M545 Partial Differential Equations
Instructor: Prof. Jennifer Mueller, 128 Weber Hall
Meeting times: 10-10:50 am MWF Engr. E206
Grading Policy: Homework 60%, Midterm Exam 20%, Comprehensive Final Exam 20%
Text: Partial Differential Equations of Mathematical Physics and Integral Equations by Guenther and Lee (Dover Edition) (The bookstore has plenty)
Office Hours: Tues. and Wed. 11 - noon or by appt.
Course outline:
Week 1: Examples and derivations of some classical examples of PDE's. Application: transport of biological substances inside the axon.
Lecture 1
Lecture 2 (revised)
Lecture 3 (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)
Lecture 1
Lecture 2 (revised)
Lecture 3 (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)
Week 2: Second order equations in two variables. Classification as hyperbolic, parabolic or elliptic, the method of characteristics.
Homework 1
Lecture 4Homework 1
Lecture 5
Week 3: Separation of Variables. The one-dimensional wave equation. Solution as a superposition of traveling waves.
Lecture 6Lecture 7
Lecture 8
Week 4: The one-dimensional wave equation. Geometric interpretation. Application: arterial pulse and shock waves in the aorta.
Homework 2Lecture 10
Lecture 11 (Arterial Pulse)
Week 5: Semilinear and quasilinear equations, general nonlinear equations. Conservation laws and jump conditions, propagation of singularities.
Lecture 12Lecture 13
Lecture 14
Week 6: Systems of hyperbolic PDE's. Application: The retracting wall problem
Homework 3: Correction: Due Friday, Oct. 10thLecture 16
Lecture 17
Week 7: Parabolic equations in one space variable. The heat equation. Adjoints and weak solutions, the maximum principle and uniqueness, Brownian motion.
Lecture 18Lecture 19
Lecture 20
Week 8: Midterm Exam and special topic: introduction to inverse problems, Application: the acoustic wave equation and ultrasound imaging.
Midterm ExamLecture 21
Lecture 22
Lecture 23
Week 9: Laplace's equation. The fundamental solution, Green's functions, the maximum principle.
Lecture 24Intro to EIT Lecture 25
Lecture 26
Homework 4: Due Friday, Oct. 31st
Week 10: Hilbert space methods and weak solutions of boundary value problems for Laplace's equation. The Helmholtz equation, the Boltzmann equation and hydrodynamic limits.
Application: ultrasound imaging.
Lecture 27Lecture 28
Lecture 29
Week 11: The generalized Laplace equation. Application: electrical impedance tomography. Integral equations of potential theory.
Lecture 30Lecture 31
Lecture 32: The Inner Ear
Homework 5: Due Friday, Nov. 14th - Complete!
Weeks 12 and 13: Derivation of the diffusion equation. The transport equation. Application: cellular homeostasis and ion transport, transfer of neural impulses.
Lecture 33Lecture 34 and 35
Lecture 36 and 37
Lecture 38 - A Model of Wound Healing
Homework 6: Due Friday, Dec. 5th
Week 14: Parabolic equations in higher dimensions. Explicit solutions, properties of solutions, thermal potentials.
Week 15: Special topic: time dependent inverse problems. Application: a skeletal muscle metabolic model.
Week 16: Final Exam, Date and time TBA
