BMA 774, Partial Differential Equation Modeling in Biology

FA 08, MF 1:30-2:45 Riddick 319, with some classes in a computer lab TBA

Prof. S R Lubkin, 5-1904, Cox 513B, lubkin@eos.ncsu.edu, www4.ncsu.edu/~lubkin

Modeling space-dependent problems in biology using partial differential equations. Conservation laws, diffusion, force balance, waves, and pattern formation. Chemotaxis and other forms of cell and organism movement. Introduction to solid and fluid mechanics/dynamics. Boundary and initial conditions. Introductory numerical methods. Scaling. Case studies and projects.

Goals: A student who successfully completes this course will be able to model a spatially-dependent biological phenomenon with a system of PDE, carefully define all terms, explain the model, classify the system, identify suitable methods of solution, implement basic solution methods, interpret the results and their biological significance, clearly present the results, and, if appropriate, revise the model.

Prerequisites: ODE and eigens: MA 341 or BMA 771 or similar. Modeling: BMA 567 or 771 or similar (such as MA 573 or 574). Intellectual maturity and independence: graduate status.

Textbook: Unfortunately there is no one book out there on this particular topic, so we will use several selected readings (either handouts or on reserve). One of them is the e-book, Numerical methods for engineers and scientists, 2nd edition, by Joe D. Hoffman, from the NCSU Library.

Grades: Grades will be determined by one or two in-class exams, homework sets, a project, and a final exam. In keeping with the graduate status of this class, the homeworks will be non-routine.

Tentative Schedule

week	topics
Aug 22	anatomy of a PDE diffusion equation, derivation from random walk
Aug 25, 29	dimensional analysis separation of variables BC, IC superposition
Sep 5	diffusion on infinite domain
Sep 8, 12	Maple and Matlab
Sep 15, 19	forward Euler method
Sep 22, 26	conservation equations advection advection-diffusion equation
Sep 29, Oct 3	nonlinear PDE: Fisher-Kolmogorov equation
Oct 6	coupled PDE
Oct 13, 17	force balance equations Navier-Stokes equations porous media
Oct 20, 24	simple examples of fluid flow
Oct 27, 31	chemotaxis
Nov 3, 7	project abstracts due FEM and 2D FDM methods
Nov 10, 14	radial coordinates Krogh cylinder
Nov 17, 21	nonlinear examples from the literature
Nov 24	integral equations project writeups due
Dec 1, 5	project presentations
Fri Dec 12, 1-4	final exam

FAQ

Q: How does this class compare with other classes available?

A: MA 401, 501, 534, and 734 are only about linear PDE and methods for linear PDE. Most of the problems you will have in biology will require nonlinear models, and the linear methods are in general not useful for them. That is why we only briefly cover the linear methods in BMA 774. MA 584, 587, and 788 cover numerical methods, but do not cover modeling. Many of the topics in BMA 774 are covered in MAE courses such as 308, 310, 314, 355, 356, 455, 505, 550, 557, 560, etc. and PY courses such as 463, but BMA 774 is specifically focused on biological problems (many of which do not involve mechanics) and the subtleties of modeling them.

Q: Do I need to know Maple and Matlab already?

A: They will be introduced in class. How much we cover depends on the background of the students in the class.