This course introduces a wide variety of mathematical tools and methods to analyze phenomena in the physical, life and social sciences. Focus of this course will be on analytical tools (the ones involving use of pen and paper) rather then the computational tools (the ones involving use of computers). Though students are encouraged to learn numerical skills with packages from programming language like Python or Matlab or C++ (or whatever else a particular student prefers) and use them in their projects.

Dimensional Analysis, Scaling, Differential Equations and Two-Dimensional Dynamical Systems. Perturbation Methods: Regular perturbation, The Poincare-Lindstedt Method, Asymptotic analysis, Singular perturbation, Boundary layers and uniform approximations, Initial layers, The WKB approximation, Asymptotic expansion of integrals, Boundary value problem. Eigenvalue Problems, Integral Equations, and Green's Functions: Sturm-Liouville problems, Orthogonal functions, Fourier Series, Integral Equations, Volterra Equations, Fredholm equations with Degenerate Kernels, Green’s function, Green’s function via eigenfunctions. Partial Differential Equations: Conservation laws, Several dimensions, Green’s identities, Energy method for uniqueness, Laplace and Poission equation, Separation of variables. Discrete Models: Difference Equations, Stochastic Models, Probability-Based Models.

MATH 23, or reasonable knowledge of differential equations.

Percentage of total grades | |
---|---|

One Midterm exam (2 hour long) | 20 % |

Homework | 20% |

Class participation | 5% |

Project | 15% |

Final Exam | 40% |

1. Midterm:May 3, 2016. Time: 4 - 6PM. Location: Carpenter Room 013

2. Project submission deadline: May 25, 2016.

3. Final Exam: June 2, 2016. Time: 3 - 6PM. Location: Kemeny 007

- Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Dominic Jordan and Peter Smith (Oxford University Press, UK, Fourth Edition, 2007)
- Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Steven Strogatz (Westview Press, Second Edition, 2015)
- Linear Partial Differential Equations for Scientists and Engineers, Tyn Myint-U and Lokenath Debnath (Birkhäuser, Fourth Edition, 2007)
- Nonlinear Partial Differential Equations for Scientists and Engineers, Lokenath Debnath (Birkhäuser, Third Edition, 2012)
- Mathematical methods for physics and engineering: A comprehensive guide, K.F. Riley, M.P. Hobson and S.J. Bence (Cambridge Univesrity Press, Third Edition, 2006)

**Reference books:**

Homework will be assigned once a week on Fridays and will be due the following Friday, unless otherwise explicitly specified by the instructor. Homework should be written neatly, clearly explaining the reasoning and feel free to use lot of extra space on the page. Please properly staple all the pages in your homework. Submit homework to the instructor after the class or during the office hours.

At the end of the course each student has to submit a research project based on the material learned during the course. Students can choose either to work on a project individually or in a team of 2 to 4 students. The main criteria for grading a project will be the originality of the idea/problem, complexity of methods, concepts and techniques used and most importantly independent learning. Students are especially encouraged to learn numerical methods and use them in their project. Once a student or group of students decide on a project then they must contact the instructor for an approval.

The final project report/document should be typed and submitted to the instructor only via the email. Print outs or written documents will not be accepted. Use of LaTex in preparing project report is highly recommended.

Download the folder containing relevant papers.

Please also read and explore section 2.5-2.6 of the textbook.

Date | Topic/Book Chapter | Slides/Class Notes | Worksheets |

03/28 | Introduction | Slides | ☺ |

03/30 | Dimensional Analysis, Ch. 1.1 |
Slides | ☺ |

04/01 | Dimensional Analysis and
Scaling Ch. 1.1-1.2 | Slides | Sheet_with_solutions |

04/04 | Scaling and Review of
Differential Equations Ch. 1.2-1.3 | Slides | ☺ |

04/06 | Review of
Differential Equations and Satbility and Bifuractions Ch. 1.3 | Slides | Sheet_with_solutions |

04/08 | Satbility and Bifuractions
Ch. 1.3 | Slides Python Notebook | ☺ |

04/11 | Phase Plane Phenomena
Ch. 2.1 | Slides | ☺ |

04/12* X-hr | TALK:
Maria Masilover
visiting grad student from TU Berlin | Slides | ☺ |

04/13 | Linear Systems
Ch. 2.2 | Slides | ☺ |

04/15 | Nonlinear Systems
Ch. 2.3 | Slides Python Notebook | Sheet_with_solutions |

04/18 | Bifurcations
Ch. 2.4 | Slides | ☺ |

04/20 | Bifurcations
Ch. 2.4 | Slides | ☺ |

04/22 | Review of HW#3 Regular perturbation Ch. 3.1 | Slides Python Notebook | ☺ |

04/25 | Poincaré-Lindstedt Method
Ch. 3.1.3 | Slides | ☺ |

04/27 | Asymptotic Analysis
Ch. 3.1.4 | ☺ | Sheet_with_solutions |

04/29 | Review of HW# 4 Singular Perturbation Ch. 3.2 | ☺ | ☺ |

05/02 | Singular Perturbation and Boundary Layer Analysis Ch. 3.2-3.3 |
Slides | ☺ |

05/06 | Boundary Layer Analysis Ch. 3.3 |
Slides | ☺ |

05/06 | Initial Layers Ch. 3.4 |
Slides
Python Notebook |
Sheet_with_solutions |

05/09 | Initial Layers and WKB Approximation Ch. 3.4 -3.5 |
Slides | ☺ |

05/11 | WKB Approximation Ch. 3.5 |
Slides | ☺ |

05/13 | WKB Approximation and Asymptotic expan. of Integrals Ch. 3.5-3.6 |
Slides | ☺ |

05/16 | Asymptotic expan. of Integrals Ch. 3.6 |
Slides | ☺ |

05/17* X-hr |
Boundary-Value Problems Ch. 5.1 |
☺ | Sheet_with_solutions |

05/18 | Sturm-Liouville Problems Ch. 5.2 |
Slides | ☺ |

05/20 | Sturm-Liouville Problems and Classical Fourier Series Ch. 5.2-5.3 |
☺ | ☺ |

05/23 | Classical Fourier Series and Basic concepts PDE Ch. 5.3 and Ch. 6.1 |
Slides | ☺ |

05/25 | Basic concepts PDE and Conservation laws Ch. 6.1 and Ch. 6.2 |
Slides | ☺ |

05/27 | Conservation laws Ch. 6.2 |
☺ | ☺ |

Students with diagnosed learning disability are encouraged to discuss with the instructor any appropriate accommodations that might be helpful. All discussions will remain confidential, although the Student Accessibility Services office may be consulted.

You are encouraged to work together on homework. However, the final writeup should be your own. On exams, all work should be entirely your own; no consultation of other persons, printed works, or online sources is allowed without the instructor's explicit permission.