BVPs - Numerical Methods for Boundary Value Problems BVPs...

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Numerical Methods for Boundary Value Problems BVPs are usually formulated for y ( x ). Along the x axis, allocate gridpoints x i , i =0 ,...,N .BC sw i l lbeimpo s eda t x 0 and x N . First and Second Derivative Matrices First and second derivatives at the interior gridpoints x 1 ,...,x N - 1 will be computed from solution values y =[ y 1 ,...,y N - 1 ]andBCs y 0 and y N by y ± = D (1) y and y ± = D (2) y For N =10(BCsared iscussedbe low) , D (1) = 1 2 h 01000000 0 - 10100000 0 0 - 1010000 0 00 - 101000 0 000 - 10100 0 0000 - 1010 0 00000 - 101 0 000000 - 10 1 0000000 - D (2) = 1 h 2 - 210000000 1 - 21000000 01 - 2100000 001 - 210000 0001 - 21000 00001 - 2100 000001 - 210 0000001 - 21 00000001 - 2
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Linear BVPs As an example, let’s discretize the linear BVP y ± +2 y ± + y =0 ,y (0) = 1 (1) = 0 The discrete equations for y =[ y 1 ,...,y N - 1 ]are D (2) y D (1) y + y Ay = b where b - y 0 /h 2 + y 0 /h, 0 ,..., 0 , - y N /h 2 - y N /h ] incorporates the coupling to the BCs. The solution to the BVP isg ivenby numerically solving Ay = b . Nonlinear BVPs and Newton’s Method As a nonlinear example, let’s compute the solution to the boundary layer
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This note was uploaded on 03/11/2012 for the course MAT 421 taught by Professor Staff during the Fall '11 term at ASU.

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BVPs - Numerical Methods for Boundary Value Problems BVPs...

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