Spring 2013: Math 322 Section 001: Applied Mathematical Analysis
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
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):
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Syllabus
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Applications of PDEs in other areas
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Chapter 1: Heat equation
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Examples of PDEs
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Conduction and convection
*Lecture 2 (02/25)
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Conduction of heat in a 1 dimensional rod
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Conservation of energy
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Heat flux
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Temperature and specific heat
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Fourier's law of heat conduction
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The diffusion equation
*Lecture 3 (01/28):
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Section 1.3 (Boundary conditions)
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Prescribed temperatures
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Insulated boundaries
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Newton's law of cooling
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Section 1.4 (Equilibrium temperature distributions)
*Lecture 4 (01/30):
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Section 1.5 (Derivation of the heat equation in two and three dimensions)
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Heat flux and the normal component
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Divergence theorem applied to the heat equation
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Fourier's law of heat conduction and the gradient
*Lecture 5 (02/01):
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Circular cylindrical coordinates and the Laplacian
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Chapter 2 (Method of separation of variables)
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Linearity
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Principle of superposition
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Homogeneity
*Lecture 6 (02/04):
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Separation of variables and Boundary Value Problems
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Eigenvalues
*Lecture 7 (02/06):
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Real and complex solutions
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Classification of PDEs
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Product solutions and the principle of superposition
*Lecture 8 (02/08):
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Superposition (extended)
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Fourier series (informal)
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Orthogonality of sines
*Lecture 9 (02/11):
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Example with constant initial conditions
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Heat conduction in a rod with insulated ends
*Lecture 10 (02/13):
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Heat conduction in a thin circular ring
*Lecture 11 (02/15):
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Laplace's equation for a circular disk
*Lecture 12 (02/18):
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Polar coordinates
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Periodic boundary conditions
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Fluid flow past a circular cylinder
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Incompressible fluids
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Stream function/ stream lines
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Irrotational flows and Laplace equation
*Lecture 13 (02/20):
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Radial and angular components of the velocity field
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Circulation
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Pressure
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Bernoulli's condition
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Drag and lift
*Lecture 14 (02/22)
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Chapter 3: Fourier series
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Piecewise continuous functions
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Piecewise smooth functions
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2L periodic functions
*Lecture 15 (02/25)
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Convergence theorem for Fourier series
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Uniform convergence and the Gibbs phenomenon
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Example
*Lecture 16 (02/27)
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Fourier cosine and sine series
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Odd functions
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The odd extension of a function
*Lecture 17 (03/01)
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Fourier cosine series
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Even extension of a function
*Lecture 18 (03/04)
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Even and odd parts
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Discussion of midterm 1
*Lecture 19 (03/06)
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Term by term differentiation of Fourier series
*Lecture 20 (03/08)
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Theorems about the term by term differentiation
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Term by term integrations of Fourier series
*Lecture 21 (03/11)
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Parseval's theorem
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Mean square error
*Lecture 22 (03/13)
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Application: Isoperimetric theorem
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Complex form of Fourier series
*Lecture 23 (03/15)
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Chapter 4: Wave equation: Vibrating Strings and membranes
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Derivation of a vertically vibrating string
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Newton's law
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Perfectly elastic strings
*Lecture 24 (03/18)
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Section 4.4 Vibrating string with fixed ends
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Vibrating strings without external forces
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Separation of variables and principle of superposition for the vibrating string
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Interpretations
*Lecture 25 (03/20)
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Interpretation of the vibrating string
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Intensity, circular frequency, first harmonic
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Traveling waves
*Lecture 26 (03/22)
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Section 4.5 Vibrating membrane
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Section 4.6: Reflection and refraction of electromagnetic and acoustic waves
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3D wave equation
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Plane-wave solutions: special traveling waves, wave vectors
*Lecture 27 (04/01)
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Dispersion relation, snell's law of refraction
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Vibrating string with external forcing
*Lecture 28 (04/03)
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Chapter 5: Sturm-Liouville and Eigenvalue Problems
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Section 5.2.1 Heat flow in a non-uniform rod
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Circularly symmetric heat flow
*Lecture 29 (04/05)
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Section 5.3 Sturm-Liouville Eigenvalue Problem
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Examples
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Boundary conditions of Sturm-Liouville type
*Lecture 30 (04/08)
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Section 5.3.2 Regular Sturm-Liouville Eigenvalue Problem
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Statement of Theorems
*Lecture 31 (04/10)
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Sturm-Liouville Problems: Examples
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Examples with Dirichlet and Neumann boundary conditions
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Equations for eigenvalues
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Illustration of the theorems using the simplest case
*Lecture 32 (04/12)
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Section 5.4 Worked example: Heat flow in a non-uniform rod without sources
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Separation of variables
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Sturm-Liouville problem for \phi(x)
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Principle of superposition
*Lecture 33 (04/15)
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Generalized Fourier coefficients
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Section 5.5 Self-adjoint operators and Sturm-Liouville Eigenvalue Problems
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Linear operators
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Lagrange's identity
*Lecture 34 (04/17)
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Green's formula
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Self-adjointness
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Proof: Orthogonal eigenfunctions
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Proof: Real eigenvalues
*Lecture 35 (04/19)
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Proof: Unique eigenfunctions (up to an scalar multiple), regular and singular case
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Nonunique eigenfunctions (periodic case)
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Section 5.6 Rayleigh Quotient
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Non-negative eigenvalues
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Minimization principle
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Trial functions
*Lecture 36 (04/22)
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Section 5.7 Worked exampleL Vibrations of a non-uniform string
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Application of the minimization principle
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Section 5.8: Boundary conditions of the third kind
*Lecture 37 (04/24)
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Section 5.9: Large eigenvalues (Asymptotic behavior)
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Highly oscillatory eigenfunctions
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Local spatial circular frequency
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Leading order approximation for \phi
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Example: Dirichlet boundary conditions
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More examples
*Lecture 38 (04/26)
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Chapter 6: Finite Difference Numerical Methods
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Classes of PDEs: Parabolic, hyperbolic and elliptic
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Section 6.2: Finite Differences and truncated Taylor series
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Polynomial approximation
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First derivative approximation
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Truncation error
*Lecture 39 (04/29)
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Centered difference approximation
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Partial difference equations. Numerical schemes
*Lecture 40 (05/01)
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Section 6.3: Heat equation
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Forward difference in time
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Truncation error of the numerical scheme
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Space-time diagram
*Lecture 41 (05/03)
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Section 6.3.3: Computations
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Stability property
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Numerical solutions shown in class for the heat equation. Matlab
*Lecture 42 (05/06)
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Section 6.3.4: Fourier-von Neumann Stability Analysis
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Stability condition
*Lecture 43 (05/08)
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Convergence
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Lax Equivalence Theorem
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Heat equation in two dimensions
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Numerical solutions shown in class for the heat equation in two dimensions. Matlab
*Lecture 44 (05/10)
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Crank-Nicholson scheme
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Implementation
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Stability analysis