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Transcript of MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 10. Ordinary differential equations. Initial value...
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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: Newton’s second law
u1 = solution y of the original equation of 2nd orderu2 = 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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Euler’s 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: Euler’s 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.
![Page 55: MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 10. Ordinary differential equations. Initial value problems.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d9c5503460f94a85240/html5/thumbnails/55.jpg)
Example (cont.)
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Example (cont.)
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Example (cont.)
![Page 59: MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 10. Ordinary differential equations. Initial value problems.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649d9c5503460f94a85240/html5/thumbnails/59.jpg)
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