Differential equations

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Differential equations There are ordinary
differential equations - functions of one
variable And there are partial differential
equations - functions of multiple variables
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Order of differential equations 1st order 2nd
order etc.
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Can always turn a higher order ode into a set of
1st order odes Example
Let
then
So solutions to 1st order are important
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Linear and nonlinear ODEs Linear No
multiplicative mixing of variables, no nonlinear
functions Nonlinear anything else
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Sometimes can linearize Example for small angles
then
which is linear
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• ODEs show up everywhere in engineering
• dynamics (Newtons 2nd law)
• heat conduction (Fouriers law)
• diffusion (Ficks law)

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• Were going to cover
• Euler and Heun's methods
• Runge-Kutta methods
• Adaptive Runge-Kutta
• Multistep methods
• Adams-Bashforth-Moulton methods
• Boundary value problems
• Goal is to get y(x) from dy/dxf(x)

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Runge Kutta methods - one step methods Idea is
that New valueold valueslopestep size or
Slope is generally a function of x, hence
y(x) Different methods differ in how to estimate
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Eulers method Use differential equation to
estimate slope, by plugging in current values of
x and y Example let
Integrate from 1 to 7. Let h0.5. Initial
condition is y(1)1. Use f for
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Begin at x1
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• Ok, not so great
• Truncation errors
• Round off errors
• There is
• local truncation error - error from application
  at a single step
• propagated truncation error - previous errors
  carried forward
• sum is Global truncation error

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Eulers method uses Taylor series with only first
order terms Error is Neglect higher order terms
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Example -
Local error at any x
See Excel sheet
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Error can be reduced by smaller h - see Excel
sheet
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Effect of reducing step size Error vs h
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• Improvements of Eulers method - Heuns method
• derivative at beginning of interval is applied
  to entire interval
• Heuns method uses average derivative for entire
  interval

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Graph of function with slope arrows explaining
Heuns method
Average the slopes
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• Heuns method is a predictor-corrector method
• predictor

• corrector

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Example of Heuns method - see Excel first few
iterations - yHeun(0)2 (given) y0yHeun(0)f(0)h
2-5000.5-248 yHeun(0.5)yHeun(0)(f(0)f(0.5))/
20.5 2(-500-245.5)/20.5-184
.375 y0yHeun(0.5)f(0.5)h yHeun(1)yHeun(0.5)(f
(0.5)f(1))/2h