Ordinary Differential Equations

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Ordinary Differential Equations
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Ordinary Differential Equations

• Example

x independent variable y dependent
variable First order ODE, dependent variable
function of one independent variable

• Other examples

Pendulum
Vibrating mass on spring

• ODEs can be used to model behavior

• Higher order ODEs can always be expressed as 1st
  order

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Numerical Solution of ODE Initial Value Problems

• Basic Concepts
• Explicit - Implicit (iterative)
• One-step (self starting) - Multi-step (non
  self-starting)
• Order
• Stability
• Basic methods to know
• Euler (simplest explicit)
• Runge-Kutta 4 (most often used RK method)
• Heun
• Iterated Heun (simplest predictor-corrector)

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Solutions
y

• General solutions or level curves

x
e.g.

• Particular solution specified by initial value

• In general

• Separable ODEs

So
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Solutions

• Closed form solutions

e.g.
where
Not a closed form solution
gives

• What should we do when no suitable closed form
  solution exists?

Answer form a numerical model of the ODE to give
approximate y values at a number of x points.

• Advantages of a numerical model

Can handle any ODE, quickly generate solutions
without extensive math, good for solving on
computers.

• Disadvantages of a numerical model

Stability (solution can be badly behaved thus,
inaccurate), accuracy (can be well-behaved but
inaccurate).
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Numerical Solutions

• Initial value problems

• Step by step methods

Use information from current position, not future

• General solution stepping from

to
is
know
dont know
?

• Modelling challenge

Represent or approximate
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Eulers Method

• Assume that from

to

• Integration becomes

so that
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Eulers Method

• Geometric interpretation

f(x,y)
y
x
polygon approx to y and constant approximation
to f(x,y)
x

• Reasonable model/approximation if y is smooth
  and not
• varying fast.

• A sequence of steps gives the solution at
  discrete points.

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Eulers Method
Explicit One-step First Order Stability (depends
very much on f and the step size)
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Heun (without iteration)

• Predictor - Corrector (but still explicit)
• One-step
• Second order
• Stability (depends on the step size, better than
  Euler)

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Iterated Heun
Predictor - Corrector (implicit,
iterative) Needs a stopping criterion One-step Sec
ond order Stability is good
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Runge-Kutta 4(RK4)
Explicit One-step Fourth Order !!! Stability
depends somewhat on f and step size)
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Final comment

• Using smaller values of h and using Eulers or
  Heuns methods probably gives better accuracy
  than using RK4 with larger values of h