Math 110B. (former 132A) Partial Differential Equations- Spring 08 - 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 10-11 Center 205. Instructor Hans Lindblad, lindblad@math.ucsd.edu. Office hour: M 1.30-2.30.
Sections: Tu 3-4 in Center 205. TA Patrik Driscoll (pdriscol@math.ucsd.edu). Office hour: Mon 3-4, Fri 1-2.
Text: Strauss Partial Differential Equations
Syllabus: The text was also used for 110 and will continue where it left off. 110 dealt with linear PDEs in one space dimensions and boundary problems using Fourier series. 132A will deal 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 in Chapter 9, starting with a review of the one dimensional case in Chapters 2 and 3. Second part will be about Laplace equations in 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 in Chapter 13 and nonlinear equations in Chapter 14.
Preliminary schedule:
wk  date  Monday  Wednesday  Friday
  1  3/31  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  4/7  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  4/14  9.2 Wave equation in space-time.  9.3 Characteristics, Singularities.  6.1,9.1 Lorentz inv. 9.3 Sources.
  4  4/21  9.4,3.5 Diffusion, Schroedinger  Exam    Pratice midterm I  6.1 Laplace Equation.
  5  4/28  6.3 Poisson's formula.  7.1-2 Green's identities.  7.3 Green's functions.
  6  5/5  7.4 Green's function; plane, sphere  12.1 Distributions.  12.1-2 Distributions.
  7  5/12  12.2 Green's functions revisited.  12.3 Fourier transform  12.3-4 Fourier Transforms.
  8  5/19  12.4 Source functions.   Review Questions   Review Questions
  9  5/26  Holiday  Exam    Review Questions  14.1 Shock Waves.
 10  6/2  14.1 Show Waves, 14.2 Solitons.  13.1 Electromagnetism.  13.2 Fluids and Acoustics.(13.5)
 11  6/9  Exam 8-11am in Center 205.    
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, 4/22, 6.30pm in Center 205, for midterm II, 5/21, 5/23 in class, for final, 6/7, 3:30-5:30pm in APM B402A. Final review questions
Homework:
wk sectiondate  Homework (tentative)  Due 6pm in box 6th floor APM  Solutions
  1  4/1  1.1: 12,    1.3: 1, 3, 6, 7, 9, 10,    
  2  4/8  1.2: 1, 10, 11,   2.1: 1, 4, 7, 8,  2.2: 4, 5, 6,  2.4: 1, 5, 6, 7, 8,  3.4:1,4,5,7,8,9  
  3  4/15  9.1: 1, 3, 4, 8,  9.2: 1, 3, 9, 11,  9.2: 15, 19,  3.2:1,  
  4  4/22  9.3: 1, 2, 3, 4, 8, 9,  9.4: 1, 2,    
  5  4/29  9.2: 6,   6.1: 2, 6, 10,  6.3: 1, 2,    
  6  5/6  7.1: 1, 2, 5, 7, 10,  7.2: 1,    7.3: 1, 2,  7.4: 1, 2, 3, 5, 6, 11, 12,  
  7  5/13  7.2: 2,  7.4: 15, 17, 19, 25,  12.1: 1, 2, 8, 9, 10, 11, 12,    
  8  5/20  12.2: 2, 5, 7, 8, 11,  12.3: 1, 2, 4b, 5, 8, 9,  12.4: 1, 2, 3, 5  
  9  5/27   Review Questions      
 10  6/3  14.1: 3, 8, 10, TBA   14.2: 1, 3, ...TBA,  
Exams: Two midterms in class and a final Mon Jun 9, 8-11. The exams will be open book but no notes.
Grade Each midterm 20%, final 50%, homeworks 10%.