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case western reserve university

Math 224: Intro. Differential Equations, Modeling Approach
 
 

 

Exams

Midterms

Thr. 11:30-1pm

I. Oct. 1

II. Nov. 12

Final

Dec. 8, 4-7pm

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Office hours:

  • Tue: 4-5pm
  • Wed: 3-4pm
  • Yost 331

Additional Help

(TA) Xiaoxia Wang

  • Mon: 4-5pm
  • Wed: 4-5pm
  • Yost 203

Description

Differential equations (DE) and Calculus in general were develop in XVII-XVIII centuries by Newton, Euler, Lagrange (and their followers). They mark the advent of modern science. Since early days DEs were widely used in diverse fields of science, engineering, technology. Today the scope of applications include life and social sciences. In this course we shall use DEs, as a modeling tool for diverse applications. Along with the traditional analytic methods, we shall extensively use modern computer technology. Many software packages are available now for solving DE: analytically, graphically and numerically.

The traditional approach aims at analytic solutions (exact or approximate), studied by mathematical tools (Algebra, Calculus, Geometry). Such matematical methods (however advanced) have severe limitations. We shall many simple problems that arise in diverse applications, which allow no analytic solutions. Even when available, analytic solutions would often require additional work to make useful predictions, so one still needs computer tools.

In this course we shall combine analytic methods and computer tools to address many problems, models that lie beyond any traditional scope of DE courses.

The emphasis in this class is placed on Mathematics as tool for Modeling and Exploration

Students will learn

1. How to formulate DE- model for a particular system, process, phenomena

2. How to solve and analyze them using analytic and computater tools;

3. How to apply mathematics (DEs) to real data analysis and predictions, draw conclusions, address open-ended problems.

Many questions and problems in homework, exams, projects will ask to explain the meaning of your work, and draw conclusion. The projects will also require critical analysis and summary.


Computer tools


Many software packages have nowadays built-in routines for solving DE. They include:

1. Mathematica - a powerful, multipurpose software, that allows symbolic, numeric and graphic manipulation of data. A special add-on package VisualDSolve by S. Wagon is handy for visualization of DE solutions.

A brief introduction to Mathematica. with demonstrations is provided in the Wolfram screencast.

2. MatLab is another multipurpose software. It has specialized DE packages designed by J. Polking. Choose the proper version of m-files dfield and pplane for your version of Matlab (e.g. 5, 6,7), download them, and place in the Matlab path (e.g. c:/../matlab../work). Another m-file odesolve solves arbitrary systems of differential equations.

Mathematica and Matlab are available through CWRU software libraries (find more details there). You can choose your own DE software, but it will be essential for many homework and projects.

Text


Differential Equations, by Blanchard, Devaney and Hall, PWS.

Supplementary material (handouts, Mathematica notebooks, etc.) are provided on class web pages: Mathematica notebooks and notes, and homework problems, solutions, tests and projects.

The lectures will cover material non included in the book, or exposition different from book. The students are responsible for all this material. I expect no prior computer experience, and most computer materials will be provided.

Lab projects


There will be few (1-2) special team-projects based on the book labs (at the end of each chapter), or other. Most of them will require both mathematics and computer. You will be given about 2 weeks for each project. The results will be presented in the report. The report should include

Title Page, with authors, project title and date.

Abstract (no more than half-page long): a brief summary of the problem, method used to solve it, and your results.

Main body: (i) Description of the problem and the mathematical (DE) model; (ii) Mathematical (analytic) solution and analysis, when available; (iii) Numeric and graphic results. Explain the significance of graphics and formulas necessary to illustrate the results (each picture, graph or formula should be clearly labeled). Use plots judiciously, do not clutter your report. Be concise and clear in your writing.

Conclusions: the meaning and significance of the problem and your results.

Appendix (optional) could include further details of calculations, methods, formulas, computer work, graphics etc., that you do not want to include in the main body. All auxiliary calculations, graphics etc. could be put in the appendix.

Syllabus

The syllabus must be regarded as a broad overview of topics (based mostly on the text). However, the presentation might be different from the text and some additional topics will be discussed in lectures.

TOPIC

SECTION

Ch. 1: First order DE
  1. Some basic DE modes
  2. Analytic, numeric, qualitative methods
  3. Eqilibria and phase-line
  4. Bifurcations
  5. Linear equations and multipliers

Sections: 1.1-1.8

Ch. 2: First order differential systems (DS)
  1. DS Models
  2. Phase plane
  3. Analytic and numeric methods
  4. The Lorenz system

2.1-2.5

Ch. 3: Matrix algebra and Linear DS
  1. Linear superposition
  2. Eigenvalue method
  3. Phase plane and bifurcations

3.1-3.8

Ch. 4: 2-nd orde DE, oscillators
  1. Forced harmonic oscillator
  2. Sinusoidal and periodic force
  3. Amplitude and phase

4.1-4.4

Ch. 6: Laplace Transform
  1. General method
  2. Perioic , discontinuous and delta sources

5.1-5.6

Ch. 5: Non-linear Systems
  1. Equilibria and stability
  2. Hamiltonian and dissipative systems
  3. Cycles and chaos
  4. Forced nonlinear oscillations

6.1-6.5 

Additional topics (time permitting).

Homework is assigned for each class, and collected once every week. Selected problems are graded and posted

Grade.

Your grade in this course is based on two midterms (1/6 each), the final exam (1/3), the homework (1/6)+ projects (1/6).