rickeyzack_285936_53810636_HW10 2850

Numerical Methods HOMEWORK 10 – Due April 17, before 11:59 pm 10.1 Application: Finite Difference Rules Go back to problem 9.4 (from your last HW) which had data for the height y i (m) of a pumpkin as a function of time t i (sec). a) Use the 3-point centered difference rule to calculate the acceleration (d 2 y /d t 2 ) of the pumpkin at t = 1.5 seconds. (Compare this value to what you already know the acceleration due to gravity should be in m/s 2 .) 3 pts Time, t i Height, y i 0 1 2 3 0 10 20 30 b) Use the 4-point backward difference rule to calculate the acceleration (d 2 y /d t 2 ) of the pumpkin at t = 3.0 seconds. Hint : you can find this rule in Table 8-1 (pg 318) of the textbook, or the back of your “Class 30/31” handout. c) IF you obtained and used measurements every 0.1 seconds (instead of 0.3 seconds, like in the table of data), quantify how much you’d expect the error in your acceleration calculation from part (a) to change. 10.2 Fundamentals: Taylor Series This is the equation for the Taylor Series expansion of f ( x + d ), as a function of f ( x ) and all its derivatives at x , that will be given to you on your midterm and exam cheat-sheets: Start with that equation and use it to derive the expression for f ( x i – 5 D x ) ( i.e. f ( x ) evaluated five nodes to the left of x i ) as a function of f ( x i ) and all the derivatives at x i (up to the FIFTH derivative). 2 pts f ( x + δ ) = f ( x ) + δ ʹ f ( x ) + δ 2 2 ʹʹ f ( x ) + δ 3 3! ʹʹʹ f ( x ) + δ 4 4! ʹʹʹʹ f ( x ) + ! 10.3 Application: Error Order and Precision The following is a 5-point difference scheme, over equally-spaced x i , for d 3 f /d x 3 at x = x i : Write out Taylor Series expressions for each of the four f i -3 , f i -2 , f i -1 , f i +1 to the FIFTH derivative, like you did in 8.4, and then combine them using the given difference scheme above to … a) Calculate the discretization error order ( i.e. write the error = O ( D x p ) for some integer p ). b) Calculate the precision of the scheme. ʹʹʹ f ( x i ) = 1 2 Δ x 3 f i − 3 − 6 f i − 2 + 12 f i − 1 − 10 f i + 3 f i + 1 ( ) + Error 7 pts

HW10 ( 10.1 – 10.6) due Monday April 17 10.4 DERIVING difference formula for ANY order derivative to ANY error order Use Taylor series expansions to derive a forward difference scheme, over equally-spaced points, using any or all of f i , f i+1 , f i+2 , f i+3 and f i+4 that approximates d 2 f /d x 2 ( 2 nd derivative at x i ) to order ( D x 3 ) discretization error. Hint : DON’T write a Taylor Series for f i . f i is just itself (or just written as f ( x i )). There’s nothing else you can do with it! Only ever create Taylor Series expansions for points other than f i . Show all your work! Write all appropriate Taylor Series out, and describe your goals – what terms do you need to keep, what terms do you need to eliminate? If you follow the methodology from class, you should end up with 4 equations for 4 unknowns ( i.e. coefficients a , b , c , d that you’re using to weight each Taylor Series). It’s fine if you then use MATLAB to solve for them – you should get nice “round” fractions in your final scheme. 10.5 Fundamentals: Characterizing ODEs (Chapter 10) For each of the following three ODEs … i. characterize its order ( e.g. 1 st -order, 2 nd -order, etc.), ii. characterize it as an IVP (Initial Value Problem) or BVP (Boundary Value Problem), iii. write the ODE in standard form. 7 pts 6 pts ( B ) y dx dy ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = sin( x ) + d ( x 3 ) dy , x − 4 ( ) = 1 ( A ) qe r = r dq dr + d 2 q dr 2 , q 2 ( ) = 0, q ' 0 ( ) = 1 ( C ) 1 + d 2 x dt 2 d 3 x dt 3 + x ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = − 4 dx dt , x 3 ( ) = 3, x ' 3 ( ) = 2, x '' 3 ( ) = 1

From 9.4 i o o 3 Goi'd 1.5 Is 21 9 10 32.2 33 32.8 31.8 29.927.1 23.4 18.8 13.4 31 a Use the 3 pt centered difference rule to calculate EIGIL tiofootgrfityPI.PK weE.tt 1.5 seconds Three point central difference k 2 f ti ft i 2ft tin E 1.5 s i 6 h big 3 30 0.3 f to f ts 2ft f E 29.9 2,13317 23.4 10 m s f yo 10m15 vs 9.81 m s expected Percent error 10298,81 100 1.936 off b Use the 4 pt backward difference rule to calculate the acceleration day Ide of the pumpkin at E3 seconds fifteen ft f ti f ti 3 4ft p 5f ti 1 2ft backward t 3 seconds 5 11

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