Solution to Mathematics Exam: Numerical Integration and Differential Equations - Prof. Bis, Exams of Mathematical Methods for Numerical Analysis and Optimization

Download Solution to Mathematics Exam: Numerical Integration and Differential Equations - Prof. Bis and more Exams Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! Solution of EXAM #2 1. (a) Derive the composite Trapezoidal rule. So, the composite trapezoidal Rule is: The error of composite trapezoidal Rule: (b) Determine h so that the composite Trapezoidal rule gives the value of 21 0 xe dx−∫ With an accuracy of . 710ε −= 2(1 0) ( ) 12 h f η ε− ′′ < 2 22| ( ) | | ( ) | | ( 2 4 ) | 2x xf e x eη −′′ ′′= = − + ⋅ ≤ 7 2 712 12 10 6 10 max{ ( )} 2 h f ε η − −⋅ ⋅< = = ′′ × , so 47.74597 10h −< × . 2. Approximate using Gaussian Quadrature with n=2. 21.5 1 xe dx−∫ Sol: Changing of variables: 1 1[(1.5 1) (1 1.5)] 2 4 x t= − + + = + 5 4 t So, 1 4 dx dt= 2 2 51.5 1 ( ) 4 4 1 1 2 21 1 1 ( ) ( ) 4 t xe dx e dt a f t a f t − +− − = = +∫ ∫ Where 25( ) 4 41( ) 4 t f t e − + = For 2n = 1 2 1a a= = , and the roots of the Legendre polynomial of degree 2 : 1,2 1 3 t = ± So, 2 2 2 1 5 1 5( ) ( )1.5 4 44 3 4 3 1 1 2 21 1 1( ) ( ) 4 4 xe dx a f t a f t e e − + − + − = + = +∫ . (b) Construct a quadrature rule of the form: 1 0 1 21 1 1( ) ( ) (0) ( ) 2 2 f x dx A A f A f − ≈ − + +∫ Which is exact for all polynomial of degree 2≤ . ( ) 1f x = 1 0 11 1 2dx A A A − = = + +∫ 2 ( )f x x= 1 0 21 1 10 2 2 xdx A A − = = − +∫ 2( )f x x= 1 2 0 21 2 1 1 3 4 4 x dx A A − = = +∫