Lecture 32 Systems of Ordinary Differential Equations IVP

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Lecture 32 - Systems of Ordinary Differential
Equations - IVP

• CVEN 302
• November 14, 2001

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Systems of Ordinary Differential Equations - IVP

• Higher order differential equations
• Systems of equations
• Runge Kutta
• Adam Bashforth
• Adam Moulton
• Stiff Equations

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Systems of ODE - Initial Value Problems
These techniques can work on large systems of
equations to do series of integration of the
problem. The equations can be solved as a series
of ODEs.
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Systems of ODE - Initial Value Problems
Given a set of initial values, y1,y2,y1 and y2.
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Systems of ODE - Initial Value Problems
The problem is constructed of 4 first order ODEs
with four variables and initial conditions.
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Systems of ODE - Initial Value Problems
The problem can be written in matrix format and
solved accordingly.
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Systems of ODE - Initial Value Problems
Forcing functions can be added and setup to solve
the equations.
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Stiff Differential Equation
All the methods for approximating the solution
to initial value problems have error terms that
involve higher order derivative of the solution.
If the derivatives can be reasonably bounded, the
method will give a predicable error bound that
can be used to estimate the accuracy of the
technique.
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Stiff Differential Equation
Problems arise when the magnitude of the
derivative increases but the solution does not.
The error will grow so large and it will dominate
the calculation. For initial value problems for
which this is likely to occur are called Stiff
Equations. Stiff equations are characterized as
those whose exact solution have an exponential
and the higher order derivatives do not decay
quickly.
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1-D Stiffness Equation
Use a step size of Dt 0.1 and y(0) 1/3.
Exact solution is y (1/3) exp(-30t).
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Summary

• Systems of ODEs
• Runge Kutta
• Adam Bashforth
• Predictor Corrector
• Stiff Differential Equations

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Homework

• Check the Homework webpage