**Phone: **825-4127 (M,W,F) or 317-5852 (Tu,Th)

**E-mail:** cratsch@math.ucla.edu

**Meeting Time:** M,W,F, 10:00 - 10:50, MS5147

**Office Hours:** M 11:00 - 12:00, W 9:00 - 10:00, W 1:00 - 2:00

**Assistant: **Frederic Gibou (fgibou@math.ucla.edu), Office Hours: tba

**Discussion:** Tu 10:00 - 10:50, MS5147

**Textbook:** Richard Haberman, * Mathematical Models*

**Prerequisites:** Lower division mathematics course sequence

**Course Description**
Mathematical Modeling is part of Applied Mathematics. It is the goal to illuminate
problems from real life by abstracting its essence to a mathematical model. It is
then hoped that the mathematical solution to the problem yields useful information
to the original problem. Thus, the follwing parts are involved:

Analyze the original problem.This step might not at all be mathematical. It requires substantial understanding of the field of the problem. Next, the essence of the problem needs to be translated into a mathematical problem, which is then solved. This might be an exact, analytic solution, or more commonly (for more complicated problems) a numerical solution. Last, the solution that has been obtained needs to be analyzed and interpreted.

This course consists of four parts: The first part is about mechanical vibrations. In this part, we will study simple linear and nonlinear ordinary differential equations, focusing on the motion of a mass-spring system, and a pendelum. In the second part, we will study population dynamics. In a simple two species model, this involves coupled ordinary differential equations. In the third part of the course, simple models of traffic flow will be discussed, which involves using partial differential equations. In the last part of the class, I will discuss some ideas of modeling epitaxial growth, which is an area of current research in the department.

**Grading Policy**

The grade of this course will be determined on the basis homeworks, one midterm, and a final. The homeworks will count 25% toward your final grade, the midterm 25% as well, and the final examination will count 50%. If you miss any of the exams or homeworks, it will be given a score of 0. In order to accommodate some problems with the collection of the homeworks, the lowest homework score will not be counted (in other words: you can afford to miss one homework assignment).

Midterm and Final Exam
There will be one midterm and one final exam. The miderm will be given on February 16,
during class; ** Please note: due to some travel I might have to do in February, I might
reschedule the midterm !.**
This will be announced as soon as possible !
The contents of the midterm will
be the material taught until the lecture before the midterm.
The final exam will be given during exam period 3, March 21, 3-6,
and will cover the material of the whole class (but with more emphasis on parts 2 to 4,
which were taught after the midterm).

It is your responsibility to be there on time. Potential time conflicts
need to be discussed with the instructor at the beginning of the course. If you do
not show up to an exam, the score will be counted as 0. If you do have an admissible,
official excuse, for the midterm, the final grade will be scaled, based on all the
other grades. However, you **must** take the final exam.

All midterms and finals will be taken without the use of books or notes. Calculators will be allowed.

**Homeworks**

Homeworks are required from every student. They are essential if you want to understand the material that is taught in the class. The homework assignments will typically be handed out on Fridays during class, and are due on Fridays, one week later. They will be collected during class as well. No late homeworks will be accepted !! If you miss a Friday class, you may turn in your homework earlier. It is your responsibility to make sure that all the pages have your name on it, and that they are securely fastened together.

Day | Date | Chapter | Topics |

M | Jan 10 | 1 | Class overview; some general remarks |

W | Jan 12 | 2,3,4 | Newton's laws, Hooke's law |

F | Jan 14 | 5,6,7 | Solution to second order ODE, harmonic oscillator |

W | Jan 19 | 8,9,10,11 | Qualitative and quantitative behaviour of harm. osc. |

F | Jan 21 | 12,13 | Damped oscillations |

M | Jan 24 | 14,15,16 | A pendelum |

W | Jan 26 | 17,18 | Linear stability analysis of a pendelum |

F | Jan 28 | 19,20 | Conservation of energy, energy curves |

M | Jan 31 | 21 | Phase planes, linear oscillator |

W | Feb 2 | 22,23,24 | Phase planes, pendelum |

F | Feb 4 | 30,31,32,33 | Population models; linear difference equations |

M | Feb 7 | 34,37 | Exponential growth or Midterm |

W | Feb 9 | Midterm or Exponential growth | |

F | Feb. 11 | 38,39 | Logistic equations |

M | Feb 14 | 40 | Growth with time delays |

W | Feb 16 | 41,42 | Linear difference equations; effect of time delays |

F | Feb 18 | 43 | Two species |

W | Feb 23 | 48,49 | Predator Prey Models |

F | Feb 25 | 50 | Solutions of Lotka Volterra equations |

M | Feb 28 | 56,57,58,59 | Traffic flow; velocity fields; traffic density |

W | Mar 1 | 60,61 | Conservation of cars |

F | Mar 3 | 63 | Traffic flow |

M | Mar 6 | 65,66 | Partial differential equations: linearization |

W | Mar 8 | 67,68 | A linear partial differential equation |

F | Mar 10 | 71 | Nonuniform traffic - method of characteristics |

M | Mar 13 | 72 | After a traffic light turns green |

W | Mar 15 | Epitaxy 1 | |

F | Mar 17 | Epitaxy 2 |

** Please note:** This is just a tentative schedule. The exact topic for each
lecture might change, as the pace of the class might get adjusted. There is no
textbook for the last two lectures.