finala97f - Final Examination MEAM 501 Analytical Methods...

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Final Examination MEAM 501 Analytical Methods in Mechanical Engineering 1997 Fall 1. State the following definitions, properties, and/or concept ( 40 points ) : (1) Define the Lagrange interpolation of a function f defined on an interval ( a , b ) by using n +1 points xx a b n 11 ,.... , , + [] . State clearly major properties of the basis functions of the Lagrange interpolation. Within a given interval (a,b), we place n+1 points, say, x n 12 1 , ,..... , + , and using the values ff x ii = (29 of the given function f ( x ) at these points, we approximate the function f by a n degree polynomial f n : fx f Lx ni i i n i j ij j ji n (29= (29 (29 = - - = + = + 1 1 1 1 , The basis functions i have the following properties : 1) they are n degree polynomials and 2) = = 1 0 if if (2) State the property of the Legendre polynomilas defined on the interval (-1,1) ? They are orthogonal with respect to the inner product fg fxgxd x , = (29(29 - 1 1 , and they are obtained from the set of polynomial basis functions 1 23 , , , ,...... , ,..... xx x x n {} by Gram-Schmidt orthogonalization process . (3) Are the two functions fx x x 1 (29=+ (29= (29 and sin π orthogonal in an interval (0,1) ? If not, orthogonalize them by adding an appropriate constant or polynomial to the function f 1 . From these make up two orthonormal basis functions. f f x x dx x d dx xd x x x d x x x x x x 0 1 0 1 0 1 0 1 2 0 1 1 1 1 31 , sin cos cos ' cos sin =+ ( 29=+ - =- + ( 29 ++ ( ∫∫ = = = = π π π π π π π π π π = 3 π Thus they are not orthogonal We shall now consider
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ff k x k 11 1 =+= + + to orthogonalize. In deed, f k f k xd x k x k x x 12 1 2 0 1 0 1 33 1 3 2 0 , , sin cos (29 =+ (29 =+- (29 =+ = = = π π ππ π that is k =- 3 2 .
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finala97f - Final Examination MEAM 501 Analytical Methods...

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