M545 Partial Differential Equations
 
Instructor: Prof. Jennifer Mueller, 128 Weber Hall
Meeting times: 11-11:50 am MWF Engr. E205
Grading Policy:  Homework 60%, Midterm Exam 20%, Comprehensive Final Exam 20%
HW policy: HW is due in class on the due date. Up to one late HW will be accepted (the following class period), after that late HW will be penalized with a deduction of 5 pts per class period. HW should be written neatly.
Text: Partial Differential Equations of Mathematical Physics and Integral Equations by Guenther and Lee (Dover Edition) (The bookstore has plenty)
Office Hours: Monday 1 - 2 pm, Wednesday 12 - 1pm
 
Solutions to HW that has been turned in can be downloaded from RAM CT
Departmental H1N1policies (applies to all students of M545)
 
Course outline:
 
Week 1: Examples and derivations of some classical examples of PDE's. Application: transport of biological substances inside the axon.
Lecture on Neurofilament Transport
Reference article for neurofilament transport (basis of lecture)
For further reading (optional):
Article by Rubinow and Blum (1980)
Article by Blum and Reed(1985)
Article by Blum and Reed (1989)
About ALS (Lou Gerig's disease)
What is ALS?
Walk to defeat ALS (Fort Collins, Sept. 20, 2009) Info and registration
Homework
Homework 1 due Fri. Sept. 4 in class
 
Week 2: Second order equations in two variables. Classification as hyperbolic, parabolic or elliptic, the method of characteristics.
Homework
Homework 2 due Fri. Sept. 18 in class
Latex file for HW 2 if you want it.
 
Week 3: Semilinear and quasilinear equations, general nonlinear equations. Conservation laws and jump conditions, propagation of singularities.
 
Week 4: Systems of quasilinear equations. Application: arterial pulse and shock waves in the aorta.
Lecture on Arterial Pulse Sept. 18th
New! Additonal Example of Quasilinear System
Homework
Homework 3 due Wed. Sept. 30 in class
Latex file for HW 3 if you want it.
New! Coordinates of the vertices of the characteristic parallelogram
New! Example of characteristic parallelograms
 
Week 5: The one-dimensional wave equation. Solution as a superposition of traveling waves.
 
Week 6: Separation of Variables.
Homework 4 Due Monday, Oct. 12th
Latex file for HW 4 if you want it.
 
Week 7: The fundamental solution and Green's functions
 
 
Week 8: Midterm Exam Wednesday October 14th. Friday: special topic: introduction to inverse problems, Application: the acoustic wave equation and ultrasound imaging.
Example of solving a nonhomogeneous PDE by separation of variables
Note: Nonhomogeneous PDEs are not on the midterm, but this includes a nice example of the method of integrating factor.
 
Week 9: Laplace's equation. The fundamental solution, Green's functions, the maximum principle.
 
Week 10: Hilbert space methods and weak solutions of boundary value problems for Laplace's equation. The Helmholtz equation. Application: ultrasound imaging.

Homework 5: Problems 4 and 5, section 5-3, p. 165 Guenther and Lee. Due Wednesday, Nov. 4th
 
Week 11: The generalized Laplace equation. Application: electrical impedance tomography. Integral equations of potential theory.
Homework 6 Due Monday, Nov. 16th
Latex file for HW 6 if you want it.
Lecture 32: The Inner Ear
 
Weeks 12 and 13: Derivation of the diffusion equation. The transport equation. Application: cellular homeostasis and ion transport, a model of wound healing
 
Week 14: Parabolic equations in higher dimensions. Explicit solutions, properties of solutions, thermal potentials.
 
Week 15: Duhamel's principle, Special topic: time dependent inverse problems (Optical Tomography)
 
Week 16: Final Exam: Monday Dec. 14 from 7 -9 am in our usual classroom