{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2280-2" }}{PARA 257 "" 0 " " {TEXT -1 23 "Maple Project 1, Part 2" }}{PARA 259 "" 0 "" {TEXT 256 35 "Numerical Methods, sections 2.2-2.4" }}{PARA 258 "" 0 "" {TEXT -1 17 "Solution Template" }}{PARA 260 "" 0 "" {TEXT -1 47 "(Two problems \+ were added on Tuesday January 30)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 152 "(1) Enter your Name and Student number \+ in this space: (You should know by now that you can mix computations \+ with word processing in a Maple document.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 408 "(2) Consider the same initial value problem as in the class handout, namely dy/dx = y, y(0)=1, which has \+ solution y=exp(x). Use the improved Euler algorithm, with n= 100, on \+ the x-interval [0,1], to approximate the solution. Print out your ap proximations at x multiples of 0.1. Compare the accuracy of the n=10 0 improved Euler approximation to e, versus the results in the handout for unimproved Euler." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 225 "(3) Do Runge-Kutta fo r the same initial value problem and x-interval, with n=20. Compare t he accuracy of your results to the work you did with improved Euler n= 100, in problem (2) above. There should be a huge improvement." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 466 "(4) Do problem #25 in Section 2.4, page 114. \+ This problem illustrates how a numerical algorithm may accidently pas s right through a place where the real solution blows up. You may use the method of linear differential equations to solve the initial valu e problem in 25(a) by hand. [If you try to use dsolve I recollect tha t Maple runs into a little trouble.] Use Maple to help answer parts \+ (b) and (c), however. Insert commands and text comments as necessary. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "(5) Do \+ problem #5 in sections 2.4, 2.5, 2.6 homework. This is the same diffe rential equation in each case, studied successively with Euler, improv ed Euler, and Runge-Kutta." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "(6) Do problem #25 in \+ section 2.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 368 "(7) \+ Go to example 4 on page 134-135. Work through and understand what th e text has to say on these two pages. Recreate the data in the last t wo columns of Figure 2.6.9 on page 134 using the Runge-Kutta routine \+ from the class handout. That is, recompute the the Runge-Kutta approx imations with h=0.05 and compare to the actual solution values, at mul tiples of 0.4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 507 " Understand (for yourself) how mathematical instability caused by the solution to the homogeneou s problem (and exhibited in the slope field in Figure 2.6.10) can lead to numerical instability like that illustrated in this important exam ple of problem , even when you're using a ``good'' numerical technique like Runge Kutta and a \"small\" time step. This example illustrates why people who create numerical solutions to problems must also be we ll-grounded in the mathematical theory behind the problems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "19 0" 44 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }