Math 301 Winter 2013-14 Midterm 1 02/06/2014 Time Limit: 60 Minutes Name (Print): Instructor This exam contains 8 pages (including this cover page) and 6 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You may not use your books, notes; calculators are highly recommended. You are required to show your work to problems 3 to 5 and choose between problems 1 and 2 to get a total of 100 points. Please indicate clearly which one you want to be graded. The last problem is optional. The following rules apply: If you see what seems to be a mistake notice it on your work and explain why and what decision you take to get around it. Organize your work, in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive no credit. Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. If you need more space, use the back of the pages; clearly indicate when you have done this. Problem Points Score 1 20 2 10 3 45 4 20 5 15 6 10 Total: 120 Electronic devices shall not be used. Laptops, phones should be kept in your bags or pocket. Nothing else that this exam, a calculator and a pen/pencil should remain on your desk. Do not write in the table to the right.
Math 301 Midterm 1 - Page 2 of 8 02/06/2014 1. Answer the following questions (make sure you justify - either providing a counter example, or an accurate proof): (a) (4 points) What does it mean for a function f(t, y) to be Lipschitz in y on a domain D = {(t, y) : a t b, α y β}? (b) (4 points) Is a Lipschitz function always differentiable? (if not, give an example, if yes, prove it by analyzing the ratio f(x+h) f(x) h ) (c) (4 points) T/F - The domain D = {(x, y) : x 2 + y 2 < 1} is convex. (d) (4 points) T/F - The domain D = {(x, y) : x 2 + y 2 1} is convex. (e) (4 points) T/F - The domain D = {(x, y) : x 2 + y 2 = 1} is convex. Remark 1. Remember, a domain is said to be convex if, for any two points u and v in the domain, and for any λ [0, 1], the new point w λ := λu + (1 λv) is in the domain.
Math 301 Midterm 1 - Page 3 of 8 02/06/2014 2. (10 points) 1. What is the advantage of Runge-Kutta methods over Taylor methods? 2. What is the fundamental difference between Adams-XX methods and RK methods? (XX stands for either Bashforth or Moulton i.e. explicit or implicit) 3. Why would you consider using explicit methods over implicit methods? And why would you consider using implicit over explicit? (in other words, what is the main advantage of each of these techniques) 3. For a certain step size h, we consider the following difference equation: y i+1 = y i + h (αf (t i, y i ) + βf (t i + h/2, y i + h/2f (t i, y i )) + γf (t i + h, y i + hf (t i, y i ))) (1) (a) (10 points) Write this expression in the form of a Butcher s tableau (see Table 1) and verify that its coefficient do not contradict the conditions to be a Runge-Kutta method seen in class.
Math 301 Midterm 1 - Page 4 of 8 02/06/2014 c A b T Table 1: General Butcher s tableau (b) (15 points) Find conditions on α, β, and γ such that the difference equation (1) corresponds to (see the given Butcher s tableaus) 1. Euler s method: 2. Midpoint formula: 3. Trapezoidal rule: 0 0 1/2 1/2 1 0 1 Euler s method Midpoint formula (c) (5 points) Show that this method is consistent if α + β + γ = 1. 0 0 1 1 0 1/2 1/2 Trapezoidal rule (d) (4 points) Calculate the degree 3 Taylor expansion of y(t i+1 ) about t i.
Math 301 Midterm 1 - Page 5 of 8 02/06/2014 (e) (6 points) Calcualte the degree 2 Taylor expansions of f(t i + h/2, y(t i ) + h/2f(t i, y(t i ))) and f(t i + h, y(t i ) + hf(t i, y(t i ))) about f(t i, y(t i )). (f) (5 points) Using the two previous questions, find conditions on α, β, and γ such that the difference method in this exercise has a local truncation error order of at least O(h 2 ). 4. Consider the following initial value problem: { ) y = f(t, y) = t (y 2 + e 2t3 /3, 0 t 1 y 0 = y(0) = 0 (IV P 1 ) (a) (10 points) Show that the unique solution is given by y(t) = y(t) = e 2t3 /3 e t3 /3 (make sure to justify and recall the theorems used).
Math 301 Midterm 1 - Page 6 of 8 02/06/2014 (b) (10 points) Write the difference equation associated to the numerical approximation using a Taylor method of order 2. 5. (15 points) Do these two Initial Value Problems have a unique solution? (make sure you answer accurately and justify accordingly) { y = f(t, y) = y+t+1 y 2 +t 2 +1, 1 t 1 (IV P a ) y 0 = y( 1) = 2 { y = f(t, y) = 2 t y + t2 e y2, 1 t 2 (IV P b ) y 0 = y(1) = 1
Math 301 Midterm 1 - Page 7 of 8 02/06/2014 6. (10 points) This question is optional and only meant to keep all of you busy for the whole hour. It will be graded only if you show you tried the rest. Using Taylor polynomials of y and f at t i 1, t i, and t i+1, derive a 3 step implicit method to solve the following IVP: { y = f(t, y), a t b y(b) = y 0 (IV P 2 ) You may use f i to denote f(t i, y i ) and h as a step size. Do you have to have a fixed step size h?
Math 301 Midterm 1 - Page 8 of 8 02/06/2014 Your go to midterm-evaluate me! (Note: You don t HAVE to fill out these questions, and whatever you say will not affect the rest of this class - you may actually rip it off and hand it in separately if you wish) 1. What interest you the most in this class? 2. What am I doing well that is helping you to understand/learn these topics? 3. What am I doing that is hindering your learning? 4. What would you like me to change or add for the 5 remaining weeks of class?