10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 10 Function Space Functional Approximation Variables are scalar in this example N f x c n n x x Figuring out x is similar to solving whole problem n 0 Increase N until function converges n x favorite set of functions length M N w cn v n vn favorite set of vectors N M n 0 vn m N Basis el d l n v n w approx N c n v n a l el a l d l n v n n 0 n 0 l l n c aTD el ej jl orthonormal We want to do the same with functions How do you take dot product Define n m dx g x n x m x works m n mn interesting range of x weighting function g x k x 0 2 m eimx cos mx i sin mx e imx e imx cos mx 2 g x 1 g x e x g x 2 x 1 1 Legendre polynomials x Hermite polynomials 2 1 x2 x 1 1 Chebyshev polynomials 1 We chose a basis n x and an inner product orthonormal m n mn 2 We re trying to solve f x q x unknown given In most problems these are all vectors but that looks too scary to start with Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Look for solutions funknown x c n n x b dx g x O f x m dx g x m x q x solution will depend on a b cn m b a a favorite range F a b cn m v m a b F cn m v m Now solve for cn If is a linear operator O c n n x fapprox x c n O n and if n n n i e n is an eigenfunction of fapprox x c n n n b x dx g x m cn n n cn n dx g m n b a m n mn a b dx g x m O f N approx c n n mn c m m n 0 a c m m dx g x m x q x bm cm 1 m dx g x m x q x f x O k 2 h x T x Often this is the operator x 2 sin cos are eigenfunctions Gives you a really messy equation Suppose 1 h x i e Schrodinger Equation Suppose 1 n n 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 10 Page 2 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY b dx g x m x O f approx a c m m dx g x m x h x c n n x c dx g x n m x h x n x H mn c m m c n H mn bm H c B m 1 N 1 mI 0 0 0 2 0 Linear Problem c H b 0 0 3 Must evaluate integrals Hmn difficult to evaluate quantum mechanics requires 6dimensional integrals H becomes a large matrix when n gets large Also have Boundary Conditions f x 0 f0 Adds another equation c n n x 0 f 0 v c f0 How to solve Can try to fit by least squares and just fit all the equations approximately Can drop larger n terms to leave space for boundary conditions Another way would be to not consider the boundary conditions and then craftily choose n so that they solve the boundary conditions To check if answer makes sense write out the series and see if cn converges Evaluate Residuals R f q max R tol R xi tol we will evaluate this later 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 10 Page 3 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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