M371-M372: Numerical solution of time-dependent differential equations in applications

Goals and objectives of the course

I will offer a new two-semester course on the numerical integration of time-dependent differential equations (DEs) with a special emphasis on applications. There is enough material for a two-semester sequence of courses. The second semester will be at a higher level than the first one, and it will complement it well. The second semester can be considered as being a preparatory step for doing research work on the numerical solution of DEs whereas the first semester will be at a lower level. In these courses I will emphasize on time-dependent DEs arising in applications, especially in engineering. I would like each student to solve an application of his/her own interest. The project will be computer-oriented and will make use of the current available software, such as Fortran or Matlab codes. Each project will be the subject of a presentation in class to develop oral skills. It will also be the subject of a short report to be written. Both the report and the oral presentation will count for the final grade. The numerical solution of time-dependent DEs is a sufficiently broad topic to attract many engineering students. This course will cover the development, the mathematical analysis, and the use of practical algorithms for the numerical solution of time-dependent DEs. While initially offered as M371-M372, the longer run plan is to have a sequence numbered like M177-M277, as suggested by Ken Atkinson.

Differential equations in applications

Here is a tentative list of some possible applications to be discussed during the courses depending on the interests of the different departments and also of the students:

Classes of differential equations and related issues

We will treat in details the following classes of DEs:

We will consider other classes of differential equations among the following ones depending on the interests raised and on the applications to be treated. We will give an emphasis on the classes of DEs with most interests in terms of applications and just a short introduction or no treatment at all on the ones with least interest. A detailed treatment will be given only in the more advanced course:

Some issues that will be discussed and which require different techniques for the diverse kinds of equations:

Some more advanced topics which could be treated only in the more advanced course:

Certain of the following topics in variational calculus could be treated in relation with applications in mechanics:

Numerical integration of differential equations

We will consider diverse classes of methods and techniques to integrate numerically the various classes of DEs. We will treat in details the following class of methods in the less advanced course:

We will also treat some of the following classes of methods in the more advanced course: The following topics will be considered: Some advanced topics in numerical integration which could be treated in the more advanced course: Some of these advanced topics could be treated from a numerical perspective in the more advanced course:

Stability of numerical integration methods

In relation with numerical stability for linear ODEs we will treat the following topics in the less advanced course: Some advanced topics in stability which could be treated in the more advanced course:

Implementation issues and software

Some issues are very important when writing software for differential equations and will be treated during the two semesters according to the main interests An overview and use of current available software will be done during the courses along the semesters with as much as possible hands-on exercises.


We require good programming skills, especially in Fortran or Matlab, but Pascal, C, or C++ are also acceptable. However, programming assistance wil be given only in Fortran and Matlab. We also require good knowledge of the theory of differential equations, of vector calculus, of linear algebra, and of numerical analysis in general (M170-M171 for example, at least M72).

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