Schedule: MWF 11:00 – 10:50 AM. Van Vleck B 119

Office hours: WF 10:00 – 11:00 AM. Th 12:00 – 1:00 PM


Syllabus: Can be found here

Textbook: Richard Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems. Pearson Prentice Hall


Homework assignments:


Hw 1 Solution Part1 Part2

Hw 2 Solution Part1 Part2

Hw 3 Solution

Hw 4 Solution


Practice problems for midterm 1


Midterm1 Solution

Hw 5 Solutions
Hw 6 Solutions Part2

Hw 7 Due March 22 Solutions

Hw 8 Due Friday, April 5 Solution Part1 Part2


Practice problems for midterm 2


Midterm 2 Solutions


Hw 9 Due Friday, April 19 Solutions

Hw 10 Due Friday, April 26 Solutions

Hw 11 Due Friday, May 3 Solutions

Hw 12 Due Friday, May 10 Solutions MathLabFile


Practice problems for chapters 5 and 6


Important information:


Midterm 1: March 1st 5:30-7:00 PM. Room: 6102 Soc Sci

Midterm 2: April 5th 5:30-7:00 PM. Room: 6102 Soc Sci

Final exam: May 16th 12:25-2:25 PM. Room: TBA


*See the syllabus for more information on exam policies.


Grading:

Midterm 1: 20 %

Midterm 2: 20 %

Final exam: 40 %

Homework: 20%


Note: Any student with a documented disability should contact me as soon as possible so that we can discuss arrangements to fit your needs.


Brief lecture outline:


*Lecture 1 (01/23):

  1. Syllabus

  2. Applications of PDEs in other areas

  3. Chapter 1: Heat equation

  4. Examples of PDEs

  5. Conduction and convection


*Lecture 2 (02/25)

  1. Conduction of heat in a 1 dimensional rod

  2. Conservation of energy

  3. Heat flux

  4. Temperature and specific heat

  5. Fourier's law of heat conduction

  6. The diffusion equation


*Lecture 3 (01/28):

  1. Section 1.3 (Boundary conditions)

  2. Prescribed temperatures

  3. Insulated boundaries

  4. Newton's law of cooling

  5. Section 1.4 (Equilibrium temperature distributions)


*Lecture 4 (01/30):

  1. Section 1.5 (Derivation of the heat equation in two and three dimensions)

  2. Heat flux and the normal component

  3. Divergence theorem applied to the heat equation

  4. Fourier's law of heat conduction and the gradient


*Lecture 5 (02/01):

  1. Circular cylindrical coordinates and the Laplacian

  2. Chapter 2 (Method of separation of variables)

  3. Linearity

  4. Principle of superposition

  5. Homogeneity


*Lecture 6 (02/04):

  1. Separation of variables and Boundary Value Problems

  2. Eigenvalues


*Lecture 7 (02/06):

  1. Real and complex solutions

  2. Classification of PDEs

  3. Product solutions and the principle of superposition


*Lecture 8 (02/08):

  1. Superposition (extended)

  2. Fourier series (informal)

  3. Orthogonality of sines


*Lecture 9 (02/11):

  1. Example with constant initial conditions

  2. Heat conduction in a rod with insulated ends


*Lecture 10 (02/13):

  1. Heat conduction in a thin circular ring


*Lecture 11 (02/15):

  1. Laplace's equation for a circular disk


*Lecture 12 (02/18):

  1. Polar coordinates

  2. Periodic boundary conditions

  3. Fluid flow past a circular cylinder

  4. Incompressible fluids

  5. Stream function/ stream lines

  6. Irrotational flows and Laplace equation


*Lecture 13 (02/20):

  1. Radial and angular components of the velocity field

  2. Circulation

  3. Pressure

  4. Bernoulli's condition

  5. Drag and lift


*Lecture 14 (02/22)

  1. Chapter 3: Fourier series

  2. Piecewise continuous functions

  3. Piecewise smooth functions

  4. 2L periodic functions


*Lecture 15 (02/25)

  1. Convergence theorem for Fourier series

  2. Uniform convergence and the Gibbs phenomenon

  3. Example


*Lecture 16 (02/27)

  1. Fourier cosine and sine series

  2. Odd functions

  3. The odd extension of a function


*Lecture 17 (03/01)

  1. Fourier cosine series

  2. Even extension of a function


*Lecture 18 (03/04)

  1. Even and odd parts

  2. Discussion of midterm 1


*Lecture 19 (03/06)

  1. Term by term differentiation of Fourier series


*Lecture 20 (03/08)

  1. Theorems about the term by term differentiation

  2. Term by term integrations of Fourier series


*Lecture 21 (03/11)

  1. Parseval's theorem

  2. Mean square error


*Lecture 22 (03/13)

  1. Application: Isoperimetric theorem

  2. Complex form of Fourier series


*Lecture 23 (03/15)

  1. Chapter 4: Wave equation: Vibrating Strings and membranes

  2. Derivation of a vertically vibrating string

  3. Newton's law

  4. Perfectly elastic strings


*Lecture 24 (03/18)

  1. Section 4.4 Vibrating string with fixed ends

  2. Vibrating strings without external forces

  3. Separation of variables and principle of superposition for the vibrating string

  4. Interpretations


*Lecture 25 (03/20)

  1. Interpretation of the vibrating string

  2. Intensity, circular frequency, first harmonic

  3. Traveling waves


*Lecture 26 (03/22)

  1. Section 4.5 Vibrating membrane

  2. Section 4.6: Reflection and refraction of electromagnetic and acoustic waves

  3. 3D wave equation

  4. Plane-wave solutions: special traveling waves, wave vectors


*Lecture 27 (04/01)

  1. Dispersion relation, snell's law of refraction

  2. Vibrating string with external forcing


*Lecture 28 (04/03)

  1. Chapter 5: Sturm-Liouville and Eigenvalue Problems

  2. Section 5.2.1 Heat flow in a non-uniform rod

  3. Circularly symmetric heat flow


*Lecture 29 (04/05)

  1. Section 5.3 Sturm-Liouville Eigenvalue Problem

  2. Examples

  3. Boundary conditions of Sturm-Liouville type


*Lecture 30 (04/08)

  1. Section 5.3.2 Regular Sturm-Liouville Eigenvalue Problem

  2. Statement of Theorems


*Lecture 31 (04/10)

  1. Sturm-Liouville Problems: Examples

  2. Examples with Dirichlet and Neumann boundary conditions

  3. Equations for eigenvalues

  4. Illustration of the theorems using the simplest case


*Lecture 32 (04/12)

  1. Section 5.4 Worked example: Heat flow in a non-uniform rod without sources

  2. Separation of variables

  3. Sturm-Liouville problem for \phi(x)

  4. Principle of superposition


*Lecture 33 (04/15)

  1. Generalized Fourier coefficients

  2. Section 5.5 Self-adjoint operators and Sturm-Liouville Eigenvalue Problems

  3. Linear operators

  4. Lagrange's identity


*Lecture 34 (04/17)

  1. Green's formula

  2. Self-adjointness

  3. Proof: Orthogonal eigenfunctions

  4. Proof: Real eigenvalues


*Lecture 35 (04/19)

  1. Proof: Unique eigenfunctions (up to an scalar multiple), regular and singular case

  2. Nonunique eigenfunctions (periodic case)

  3. Section 5.6 Rayleigh Quotient

  4. Non-negative eigenvalues

  5. Minimization principle

  6. Trial functions


*Lecture 36 (04/22)

  1. Section 5.7 Worked exampleL Vibrations of a non-uniform string

  2. Application of the minimization principle

  3. Section 5.8: Boundary conditions of the third kind


*Lecture 37 (04/24)

  1. Section 5.9: Large eigenvalues (Asymptotic behavior)

  2. Highly oscillatory eigenfunctions

  3. Local spatial circular frequency

  4. Leading order approximation for \phi

  5. Example: Dirichlet boundary conditions

  6. More examples


*Lecture 38 (04/26)

  1. Chapter 6: Finite Difference Numerical Methods

  2. Classes of PDEs: Parabolic, hyperbolic and elliptic

  3. Section 6.2: Finite Differences and truncated Taylor series

  4. Polynomial approximation

  5. First derivative approximation

  6. Truncation error


*Lecture 39 (04/29)

  1. Centered difference approximation

  2. Partial difference equations. Numerical schemes



*Lecture 40 (05/01)

  1. Section 6.3: Heat equation

  2. Forward difference in time

  3. Truncation error of the numerical scheme

  4. Space-time diagram


*Lecture 41 (05/03)

  1. Section 6.3.3: Computations

  2. Stability property

  3. Numerical solutions shown in class for the heat equation. Matlab


*Lecture 42 (05/06)

  1. Section 6.3.4: Fourier-von Neumann Stability Analysis

  2. Stability condition


*Lecture 43 (05/08)

  1. Convergence

  2. Lax Equivalence Theorem

  3. Heat equation in two dimensions

  4. Numerical solutions shown in class for the heat equation in two dimensions. Matlab


*Lecture 44 (05/10)

  1. Crank-Nicholson scheme

  2. Implementation

  3. Stability analysis