Solving differential equations
Marek Trtík
PA199

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uInitial value problem for ordinary differential equations.
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uForward Euler’s method.
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uBackward Euler’s method.
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uMidpoint method.
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uRunge-Kutta methods.
Outline

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Initial value problem

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Initial value problem


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Initial value problem


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Direction field
Step 1
Step 2
Step 3
Draw more arrows.
Step 4
Predict solutions.

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Numerical solution


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Taylor theorem


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uExamples (Taylor approximation):
Numerical solution

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Forward Euler’s method


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Forward Euler’s method


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Forward Euler’s method


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uInstability issue:
u The iteration process may diverge.
uExample:
Forward Euler’s method

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Forward Euler’s method


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Forward Euler’s method


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Backward Euler’s method

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Backward Euler’s method


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Backward Euler’s method


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Backward Euler’s method


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uWe can plot out result and compare it with forward Euler’s method:
Backward Euler’s method

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Backward Euler’s method


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uWe can plot out result and compare it with forward Euler’s method:
Backward Euler’s method

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Backward Euler’s method


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u
Midpoint method

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Midpoint method
(*)

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Midpoint method
(*)

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Midpoint method


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Midpoint method


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Midpoint method


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Midpoint method


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Midpoint method


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u
Runge-Kutta methods

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Runge-Kutta methods


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Runge-Kutta methods


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Runge-Kutta methods


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Runge-Kutta methods


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Runge-Kutta methods


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Runge-Kutta methods


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Runge-Kutta methods


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Schema of numerical methods
Picture source: [2]

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u[1] A. Witkin, D. Baraff; Differential Equation Basics; Physically Based Modeling: Principles and
Practice, 1997
u[2] J.C.Butcher; Numerical methods for ordinary differential equations; 3rd edition, Wiley, 2016.
u[3] https://tutorial.math.lamar.edu/Classes/DE/DE.aspx
References