Functional Approximation Evaluate Residuals10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #10: Function Space. Functional Approximation (Variables are scalar in this example) )()()(0xxcxfNnnnΔ+≈∑=φ Figuring out Δ(x) is similar to solving whole problem Increase N until function converges {φn(x)} favorite set of functions length M {vn} favorite set of vectors ∑=≈Nnnnvcw0 N<M vn {}mℜ∈ Basis: el = ∑=Nnnnlvd0, ∑∑∑==≈= nlnnlllllNnnnapproxvdaeavcw,,0 el·ej = δjl Î orthonormal c = aTD We want to do the same with functions. How do you take dot product? Define “φn·φm” = “works”: <φ∫ xof rangeginterestin*)()()( xxxgdxmnφφm|φn> = δmn weighting function g(x) = k x: 0 Æ 2π φm = eimx = cos(mx) + i·sin(mx) )cos(2mxeeimximx=+− g(x) = 1 x: -1 Æ +1 Legendre polynomials 2)(xexg−= x: -∞ Æ +∞ Hermite polynomials 22)(xxg−1=π x: -1 Æ +1 Chebyshev polynomials 1) We chose a basis {φn(x)} and an inner product orthonormal: <φm|φn> = δmn 2) We’re trying to solve Ôf(x) = q(x) (“In most problems, these are all vectors, unknown given 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) ≈ ∑ )(xcnnφ[]∫∫=bambamxqxxgdxxfOxgdx )()()( )()()( *rangefavorite *φλφ solution will depend on a,b,cn,m. F(a,b,cn,m) = v(m,a,b) F(cn,m) = v(m) Now solve for cn. If Ô is a linear operator: Ôfapprox(x) = ∑∑= )ˆ()(ˆnnnnOcxcOφφand if Ôφn = λnφn (i.e. φn is an eigenfunction of Ô) Ôfapprox(x) = ∑= )(xcnnnφλ∫∑∫∑=banmnnbannnmgdxccxgdxφφλφλφ** )( <φm|φn> = δmn ∫∫∑∫=≡====)()()( 1)()()( ˆ)( **0*xqxxgdxcbxqxxgdxcccfOxgdxmmmmmmmmmNnmnnnbaapproxmφλφλλδλφ f(x) )()(ˆ22xTxhxkO⎥⎦⎤⎢⎣⎡+∂∂= Often this is the operator 10.34, Numerical Methods Applied to Chemical Engineering Lecture 10 Prof. William Green 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]. sin are eigenfunctions cos Gives you a really messy equation: Suppose Ô = Ô1 + h(x) {i.e. Schrodinger Equation} Suppose Ô1φn = λnφ10.34, Numerical Methods Applied to Chemical Engineering Lecture 10 Prof. William Green 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]. ()( ˆ)()( ∑∫∫∫+=xxhxxgdxcxxgdxcfOxxgdxnmnmmmbaapproxmφφφλφmn***H )()()()( )()()∑xcxhnnφ mmnnmmbHcc =+∑λ (H+Λ)c = B m=1,…N Linear Problem: c = (H+Λ)\bλmI = ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛31λλλ0000002Must evaluate integrals Hmn: difficult to evaluate, quantum mechanics requires 6-dimensional integrals. H becomes a large matrix when n gets large. Also have Boundary Conditions: f(x = 0) = f0Adds another equation: 0)0( fxcnn==∑φ v·c = f0How 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 Φn 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 = Ôf – q max(R) < tol? ||R(xi)|| < tol? we will evaluate this
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