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Jack Baskin School of EngineeringUC Santa Cruz

Lectures Notes - Winter 2010

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Lecture Notes, Winter 2010

WARNING: downloading the lecture notes and putting them under your pillow at night will not help you learn the material.

Preparation for class: This graduate class relies heavily on undergraduate mathematics including
  • Calculus of several variables, including partial differentiation, vector calculus and spherical/cylindrical coordinate systems
  • Concepts of linear algebra
  • Ordinary differential equations
  • Fourier series, Laplace transforms
If you do not master these concepts you will struggle with this course. To prepare for it adequately, please see the "Preparation for class" items below in the week by week lectures. RHB refers to the Riley, Hobson & Bence textbook (see Syllabus page).

Week 1:
Preparation for class:
  • Preparation for Tuesday class: master RHB Chapter 5 (in particular 5.1, 5.2, 5.3, 5.4, 5.5, 5.6) and Chaper 10 (in particular 10.7 and 10.9)
  • Preparation for Thursday class: master RHB Chapter 14 (in particular 14.1 and 14.2)
Lecture notes:
  • Lecture 0 : Reviews (not done in class). This contains the bare minimum you need to know for the next few weeks. Browse and study RHB Ch. 5 for complement.
  • Lecture 1 : Definitions and basic examples.
  • Lecture 2 : First order PDEs. Warm-up example.
Week 2:
Preparation for class:
  • Preparation for Tuesday class: make sure you understood really well the warm-up example of the last lecture.
  • Preparation for Thursday class: Read textbook 2.1 through to 2.5 and relate what you read to the lectures. The textbook presents a summary of what we have just done.
Lecture notes:
  • Lecture 3 : Method of characteristics for semilinear equations and examples. These notes are my "old" notes which introduce the method specifically through the example of linear equations with constant coefficients, while the in-class lectures was on the general method. See which one makes more sense!
  • Lecture 4 and Lecture 5 : Method of characteristics for quasilinear equations and existence and uniqueness theorem. The two lectures contain more material than what we covered in class - please read if you are interested in the details.
Week 3:
Preparation for class:
  • Preparation for Tuesday class: make sure you understood really well the method of characteristics for quasilinear equations.
  • Preparation for Thursday class: finish reading Textbook Chapter 2. Read and try to understand handout on the use of first order PDEs for probability generating functions.
Lecture notes:
  • Lecture 6 : Conservation laws (continued) and example of traffic flow.
  • Lecture 7 : Weak problems and weak solutions: shocks
Week 4:
  • Preparation for Tuesday class: make sure you MASTER partial differentiation chain rule for change of coordinate systems (RHB Chapter 5.5 and 5.6). Try questions 5.9 and 5.10 for example.
  • Preparation for Thursday class: Read textbook Chapter 3.
  • Lecture 8 : Introduction to canonical forms for 2nd order linear PDEs
  • Lecture 9 : Canonical forms, part II.
Week 5:
  • Preparation for Tuesday class: make sure you MASTER Fourier Series (RHB Chapter 12. Try questions like 12.13, 12.14, and similar questions. This handout contains the bare minimum you need to know.
  • Preparation for Thursday class: REALLY work on those Fourier Series.
  • Lecture 10 : Generic behavior of the fundamental equations, method of separation of variables. Part I: the wave equation.
  • Lecture 11 : Generic behavior of the fundamental equations, method of separation of variables. Part II: the heat equation
Week 6:
  • Preparation for Tuesday class: Read handout sections 2, 3 and 4. Really try to understand the examples.
  • Preparation for Thursday class: Work on your ODEs with forcing terms. Like y'' + a y = f(t) See RHB Chapter 15.1.1-3, and questions 15.1, 15.2.
  • Lecture 12 : Generic behavior of the fundamental equations, method of separation of variables. Part III: Laplace's equation
  • Lecture 13 : Forced linear equations. The wave equation and the diffusion equation.
Week 7:
  • Preparation for Tuesday class: work on Take-Home midterm.
  • Preparation for Thursday class: review Laplace transforms (RHS 13.2)
  • Lecture 14 : Forced linear equations. The Poisson equation.
  • Matlab routines:
    • Forced_wave.m : Matlab routine to animate the solution of the Forced Wave problem
    • Forced_diffusion.m : Matlab routine to animate the solution of the Forced Diffusion problem
    • Poisson.m : Matlab routine to plot the solution of the Poisson problem.
  • Lecture 15 : Chapter 5: Introduction to Generalized Eigenfunction expansions. Sturm Liouville theory.
Week 8:
  • Preparation for Tuesday class: Read textbook Chapter 6.1-6.4
  • Preparation for Thursday class: TBA
  • Lecture 16 and 17 : Chapter 5: Introduction to Generalized Eigenfunction expansions. Sturm Liouville theory part II and III.
Week 9:
  • Lecture 18 : Chapter 5: Forced problems and introduction to Green's functions.
  • Lecture 19 : Chapter 5: Forced problems and introduction to Green's functions part II. Example of the 3D wave equation.
  • Note: Thursday lecture moved to Friday. Preparation for class: Read Textbook 6.5.
Week 10:
  • Lecture 20 : Chapter 6: Introduction to Elliptic problems. Green's functions in 2D (Note: this was not taught in class, FIY if you ever need it).
  • Lecture 21 : Chapter 6: Elliptic equations in 2 or more dimensions. Existence/uniqueness of solutions. Fundamental solution of the Laplace equation, and Greens' functions for the infinite plane.
  • Lecture 22 : Chapter 6: Elliptic equations in 2 or more dimensions. What to do with bounded problems. (Note: this was not taught in class, FIY if you ever need it).