Slope Fields and Numerical Solutions

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Slope Fields and Numerical Solutions

• MATH 224

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First-Order Differential Equations

• Form of DE 1st order
• Ideal goal find functions y(t) that satisfy the
  differential equation
• Last class slope fields give rough shape of
  curves
• Can we connect the dots to make a curve?

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Temperature Example

• Return to heating and cooling example

• Look at particular k, Te values

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Slope Field
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Wait isn't this easy???
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Continued
Always check your answer against the original DE!
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Modeling

• Writing down model DEs can be almost trivial is
  some cases
• See some assignment problems
• Solving those same DEs to find an exact formula
  for y(t) prediction can be difficult or
  impossible (no closed form)
• Today, we find numerical approximations to the
  exact y(t) functions

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Back to Slope Fields

• Slope fields give us an idea for a computer-based
  approximation strategy
• Computer can compute slopes at any (t,y)
• Why not approximate curved solution by straight
  line segments, based on calculated slopes?

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Euler's method

• Pick starting point, (t, y), and step size, delta
  x
• Find slope there
• sub (t, y) into dy/dt formula
• Follow that slope for delta x
• Treat new point as new starting point
• Repeat

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Graphically
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Slope Field vs Estimated Curve

• Both use dy/dt . formula
• For slope field, compute slopes on a grid to see
  overall shape of solution family
• For Euler's method, compute slopes only along a
  path to estimate a particular solution

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Euler's method limitations

• Accuracy of estimate depends on step size
• smaller steps means more accurate
• but longer to compute
• Euler's method needs very small steps to be
  accurate
• Can be generalized to more efficient algorithms
• Runge-Kutta (Section 2.6)
• We'll jump straight to the good stuff!

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Adaptive Runge-Kutta Solvers

• MATLAB has many built in ODE 'solvers'
• don't give formulas, just give a predicted curve
  based on DE and a starting point
• predicted solution still has numerical errors in
  addition to any assumptions, simplifications in
  model
• We'll use ode45 (general-purpose solver)

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Using ode45

• You need
• a function that represents the f(t,y) in

• an initial y value, y0 where you'll start the
  solution
• an interval of time you want to predict over,
  t0, tn
• an error tolerance you'll accept in the estimated
  final y value, y(tn)

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Example
y0 30 t0 0 tn 48 Define the function
dy/dt f _at_(t, y) -0.05(y5) run the
solver to generate approximate temperature
curve t, y ode45(f, t0, tn, y0, 1e-5)
the actual predicted points, and lines connecting
them plot(t, y, '.-k', 'markersize',
15) title('ODE 45 Prediction for Temp DE')
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Graph over Slope Field
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Numerical Solutions (the good)

• Single solution curve is easy to generate
• relatively simple MATLAB coding required
• syntax is very picky don't expect to guess at it
  and have it work
• Can generate multiple curves by looping over
  different starting points
• We'll see how same approach works with more than
  just single 1st order DEs

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Numerical Solutions (the not so good)

• Solution found is not general
• can only pick one starting point, not find
  general family of solutions
• Solution is not a function
• it's a list of points that approximate the
  function
• can't take limits, derivatives, modify
  parameters
• Still, numerical solutions are
• relatively easy to generate
• way better than nothing

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Before next week

• Try that solving technique on some of the DEs you
  create in the assignment, to see if
• can generalize MATLAB to other examples
• predictions make sense in the problem
• creating DEs is easy
• numerical solving is easy (with MATLAB), but
  limited
• Rest of the course solving analytically