MATH 685 CSI 700 OR 682 Lecture Notes

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MATH 685/ CSI 700/ OR 682 Lecture Notes

• Lecture 10.
• Ordinary differential equations.
• Initial value problems.

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Differential equations

• Differential equations involve derivatives of
  unknown solution function
• Ordinary differential equation (ODE) all
  derivatives are with respect to single
  independent variable, often representing time
• Solution of differential equation is function in
  infinite-dimensional space of functions
• Numerical solution of differential equations is
  based on finite-dimensional approximation
• Differential equation is replaced by algebraic
  equation whose solution approximates that of
  given differential equation

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Order of ODE

• Order of ODE is determined by highest-order
  derivative of solution function appearing in ODE
• ODE with higher-order derivatives can be
  transformed into equivalent first-order system
• We will discuss numerical solution methods only
  for first-order ODEs
• Most ODE software is designed to solve only
  first-order equations

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Higher-order ODEs
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Example Newtons second law
u1 solution y of the original equation
of 2nd order u2 velocity y
Can solve this by methods for 1st order
equations
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ODEs
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Initial value problems
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Initial value problems
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Example
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Example (cont.)
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Stability of solutions

• Solution of ODE is
• Stable if solutions resulting from perturbations
  of initial value remain close to original
  solution
• Asymptotically stable if solutions resulting from
  perturbations converge back to original solution
• Unstable if solutions resulting from
  perturbations diverge away from original solution
  without bound

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Example stable solutions
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Example asymptotically stable solutions
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Example stability of solutions
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Example linear systems of ODEs
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Stability of solutions
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Stability of solutions
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Numerical solution of ODEs
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Numerical solution to ODEs
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Eulers method
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Example
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Example (cont.)
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Example (cont.)
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Example (cont.)
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Numerical errors in ODE solution
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Global and local error
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Global vs. local error
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Global vs. local error
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Global vs. local error
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Order of accuracy. Stability
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Determining stability/accuracy
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Example Eulers method
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Example (cont.)
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Example (cont.)
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Example (cont.)
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Example (cont.)
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Stability in ODE, in general
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Step size selection
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Step size selection
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Implicit methods
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Implicit methods, cont.
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Backward Euler method
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Implicit methods
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Backward Euler method
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Backward Euler method
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Unconditionally stable methods
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Trapezoid method
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Trapezoid method
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Implicit methods
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Stiff differential equations
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Stiff ODEs
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Stiff ODEs
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Example
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Example, cont.
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Example (cont.)
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Example (cont.)
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Example (cont.)
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Numerical methods for ODEs
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Taylor series methods
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Taylor series 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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Extrapolation methods
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Multistep methods
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Multistep methods
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Multistep methods
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Example of multistep methods
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Example (cont.)
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Multistep Adams methods
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Properties of multistep methods
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Properties of multistep methods
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Multivalue methods
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Multivalue methods
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Example
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Example (cont.)
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Example (cont.)
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Example (cont.)
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Multivalue methods, cont.
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Variable-order/Variable-step methods