MTH5201 Mathematical Methods in Science and Engineering 1 Fall 2014 Syllabus Instructor: Dr. Aaron Welters; O ce: Crawford Bldg., Room 319; Phone: (321) 674-7202; Email: awelters@fit.edu O ce hours: Mon. 2pm-5pm, Wed. 4pm-5pm, and by appointment. Course webpage: http://www.fit.edu/~awelters/teaching/2014/fall/mth 5201.html Course description: MTH5201 (3 credits) Fourier series and their convergence properties; Sturm-Liouville eigenfunction expansion theory; Bessel and Legendre functions; solution of heat, wave and Laplace equations by separation of variables in Cartesian coordinates. Prerequisites: MTH 2001, MTH 2201. Lecture time: Mon. and Wed. at 5pm-6:15pm in Crawford Bldg., Room 403. The course will run from Aug. 18, 2014 to Dec. 8, 2014 & the last day of class is Wed., Dec. 3. We have in total 29 days (14 Mondays, 15 Wednesdays) of lectures and exams (excluding the nal exam). Breaks and holidays are: Labor Day, Sept. 1 Mon.; Fall Break & Columbus Day, Oct. 13 Mon.; Thanksgiving break, Nov. 26 Wed.; Final Exams Week Dec. 8 Mon. & Dec. 11 Wed. Important note: Attendance at lectures is mandatory. A sign-up sheet will be passed around in each class. Course textbook: [EK11] Kreyszig, Erwin. Advanced Engineering Mathematics. 10th Ed., John Wiley & Sons, 2011. Note: We will cover parts of Chapters 2, 5, 11, and 12 of this textbook and supplement the material when necessary. Important note: The bookstore will not carry this book for this semester. Thus, you must purchase it elsewhere (such as through Amazon). I have placed one copy in Evans library on reserve for use by the whole class. Reference books: [HC07] Cheng, Hung. Advanced Analytic Methods in Applied Mathematics, Science, and Engineering. Boston, MA: Luban Press, 2007. [BO78] Bender, C. and Orszag, S. Advanced Mathematical Methods for Scientists and Engineers. New York, NY: McGraw-Hill, 1978. Reprinted by Springer-Verlag, New York, 1998. [CM95] Mei, C. C. Mathematical Analysis in Engineering. Cambridge, England: Cambridge University Press, 1995. [FB76] Hildebrand, F. B. Advanced Calculus for Applications. 2nd Ed., Prentice-Hall, 1976. Exams: There will be 3 exams the last of which is a comprehensive nal exam. The rst two exams are scheduled for Mon. Sept. 22 and Wed. Oct. 29, 5pm-6:15pm, Crawford Bldg., Room 403. The nal exam is Mon. Dec. 8, 6pm-8pm, Crawford Bldg., Room 1
403. Homework: Homework will consist of weekly problem sets which will be posted on the course webpage each Wed. by 11:59pm and due the following Wed. in class. Late problem sets are not accepted. See below for more details on this. Additionally, extra credit problem sets will occasionally be given out which can improve your homework scores. Grading Policy: Your grade will be based on 20% prob. sets + 40% midterms (20% each) + 40% nal. Your nal grade will be determined by your homework (including extra credit) after dropping your lowest homework score and by curving all nal scores of the exams. Drop dates: Aug. 29 Fri. last day to drop class with full tuition refund & without receiving a grade of W; Oct. 24 Fri. Last day to withdraw from class with a nal grade of W. Last day of classes: Dec. 3 Wed. The small print Collaboration policy: Independently of whether you collaborate or not, any homework submitted must be formulated by you in your own words. Word-by-word copying is strictly forbidden, and may result in a 0 point score for all concerned. Missed homeworks: If you have to miss a homework deadline for some valid reason, contact the lecturer. An attempt will then be made to deal with the matter satisfactorily. The arrangement will be con rmed by email to you, which you should keep for your records. Do not count on anyone else to make such arrangements for you: you re in charge of getting things done. All such matters must be resolved before the last day of classes: no further changes to homework scores will be made after that. The same applies to midterm scores. Exams: Closed book, no notes allowed, and NO electronic devices will be allowed in the exams. Finally: The material covered in this class is great, make sure you stay on top of it and you will do very well!
MTH5201 Fall 2014 Syllabus I. Linear second-order ODEs: (1.5 lectures; review) Based on [EK11] Chap. 2, 1, 2, 4-6. Second-order linear ODEs; homogeneous vs. nonhomogeneous; superposition principle; initial value problem; basis of solutions; general solution; method of reduction of order. Euler-Cauchy equations and example of a boundary value problem of the electric potential eld between two concentric spheres. Existence and uniqueness theorem for initial value problems; linear independence of solutions and the Wronksian. HW Assign. 1; Due Date: 08/27/14; From [EK11]: Sec. 2.1, #11-19; Sec. 2.2, #1, 8, 13, 14, 16, 18, 24-26, 31-34; Sec. 2.4, #5-10; Sec. 2.5, #2, 3, 12, 13, 20; Sec. 2.6, #9-16; Sec. 2.9, #7, 14, 15. Complete solution to problem Sec. 2.5, #2 will be posted on course website after due date. II. Series solutions of ODEs and special functions including Bessel and Legendre functions: (8.5 lectures) Based on [EK11] Chap. 5, 1-5. Introduction to power series (radius of convergence, uniqueness of coe cients, operations on power series, power series for common elementary functions); on the existence of power series solutions (and their radius of convergence) to second-order linear ODEs in standard form with analytic coe cients; the power series method. Legendre s di erential equation; Legendre functions; recurrence relations (recursion formula); Legendre polynomials (including the generating function, Rodrigues s formula, and Bonnet s recursion). HW Assign. 2; Due Date: 09/03/14; From [EK11]: Sec. 5.1, #1-3, 6-8, 15-20; Sec. 5.2, #1-5, 10-12, 14, 15. Complete solution to problem Sec. 5.2, #2 will be posted on course website after due date. Regular and singular points of second-order linear ODEs; Frobenius method as an extended power series method, the indicial equation, and basis of solutions in the three cases depending on the di erence of the roots of the indicial equation; example using Euler-Cauchy equations as a special case. HW Assign. 3; Due Date: 09/10/14; From [EK11]: Sec. 5.3, #1-5, 9-13. Complete solution to problem Sec. 5.3, #12 will be posted on course website after due date. Bessel s equation; Bessel functions of the rst kind (gamma function, series representation and coe cient formula, derivative recursions, Bessel functions of half-integer order as elementary functions); general solution of Bessel s equation for noninteger roots of indicial equation. HW Assign. 4; Due Date: 09/17/14; From [EK11]: Sec. 5.4, #2-7, 12-15, 19, 20. Extra credit #1 (to posted on course website). Complete solution to problem Sec. 5.4, #6 will be posted on course website after due date. Bessel functions of the second order; general solution of Bessel s equation for integer roots of indicial equation; Hankel functions. Assignment 5; Due Date: 10/01/14; From [EK11]: Sec. 5.4, #16-18; Sec. 5.5, #2, 4, 6-9, 11. Team project: Modeling a Vibrating Cable and solutions in terms of
Bessel functions of order zero. Complete solution to problem Sec. 5.5, #6 will be posted on course website after due date. III. Exam 1: Mon. Sept. 22, 5pm-6:15pm, Crawford Bldg., Room 403. Material covered: Cauchy-Euler equations; method of reduction of order; series solutions of ODEs and special functions including Legendre and Bessel functions up to and including HW Assign. 4. Practice exam 1 and solutions will be posted on course website before exam. IV. Fourier series and their convergence properties, Fourier integrals, and Sturm- Liouville eigenfunction expansion theory: (9 lectures) Based on [EK11] Chap. 11, 1-9. Periodic functions; trigonometric functions; orthogonality of the trigonometric system; Fourier series; Euler formulas for Fourier coe cients; representation theorem of periodic piecewise-smooth functions as Fourier series; even and odd functions; Fourier sine and cosine series; odd and even periodic extensions of functions and their Fourier series representation. HW Assign. 6; Due Date: 10/08/14; From [EK11]: Sec. 11.1, #7-9, 13, 15, 16, 21, 23; Sec. 11.2, #8-10, 19, 23-25. Complete solution to problem Sec. 11.1, #16 will be posted on course website after due date. Periodic forced oscillations of (one-degree-of-freedom) spring-mass-damper systems and the analogy for RLC circuits; steady-state solutions using Fourier series; general solutions. HW Assign. 7; Due Date: 10/15/14; From [EK11]: Sec. 11.2, #26, 27, 29, 30; Sec. 11.3, #6-9, 13, 14, 17, 19. Complete solution to problem Sec. 11.3, #17 will be posted on course website after due date. Approximation by trigonometric polynomials; minimum square error; Bessel s inequality; Parseval s identity. Sturm-Liouville equation; Sturm-Liouville problems; eigenvalues and eigenfunctions; orthogonality of eigenfunctions; the trigonometric functions and Legendre polynomials as important examples. HW Assign. 8; Due Date: 10/22/14; From [EK11]: Sec. 11.4, #2, 4, 5, 7, 11; Sec. 11.5, #2, 5, 6, 8-10, 13, 14. Team project: Special functions and orthogonal polynomials the Chebyshev and Laguerre polynomials. Complete solution to problem Sec. 11.5, #5 will be posted on course website after due date. Generalized Fourier series; eigenfunction expansions; Fourier series, Fourier-Legendre series, and Fourier-Bessel series as important examples; Bessel s inequality and Parseval equality; mean square convergence and completeness. Fourier Integral (as limit of Fourier series as period becomes in nite); representation theorem of piecewise-smooth absolutely integrable functions as a Fourier integral; Fourier sine and cosine integrals; sine and laplace integrals. V. Exam 2: Wed. Oct. 29, 5pm-6:15pm, Crawford Bldg., Room 403. Material covered: Bessel functions of the second order and general solutions of Bessel s equation for integer roots of indicial equation; Fourier series and their convergence properties, Fourier integrals, and Sturm-Liouville eigenfunction expansion theory; material after exam 1 up to and including HW Assign. 8. Practice exam 2 and solutions posted on course website before exam.
VI. Solutions of heat, wave and Laplace equations by separation of variables in Cartesian coordinates: (8 lectures) Based on [EK11] Chap. 12, 1-10. Second-order linear PDEs; homogeneous vs. nonhomogeneous; superposition principle; initial and boundary conditions. Physical derivation of the one-dimensional wave equation in modeling a vibrating string (both the continuum derivation and the multiscale derivation of a 1D lattice model in the limit). Solution of the one-dimensional wave equation; separation of variables; the Sturm- Liouville problem from the boundary conditions; eigenfunction expansion; solution of the initial value problem using Fourier series; physical interpretation as superposition of forward and backward traveling waves for zero intitial velocity. HW Assign. 9; Due Date: 11/12/14; From [EK11]: Sec. 12.1, #2, 3, 6, 7, 14, 16, 17; Sec. 12.3, #5, 7, 9, 11, 14. Complete solution to problem Sec. 12.3, #7 will be posted on course website after due date. Physical derivation of the heat equation in modeling heat ow from a body in space (via Gauss s theorem); the Laplacian. Solution of the one-dimensional heat equation; separation of variables; the Sturm- Liouville problem from the boundary conditions; eigenfunction expansion; solution of the initial value problem using Fourier series; physical interpretation. Steady two-dimensional heat problems and Laplace s equation (equation for the electrostatic potential); boundary value problems: Dirichlet, Neumann, and Robin problems. Solution of the Dirichlet problem in a rectangle; separation of variables; the Sturm- Liouville problem from the boundary conditions; eigenfunction expansion; solution of the boundary value problem using Fourier series. HW Assign. 10; Due Date: 11/19/14; From [EK11]: Sec. 12.6, #1, 5-7, 11-15, 18, 19, 21, 23. Complete solution to problem Sec. 12.6, #18 will be posted on course website after due date. Physical derivation of the two-dimensional wave equation in modeling a vibrating membrane (both the continuum derivation via the tensile force using Stokes theorem and the derivation as an Euler-Lagrange equation using a Lagrangian). Solution of the two-dimensional wave equation for a rectangular membrane; separation of variables; Helmholtz equation, the Sturm-Liouville problems from the boundary conditions; eigenfunction expansion; solution of the initial value problem using double Fourier series. HW Assign. 11; Due Date: 12/03/14; From [EK11]: Sec. 12.9, #4-6, 8, 11-13, 17-20. Complete solution to problem Sec. 12.9, #19 will be posted on course website after due date. Extra credit #2 (to be posted on course website); Due Date and Time: 12/08/14 by 8pm. VII. Comprehensive Final Exam: Mon. Dec. 8, 6pm-8pm, Crawford Bldg., Room 403. Two part exam. Material covered in part I (new material): Generalized Fourier series; eigenfunction expansions; Fourier-Legendre series and Fourier-Bessel series; Solutions of heat, wave and Laplace equations by separation of variables in Cartesian coordinates; material after exam 2 up to and including HW Assign. 11. Material covered in part II
(older material): all material up to and including exam 2. Two practice exams (Exam A and B) with solutions posted on course website before exam.