Math 110B. Partial Differential Equations- Spring 10 - Hans Lindblad

Description: Partial differential equations describe many phenomena in Physics such as fluid and heat flow and wave propagation. PDEs are used in Engineering and Economics and occur in other branches of mathematics such as geometry and probability. Questions about PDE motivated many of the developments analysis. These days one can sometimes numerically solve a PDE approximately, but knowledge of special solutions and the theory is essential.
Lectures: MWF 4-5 HSS 2305B. Instructor Hans Lindblad, lindblad@math.ucsd.edu. Office hour: M 3-4 in Cafe Espresso.
Sections: Tu 10-11 in HSS 2305B. TA Chad Wildman (cwildman@math.ucsd.edu). Office hour: W 10-11 Th 1-2 APM 6442.
Text: Strauss Partial Differential Equations
Syllabus: The text was also used for 110A and will continue where it left off. 110A dealt with linear PDEs in one space dimensions and boundary problems using Fourier series. 110B deals with PDEs in more space dimensions. We will derive the fundamental solutions for the the Wave, Heat and Laplace equations. The first part is about the Wave and Heat equations in several space dimensions, Chapter 9, starting with a review of the one dimensional case, Chapters 2-3. Second part will be about Laplace equations, Chapters 6 (review) and 7. The third part will be about distributions and Fourier transforms which provide alternative derivations of the fundamental solutions. The fourth part will deal with applications, to Fluids and Electromagnetism, Chapter 13, and nonlinear equations, Chapter 14.
Preliminary schedule:
wk  date  Monday  Wednesday  Friday
  1  3/29  1.1-6 The classical PDEs, Initial value & Boundary value problem  2.1 Wave equation on the line.
 2.2 Causality and Energy.		  2.4 Diffusion on the line.
  2  4/5  3.3 Diffusion with sources.
 3.4 Wave equation with source. 		 9.1 Energy and causality of waves 3.2 Reflected and spherical waves 9.2 Wave equation in space-time.
  3  4/12  9.2 Wave equation in space-time.  9.3 (Characteristics, Singularities)  6.1,9.1 Lorentz inv. 9.3 Sources.
  4  4/19  9.4,3.5 Diffusion, Schroedinger  Exam    Pratice midterm I  6.1 Laplace Equation.
  5  4/26  6.3 Poisson's formula.  7.1-2 Green's identities.  7.3 Green's functions.
  6  5/3  7.4 Green's function; plane, sphere  12.1 Distributions.  12.1-2 Distributions.
  7  5/10  12.2 Green's functions revisited.  12.3 Fourier transform  12.3-4 Fourier Transforms.
  8  5/17  12.4 Source functions.   Review Questions   Review Questions
  9  5/24  14.1 Shock Waves.  13.1 Electromagnetism.  13.2 Fluids and Acoustics.
 10  5/31  Holiday  14.5 Water waves.  14.2 Solitons
 11  6/7  Exam Thursday 6/10 3-6pm    
Links to a previous syllabus using a different book and a corresponding graduate course, that I taught.
Review: Practice exams, homework solutions and lectures notes will be posted. Review for midterm I, 4/20, 6.30pm in Center 205, for midterm II, 5/19, 5/21 in class, for final, 6/5, 3:30-5:30pm in APM B402A. Final review questions
Homework:
wk sectiondate  Homework due 6pm on Thursday after Tuesday section in box 6th floor APM  Solutions
  1  3/30  1.1: 12,    1.3: 1, 7, 10,    hw1.pdf
  2  4/6  1.2: 1, 10,   2.1: 1, 7, 8,  2.2: 5,  2.4: 1, 6, 7, 8,  3.4:1,5,8,9  hw2.pdf
  3  4/13  9.1: 1, 3, 4, 8,  9.2: 1, 3, 11,  3.2:1,  hw3.pdf
  4  4/20  9.2: 9, 15, 19, 9.3: 8, 9,  9.4: 1, 2,    hw4.pdf
  5  4/27  9.2: 6,   6.1: 2, 6, 10,  6.3: 1, 2,    hw5.pdf
  6  5/4  7.1: 1, 2, 5, 7, 10,  7.2: 1,    7.3: 1, 2,  7.4: 1, 2, 3, 5, 6, 11, 12,  hw6.pdf
  7  5/11  7.2: 2,  7.4: 15, 17, 19,  12.1: 1, 2, 8, 9, 10, 12,    hw7.pdf
  8  5/18  12.2: 2, 5, 7, 8,  12.3: 1(i)-(ii), 2(i)-(iii), 4b, 5, 8, 9,  12.4: 1, 2, 3  hw8.pdf
  9  5/25   Review Questions      
 10  6/1  14.1: 3, 5, 10,   14.2: 1, 3, 13, 14.5: 1, 2, 3,    
Exams: Two midterms in class and a final Thu 6/10, 3-6. The exams will be open book but no notes.
Grade Each midterm 20%, final 50%, homeworks 10%.