Numerical techniques for ordinary differential equations I
Numerical techniques for ordinary differential equations I
Numerical techniques for ordinary differential equations I
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Advanced Mathematical Methods<br />
<strong>for</strong> Civil Engineering Applications<br />
Ordinary Differential Equations (ODEs)<br />
(Section I)<br />
Wonsiri Punurai, PhD<br />
Department of Civil Engineering<br />
Room 6391, EG Building 3<br />
Faculty of Engineering, Mahidol University<br />
Class Web: www.egmu.net/~civil/wonsiri
Outline<br />
• Basic Ideas<br />
• Approximations and Errors Review<br />
• Taylor Series and Order Notation Review<br />
• Euler’s Method<br />
• Predictor-Corrector Approach<br />
• Runge-Kutta Method<br />
• Adaptive step size Algorithm
Basic ideas - ODEs in Civil Engineering
Basic ideas - ODEs in Civil Engineering
Approximations and Errors Review
Outline<br />
• Basic Ideas<br />
• Approximations and Errors Review<br />
• Taylor Series and Order Notation Review<br />
• Euler’s Method<br />
• Predictor-Corrector Approach<br />
• Runge-Kutta Method<br />
• Adaptive step size Algorithm
Use of significant figures
Round-off errors
Approximations and Errors Review
Error Definition
Example – error estimates <strong>for</strong> iterative methods<br />
See more in later
Example – error estimates <strong>for</strong> iterative methods
Effect of order of operations
Taylor Series
Outline<br />
• Basic Ideas<br />
• Approximations and Errors Review<br />
• Taylor Series and Order Notation Review<br />
• Euler’s Method<br />
• Predictor-Corrector Approach<br />
• Runge-Kutta Method<br />
• Adaptive step size Algorithm
Taylor Series
Taylor series of a function
Taylor series expansions <strong>for</strong> some common functions
MATLAB example – evaluating the series <strong>for</strong> sin(x)
Order Notation
Outline<br />
• Basic Ideas<br />
• Approximations and Errors Review<br />
• Taylor Series and Order Notation Review<br />
• Euler’s Method<br />
• Predictor-Corrector Approach<br />
• Runge-Kutta Method<br />
• Adaptive step size Algorithm
Euler’s Method
Example - Hand Calculation with Euler’s Method
Implementing Euler’s Method in MATLAB
Solving Previous Example using MATLAB
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