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:
Practice problems for midterm 1
Hw
5 Solutions
Hw
6 Solutions
Part2
Hw 8 Due Friday, April 5 Solution Part1 Part2
Practice problems for midterm 2
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):
Syllabus
Applications of PDEs in other areas
Chapter 1: Heat equation
Examples of PDEs
Conduction and convection
*Lecture 2 (02/25)
Conduction of heat in a 1 dimensional rod
Conservation of energy
Heat flux
Temperature and specific heat
Fourier's law of heat conduction
The diffusion equation
*Lecture 3 (01/28):
Section 1.3 (Boundary conditions)
Prescribed temperatures
Insulated boundaries
Newton's law of cooling
Section 1.4 (Equilibrium temperature distributions)
*Lecture 4 (01/30):
Section 1.5 (Derivation of the heat equation in two and three dimensions)
Heat flux and the normal component
Divergence theorem applied to the heat equation
Fourier's law of heat conduction and the gradient
*Lecture 5 (02/01):
Circular cylindrical coordinates and the Laplacian
Chapter 2 (Method of separation of variables)
Linearity
Principle of superposition
Homogeneity
*Lecture 6 (02/04):
Separation of variables and Boundary Value Problems
Eigenvalues
*Lecture 7 (02/06):
Real and complex solutions
Classification of PDEs
Product solutions and the principle of superposition
*Lecture 8 (02/08):
Superposition (extended)
Fourier series (informal)
Orthogonality of sines
*Lecture 9 (02/11):
Example with constant initial conditions
Heat conduction in a rod with insulated ends
*Lecture 10 (02/13):
Heat conduction in a thin circular ring
*Lecture 11 (02/15):
Laplace's equation for a circular disk
*Lecture 12 (02/18):
Polar coordinates
Periodic boundary conditions
Fluid flow past a circular cylinder
Incompressible fluids
Stream function/ stream lines
Irrotational flows and Laplace equation
*Lecture 13 (02/20):
Radial and angular components of the velocity field
Circulation
Pressure
Bernoulli's condition
Drag and lift
*Lecture 14 (02/22)
Chapter 3: Fourier series
Piecewise continuous functions
Piecewise smooth functions
2L periodic functions
*Lecture 15 (02/25)
Convergence theorem for Fourier series
Uniform convergence and the Gibbs phenomenon
Example
*Lecture 16 (02/27)
Fourier cosine and sine series
Odd functions
The odd extension of a function
*Lecture 17 (03/01)
Fourier cosine series
Even extension of a function
*Lecture 18 (03/04)
Even and odd parts
Discussion of midterm 1
*Lecture 19 (03/06)
Term by term differentiation of Fourier series
*Lecture 20 (03/08)
Theorems about the term by term differentiation
Term by term integrations of Fourier series
*Lecture 21 (03/11)
Parseval's theorem
Mean square error
*Lecture 22 (03/13)
Application: Isoperimetric theorem
Complex form of Fourier series
*Lecture 23 (03/15)
Chapter 4: Wave equation: Vibrating Strings and membranes
Derivation of a vertically vibrating string
Newton's law
Perfectly elastic strings
*Lecture 24 (03/18)
Section 4.4 Vibrating string with fixed ends
Vibrating strings without external forces
Separation of variables and principle of superposition for the vibrating string
Interpretations
*Lecture 25 (03/20)
Interpretation of the vibrating string
Intensity, circular frequency, first harmonic
Traveling waves
*Lecture 26 (03/22)
Section 4.5 Vibrating membrane
Section 4.6: Reflection and refraction of electromagnetic and acoustic waves
3D wave equation
Plane-wave solutions: special traveling waves, wave vectors
*Lecture 27 (04/01)
Dispersion relation, snell's law of refraction
Vibrating string with external forcing
*Lecture 28 (04/03)
Chapter 5: Sturm-Liouville and Eigenvalue Problems
Section 5.2.1 Heat flow in a non-uniform rod
Circularly symmetric heat flow
*Lecture 29 (04/05)
Section 5.3 Sturm-Liouville Eigenvalue Problem
Examples
Boundary conditions of Sturm-Liouville type
*Lecture 30 (04/08)
Section 5.3.2 Regular Sturm-Liouville Eigenvalue Problem
Statement of Theorems
*Lecture 31 (04/10)
Sturm-Liouville Problems: Examples
Examples with Dirichlet and Neumann boundary conditions
Equations for eigenvalues
Illustration of the theorems using the simplest case
*Lecture 32 (04/12)
Section 5.4 Worked example: Heat flow in a non-uniform rod without sources
Separation of variables
Sturm-Liouville problem for \phi(x)
Principle of superposition
*Lecture 33 (04/15)
Generalized Fourier coefficients
Section 5.5 Self-adjoint operators and Sturm-Liouville Eigenvalue Problems
Linear operators
Lagrange's identity
*Lecture 34 (04/17)
Green's formula
Self-adjointness
Proof: Orthogonal eigenfunctions
Proof: Real eigenvalues
*Lecture 35 (04/19)
Proof: Unique eigenfunctions (up to an scalar multiple), regular and singular case
Nonunique eigenfunctions (periodic case)
Section 5.6 Rayleigh Quotient
Non-negative eigenvalues
Minimization principle
Trial functions
*Lecture 36 (04/22)
Section 5.7 Worked exampleL Vibrations of a non-uniform string
Application of the minimization principle
Section 5.8: Boundary conditions of the third kind
*Lecture 37 (04/24)
Section 5.9: Large eigenvalues (Asymptotic behavior)
Highly oscillatory eigenfunctions
Local spatial circular frequency
Leading order approximation for \phi
Example: Dirichlet boundary conditions
More examples
*Lecture 38 (04/26)
Chapter 6: Finite Difference Numerical Methods
Classes of PDEs: Parabolic, hyperbolic and elliptic
Section 6.2: Finite Differences and truncated Taylor series
Polynomial approximation
First derivative approximation
Truncation error
*Lecture 39 (04/29)
Centered difference approximation
Partial difference equations. Numerical schemes
*Lecture 40 (05/01)
Section 6.3: Heat equation
Forward difference in time
Truncation error of the numerical scheme
Space-time diagram
*Lecture 41 (05/03)
Section 6.3.3: Computations
Stability property
Numerical solutions shown in class for the heat equation. Matlab
*Lecture 42 (05/06)
Section 6.3.4: Fourier-von Neumann Stability Analysis
Stability condition
*Lecture 43 (05/08)
Convergence
Lax Equivalence Theorem
Heat equation in two dimensions
Numerical solutions shown in class for the heat equation in two dimensions. Matlab
*Lecture 44 (05/10)
Crank-Nicholson scheme
Implementation
Stability analysis