# W09hw1 - Math 623 Fall 2011 Homework 1 For full credit your...

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Math 623, Fall 2011: Homework 1. For full credit, your solutions must be clearly presented and all code included. (1) Consider the following initial value problem for the function u = u ( x ) deﬁned for 0 x 1. u xx + (1 + x 2 ) u x - (1 + x ) u = 0 and u (0) = 1 ,u x (0) = - 0 . 5 . (a) Write down the ﬁnite diﬀerence scheme for the ODE above, using a forward diﬀerence for u x and a symmetric diﬀerence for u xx . Implement the corresponding numerical algorithm with Δ x = 2 - 10 , and plot the graph of u ( x ) , 0 x 1. Also give your computed value of u (1) correct to six decimal places. (You should get u (1) 1 . 138). (b) Same question as in (a) but use a backward diﬀerence for u x . (c) Same question as in (a) but use a central diﬀerence for u x . Make sure that the initial conditions you use in the numerical algorithm are second order accurate. (d) For Δ x = 2 - N , N = 1 , 2 .., 15, let u Δ x (1) be the computed value of the solution u ( x ) at x = 1. Set ± x ) = u Δ x (1) - u

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W09hw1 - Math 623 Fall 2011 Homework 1 For full credit your...

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