errors: ● The loss of precision due to computer rounding of decimal

quantities● Example, standard arithmetic is O(10-16/h)

● Total error: Best accuracy is usually obtained when these different error types match

– O(h) matches O(10 16− /h) when h≈10 8− , producing a total error around 10 8− . – Higher derivatives: pth derivative rounding error typically O(10 16− /hp)

● For this reason rarely use FD for derivatives beyond the third or fourth

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Error versus discretization order

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Structure of the FD matrix1st order

2nd order

Always tridiagonal – can use fast matrix solvers

– Thomas' algorithm

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Summary● Finite difference (FD) methods can be used

for partial differential equations, too● There are a wide range of FD methods,

which use different approximations for the derivatives, different matrix techniques, different strategies for creating intervals or grids/meshes for PDEs

● More at http://www.scholarpedia.org/article/Finite_difference_method