Remaining examinations Project: suggestion of problems for future years (i) voluntary (ii) quality is more important than quantity (iii) at most two

Size: px

Start display at page:

Download "Remaining examinations Project: suggestion of problems for future years (i) voluntary (ii) quality is more important than quantity (iii) at most two"

Lilian Marion Chandler
5 years ago
Views:

1 Remaining examinations Project: suggestion of problems for future years (i) voluntary (ii) quality is more important than quantity (iii) at most two future home problems and two future exam problems, with solutions, not of essay type, and of comparable difficulty as current problems (iii) Be sure to cite the references you have used (iv) in Finnish, English, or Swedish (v) deadline Friday January 12 at (vi) examples of last year s projects are problems 23, 34, 35, and 46 in the problem collection. 1

2 Final Tuesday December 19 at 08:30-12:30 in V&V 5 problems, i.e. 48 minutes per problem. Covers material covered by: (i) lectures (not appetizers) (ii) the sections of the Byron and Fuller book listed in the schedule (iii) the sections of the Bender and Orszag book listed in the schedule (iv) the extra notes available on the home page (not Riemann ζ-function) (v) recitations (vi) home problems If a question asks you to describe a method to solve it, you are expected to outline how the solution will proceed and not actually carry through the calculations. However, the problems are so chosen that the methods we have discussed do work on them, so if you wish to actually solve a problem, you are welcome to do so. Look it up or equivalents thereof do not constitute a solution. My written examinations tend to be difficult, but the grading tends to be easy. 2

3 Examples of questions: 1. How would you go about evaluating 0 dx eiαx2 for α R Expect an application of residue method with a slice-of-pie contour to relate the integral to 0 dx e α x2, and the standard trick to evaluate Gaussian integrals. 2. Describe methods to sum infinite series. Expect (i) exact methods based on Taylor series and sums of residues, (ii) approximate methods as interpretation as Riemann series, convergenceimproving transformations such as Poisson summation formula, and (iii) summation techniques to continue series to domains in which the original expressions diverge (Abel, Borel). 3. What is an irregular singular point of a differential equation, and why is that an important concept? Expect a definition, and statement that the standard power series (Frobenius) 3

4 method often fails near an isp, making it necessary to find other ways to solve the equation in that range. 4. Why are Green s functions useful? Expect a statement that they allow us to construct solutions of differential equations with arbitrary inhomogeneous (source) terms, and that the Green s function can (in principle) be constructed from the solution of the homogeneous equation alone. 5. How would you go about finding approximate solutions of the following differential equations: a) y + (h 2θ cos(2x))y = 0 for θ h Expect a discussion of regular perturbation theory. b) x 4 y 3x 2 y + 2y = 0 for x 0 + Expect a discussion of approximate solutions of ODEs near an ISP (e.g. based on y = e S(x) ). 4

5 c) ɛy + (cosh x)y y = 0, y(0) = y(1) = 1, 0 x 1, ɛ 0 + Expect a discussion of boundary layer method. 6. What is the WKB method? Give an example of an application of the method. Expect a brief discussion of the method stating the conditions of validity and explaining, qualitatively, how to connect solutions on two sides of a region in which WKB is not expected to work (connection formulas). As an example, expect, e.g, approximate evaluation of energies of bound states in quantum mechanics. 7. Give a physical application of a variational problem, and outline a method to solve it. Expect for example an application in mechanics (minimal action), geometrical optics (ligth travels along the fastest path), electrostatics (charge distributions determined by energy minimization), statistical physics (equilibrium distributions determined by entropy maximization), or one of many 5

6 others. Expect Euler-Lagrange equations. 8. Evaluate dx δ (x 1)x 2. Expect 2. 6

7 I am not interested in the answers (I know them, I made the exam!), I am interested in seeing how you intend to get there. Result statistics will be posted on the wall (window) of my office as soon as the exam has been graded. At that point you can view your own exam and its grading. Grading estimated to be finished: between Jan. 2 and Jan. 19 7

Advanced Mathematical Methods for Scientists and Engineers I

Carl M. Bender Steven A. Orszag Advanced Mathematical Methods for Scientists and Engineers I Asymptotic Methods and Perturbation Theory With 148 Figures Springer CONTENTS! Preface xiii PART I FUNDAMENTALS