Syllabus for Partial Differential Equations in Biomathematics

Dr Abdul Hassen, Robinson 229E, (856) 256-4500 Ext. 3888, hassen@rowan.edu

Office Hours: W 5:30 – 6:30pm (you are also welcome to just email me questions that you have, make an appointment outside of office hours, or else stop by my office whenever I'm around).

Textbook: Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems, Richard Haberman, Prentice-Hall, 2003 (5th Edition). ISBN:  0321797051

Prerequisites: Ordinary differential equations, vector calculus, linear algebra, Laplace transforms, knowledge of a computer algebra system (either Mathematica or MATLAB)

Course description: Students in this course will become familiar with various analytical and numerical techniques for solving partial differential equations (PDEs).  At the end of this course, students will be able to:

1. Identify mathematical models for describing various biological applications including, reaction-diffusion (R-D) systems, and spatial spread of genes and of diseases, random dispersal of population, random and chemotactic motion of microorganisms, cellular maturation, pattern formations in developmental biology and morphogenesis, and animal coat patterns.

2. Relate mathematical theory and methods like, diffusion mechanism, waves, bifurcation theory

3. Model to real problems in biology with partial differential equations.

4. Use analytical techniques such as separation of variable, Fourier series, Green’s functions for solving linear multi-dimensional PDEs.

5. Use numerical methods such as finite-difference, finite-element, and Monte-Carlo to solve PDEs.

6. Use method of characteristics and method of inverse scattering to solve nonlinear PDEs.

7. Compute and simulate solutions of Partial differential equations using computer software such as Mathematica.

Course Content Topics that may be covered include:

1. Hyperbolic, elliptic, and parabolic partial differential equations

2. Review of diffusion and conservation laws, waves and pattern formation.

3. Chemotaxis and other forms of cell and organism movement.

4. Separation of variables, superposition principle, Fourier series.

5. Green’s functions, nonhomogeneous problems.

6. Finite-difference methods, finite-element method, Monte-Carlo.

7. Fourier transforms, Laplace transforms, infinite-domain problems.

8. Nonlinear partial differential equations. 

Grading policy: Grading is based on four homework assignments (30%), student lecture (10%), a mid-term and a final exam (40%) and student expository paper and presentation (20%).

Letter Grade: A(-) 85 -100, B(-,+) 70 – 84, C(-,+) 60 – 69, D(-,+) 50- 59, F <49

Homework Policy: There will be four weekly homework assignments (posted on the WebCT course website).  Each assignment will have a written component and a CAS component. Here are instructions for handing in homework:

I. Written Component: List all exercises assigned on the front cover page of your written solution set.  Circle those exercises that you wrote complete solutions for; if you wrote only a partial solution (e.g. you completed 2 out of 4 parts), then circle the exercise number and write the corresponding fraction that you completed next to it.

II. CAS Component: You can submit your programs via e-mail as attachment.  

Student Lecture: Students in teams of 2 are required to lecture for half an hour in class period on certain sections of the material from the textbook as described in the student lecture assignment. Here are some guidelines for preparing your lecture:

1. You (as a team) are required to schedule a 20-minute meeting with me at least one week before your lecture to make sure that your group understands the material and to outline the format of your lecture.  Please don’t forget to make this appointment.

2. Please divide your lecture into two fifteen-minute sessions with a 5-minute break in between. There will be a discussion following the lectures. I will let you decide on the format that you think best serves your objectives. However, all team members should participate in lecturing and be prepared to answer questions from the class.

3. Please make sure that you work out examples during your lecture.

4. Try to describe the main mathematical and physical ideas involved in your technique without having to get into technical details.  I will be there to guide you through your lecture and to help answer questions so don't worry about the little details.  You are encouraged to keep the lecture informal and to provide a written outline of any techniques discussed so that students can follow it when doing the homework.

5. You will receive a score based on the quality of your lecture and on the average class score from homework assigned that week.

Expository Paper and Student Presentation:

Students are required to write an expository paper and to give a 15-minute in-class presentation (e.g. Powerpoint) on an advanced topic involving PDEs.  Topics must be pre-approved by the instructor.  You are highly encouraged to select a topic stemming from your current research if it involves PDEs.  You may also choose from the list of

Your talk should focus on the following aspects:

• Short explanation of mathematical model used and its validity based on physical/engineering assumptions.
• Short explanation of both numerical and analytical solutions (if available) and their validity/feasibility based on physical/engineering expectations.  This is best achieved by running through an explicit example.
• Current research status of your topic and its applications.

Your paper should focus on the following aspects:

• In-depth explanation and derivation of mathematical model used and its validity based on physical/engineering assumptions.
• In-depth explanation of both numerical and analytical solutions (if available) and their validity/feasibility based on physical/engineering expectations.  This is best achieved by running through an explicit example.
• Current research status of your topic and its applications, including a broad list of current references to peer-reviewed journal articles

Deadlines for Expository Paper and Student Presentation:

To be announced in class