Lecture 23 Systems of Ordinary Differential Equations IVP

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Lecture 23 - Systems of Ordinary Differential
Equations - IVP

• CVEN 302
• August 2, 2002

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Systems of Ordinary Differential Equations - IVP

• Higher order differential equations
• Systems of equations
• Taylor series
• Euler Method
• Runge Kutta
• Stiff Equations

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Higher order differential equations
Higher order initial value problems can be
transformed from a higher order equation into a
system of first-order differential equations.
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Higher order differential equations
Rewrite the equation so x is form
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Higher-order Differential Equations
Rewrite the equations in the form of series of
first order differential equations
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Higher-order Differential Equations
Consider a simple second order differential
equation for a vibrating spring mass
system. The initial conditions are x(0) x0
and x(0) 0.
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Higher-order Differential Equations
Rewrite the equation The first derivative can
be written as
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Higher-order Differential Equations
The equation can be written as a set of two first
order equations. The boundary conditions,
x(0) x0 and v(0) 0.
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Higher-order Differential Equations
The first order equations can be solved using the
techniques we developed in one dimensional
initial value problems.

• Taylor Series
• Euler
• Runge-Kutta
• Adam Bashforth

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System of Initial Value Problems
The equations can be defined as
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System of Initial Value Problems
The single step, Euler
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Higher-order Differential Example Problem
Consider a simple second order differential
equation for a vibrating spring mass
system. The initial conditions are x(0) 0.2,
x(0) 0 and Dt 0.02. (Exact solution 0.2
cos(2t))
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Example Problem
The equation can be written as a set of two first
order equations. The boundary conditions,
x(0) 0.2 and v(0) 0.
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Example Problem
The breakdown of the Euler method.
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Example Problem

• The example
• There is problem with the step sizes causing an
  instability.

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Example Problem
The problem is that an instability is created by
the delay in the update of the velocity step. An
alternative is to use a more accurate scheme such
as the fourth order Runge Kutta.
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Example Problem
The equations are defined as functions. The
boundary conditions, x(0) 0.2 and v(0) 0.
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Example Problem
The components of the Runge-Kutta ki,j
where i is the step and j is the function.
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Example Problem
The upgraded single step Use the initial
values x(0) 0.02 and v(0) 0
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4th Order Runge-Kutta Method Example
The values for the step size where,
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4th Order Runge-Kutta Method Example

• The points are not delayed as the Euler method.
• Accuracy is dependent on the step size of the
  problem.

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Multi-step Methods
The principle behind a multi-step method is to
use past values, t, y , dn-1y/dxn-1 to construct
a polynomial that approximate the derivative
function.
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Multi-step Methods
The three point Adam Bashforth
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Multi-step Methods
These methods are known as explicit schemes
because the use of current and past values are
used to obtain the future step. The method is
initiated by using either a set of know results
or from the results of a Runge-Kutta to start the
initial value problem.
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Example Problem
The equations are defined as functions. The
boundary conditions, x(0) 0.2 and v(0) 0.
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Example Problem
The equations are defined as functions..
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Adam Bashforth 3 Point Example
The values for the step size
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Adam Bashforth 3 Point Example

• The program is initialized by a 4th order
  Runge-Kutta.
• Accuracy is dependent on the step size of the
  problem.

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Implicit Multi-step Methods
The method uses what is known as a
Predictor-Corrector technique. It uses the
explicit scheme to estimate the initial guess and
uses the value to guess the future y and dy/dx
f(x,y) values. Using these results, the Adam
Moulton method can be applied.
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Implicit Multi-step Methods
Adams third order Predictor-Corrector
scheme. Use the Adam Bashforth three point
explicit scheme for the initial guess.
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Implicit Multi-step Methods
Adams third order Predictor-Corrector
scheme. Use the Adam Moulton three point
implicit scheme to take a second step.
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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
• Taylor Series
• Euler
• Runge Kutta
• Adam Bashforth
• Predictor Corrector
• Stiff Differential Equations

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Homework

• Check the homework webpage