Math 118A: Partial Differential Equations, Fall 2013

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The fact is that PDEs, in particular those that are nonlinear, are too subtle to fit into a too general scheme; on the contrary each important PDE seems to be a world in itself. (Sergiu Klainerman, 2000)

Announcements

Final scores and course grades are posted on SmartSite.

Other Class Webpages

The SmartSite page for this class is here.

The Piazza webpage for this class is here. Please post homework questions to Piazza so everyone can see the discussion. Thanks!

Important Dates

• First class: Fri, Sept 27
• Last day to add: Fri, Oct 11
• Last day to drop: Wed, Oct 23
• Last class: Fri, Dec 6
• Academic holidays: Mon, Nov 11; Fri Nov 29

Exams

There will be two Midterms and a Final

• Midterm 1: Wed, Oct 23 (in class)
• Midterm 2: Wed, Nov 20 (in class)
• Final: Tue, Dec 10, 10:30 a.m. – 12:30 p.m. (Exam Code G)

Grade

Grade will based on exams and homework, weighted as follows:

• 10% Homework
• 25% Each Midterm
• 40% Final

Text

Text: Partial Differential Equations, Walter Strauss, 2nd Ed., 2008.

Syllabus

The department syllabus for the class is here.

MATLAB scripts

A MATLAB program for the the one-dimensional wave equation is here. The MATLAB file is here.

Midterm 1

Midterm I will be in class on Wednesday, Oct 23. Here is a list of topics, with corresponding sections in the text:

• Sec 1.1: Definition of PDEs. Linear vs. nonlinear PDEs. Superposition principle for linear PDEs.
• Sec 1.2: First-order linear PDEs. Solution of the advection equation.
• Sec 1.3: Derivations of the advection, wave, and heat equations in one space dimension. (We didn't cover the case of several space dimensions, and that won't be on the exam.)
• Sec 1.5: Well-posed problems. Existence and uniqueness of solutions.
• Sec 2.1: One-dimensional wave equation. General solution. D'Alembert's solution of the initial value problem.
• Sec 2.2: Causality and energy. Domains of dependence and influence. Characteristics. Conservation of energy and uniqueness.
• Sec 2.4: One-dimensional heat equation. Green's (or source) function for the heat equation. Solution of the initial value problem.
• Sec 2.5: Comparison of waves vs. diffusions.

Some sample midterm questions are here.

Solutions to the sample midterm questions are here, but I strongly recommend that you try the problems yourself before looking at the solutions.

Solutions to Midterm 1 are here.

Midterm 2

The second midterm is next Wednesday, Nov 20. It will cover separation of variables and Fourier series. Here's a list of topics, with corresponding sections in the text:

• Sec 4.1: Separation of variables for the heat and wave equations with Dirichlet BCs.
• Sec 4.2: Neumann BCs.
• Sec 5.1: Fourier coefficients.
• Sec 5.2: Fourier sine, cosine, and full Fourier series. Connection with even, odd and periodic functions.
• Sec 6.2: Separation of variables for Laplace's equation in a rectangle.
• Sec 6.3: Separation of variables for Laplace's equation in a circle.

As usual, this list isn't exclusive. There may be questions on other PDEs, BCs, or related topics, which can be answered if you understand the basic ideas.

Some sample midterm questions are here. (The actual midterm will probably have only 4 questions.)

Solutions to the sample midterm questions are here, but I strongly recommend that you try the problems yourself before looking at the solutions.

Solutions to Midterm 2 are here.

Final

The final exam will be comprehensive. A brief outline of the topics and corresponding sections in the text is as follows:

• Sec 1.1: Definition of PDEs. Linear vs. nonlinear PDEs. Superposition principle for linear PDEs.
• Sec 1.2: First-order linear PDEs. Solution of the advection equation.
• Sec 1.3: Derivations of the advection, wave, and heat equations in one space dimension.
• Sec 1.5: Well-posed problems. Existence and uniqueness of solutions.
• Sec 2.1: One-dimensional wave equation. General solution. D'Alembert's solution of the initial value problem.
• Sec 2.2: Causality and energy. Domains of dependence and influence. Characteristics. Conservation of energy and uniqueness.
• Sec 2.3: The diffusion equation.
• Sec 2.4: One-dimensional heat equation. Green's (or source) function for the heat equation. Solution of the initial value problem.
• Sec 2.5: Comparison of waves vs. diffusions.
• Sec 4.1: Separation of variables for the heat and wave equations with Dirichlet BCs.
• Sec 4.2: Neumann BCs.
• Sec 5.1: Fourier coefficients.
• Sec 5.2: Fourier sine, cosine, and full Fourier series. Connection with even, odd and periodic functions.
• Sec 5.3: Orthogonality and eigenfunction expansions.
• Sec 5.4: Completeness and convergence of Fourier series. The Gibbs phenomenon (but not the proof).
• Sec 6.1: Laplace's equation. The maximum principle.
• Sec 6.2: Separation of variables for Laplace's equation in a rectangle.
• Sec 6.3: Separation of variables for Laplace's equation in a circle. Poisson's formula. The mean value property.

Solutions to the final are here.

Homework

Homework will be assigned weekly and a hard copy will be due in class on Fridays. Please write clearly or type and staple your solutions. If you want to type good-looking mathematics, the standard tool is LaTeX, or one of its many variants. See here to get started.

Problem numbers refer to the exercises in the text.

Set 1 (Fri, Oct 4)
Sec 1.1, p. 5: 2, 3, 10, 12
Sec 1.2, p. 9: 1, 4, 9
In case you don't have the text yet, a copy of the exercises is here.

Some solutions are here.

Set 2 (Fri, Oct 11)
Sec 1.3, p. 19: 1
Sec 1.5, p. 27: 1
Sec 2.1, p. 38: 1, 5, 6
Sec 2.2, p. 41: 2

Some solutions are here.

Set 3 (Fri, Oct 18)
Sec 1.3, p. 19: 3, 5
Sec 2.2, p. 41: 6
Sec 2.3, p. 45: 1
Sec 2.4, p. 52: 6, 7, 9

Some solutions are here.

Set 4 (Mon, Oct 28)
Sec 2.4, p. 52: 1, 10
Sec 2.5, p. 55: 2, 3 (there is a typo in 3: it should be u on the right-hand side)

Some solutions are here.

Set 5 (Fri, Nov 1)
Sec 2.3, p.45: 2, 5, 6, 8
Sec 2.5, p.55: 1

Some solutions are here.

Set 6 (Fri, Nov 8)
Sec 4.1, p.89: 1, 2, 3, 6
Sec 5.1, p.111: 1, 2(a), 3(a), 8

Some solutions are here.

Set 7 (Fri, Nov 22)
Sec 4.1, p. 89: 4
Sec 4.2, p. 92: 1, 4
Sec 5.1, p. 111: 4, 5
Sec 5.2, p. 117: 4, 11
Sec 6.2, p. 164: 4, 7
Sec 6.3, p. 172: 2, 4

Some solutions are here.

Set 8 (Fri, Dec 6)
Sec 5.3, p. 122: 2, 6, 8, 12, 15
Sec. 6.1, p. 160: 5, 6, 10, 12, 13