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