review2_compiled - Review for Exam 2 Ben Wang and Mark...

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Review for Exam 2 Ben Wang and Mark Styczynski This is a rough approximation of what we went over in the review session. This is actually more detailed in portions than what we went over. Also, please note that this has not been proofread fully, so it may have typos or other things slightly wrong with it, but it’s a good guide towards what you should know about. Finally, be familiar with what is in your book so that if there’s something on the exam that you don’t know well (e.g., LU decomposition from the last exam), you at least know where to look for it. Chapter 4: IVPs Objective: develop iterative rules for updating trajectory with the objective of getting numerical solution to equal to the integration of the exact solution Converting higher-order differential equations into systems of coupled first-order differential equations. What is quadrature? Numerical integration will involve ‘integrating’ a polynomial Æ that is why polynomial interpolation is important Polynomial interpolation : Polynomial interpolation is accomplished by definining N support point which the polynomial will equal the function at. We define the polynomial: N N o x a x a x a a x p + + + + = ... ) ( 2 2 1 such that ) ( ... ) ( 2 2 1 j N j N j j o j x f x a x a x a a x p = + + + + = for j = 0,1,2,…N You can solve this like way back in the day with a linear system. But let’s find a better way of doing this. Lagrange interpolation For N support points, Lagrange interpolation is the sum of N lagrange polynomials, each of which are of order x N-1 , designed to fit closely (exactly?) at a specified support point. Now that we have a sum of polynomials or a single polynomial, we can integrate this numerically, using trapezoidal rule, Simpson’s rule etc. Integration: Newton-Cotes Æ uniformly spaced points
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Different methods: trapezoid rule, simpson’s rule () [] 01 2 4 6 b a b a ba fxd x f f 2 f xd x f ≈+ + with errors ( ) 3 4 Ob a How to get more accurate? Break into smaller segments before making too many more points How to integrate in two dimensions? Make a rectangle containing all points, and multiply the function you are integrating by an “indicator” function that says whether you are in the area to be integrated (indicator = 1) or not (indicator = 0) This all relates to time integrals Linear ODE systems and dynamic stability: Linear system, begin with a known ode problem x_dot = Ax.
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.

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review2_compiled - Review for Exam 2 Ben Wang and Mark...

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