Project 3: Introduction to Singular Perturbations

To introduce, understand, and apply perturbation methods in problems with singular perturbations, thereby gaining insights into the behavior of solutions in the presence of small parameters. Based on the section 2.4 Introduction to Singular Perturbations of the text book, to study this problem.

Singular perturbation problems often arise in various fields of science and engineering where the behavior of a system changes dramatically as a small parameter approaches zero.

  1. Introduction to Singular Perturbations Understand the concept of singular perturbations and identify scenarios where singular perturbations are applicable. Common examples include boundary layer problems in fluid mechanics and the behavior of electrical circuits with small inductances or capacitances.

  2. Mathematical Formulation Identify a simple singular perturbation problem that can be analytically solved. Typical problems may be of the form: \(\epsilon y'' + A y' + By = f(x), \quad y(a) = \alpha, \, y(b) = \beta\) where \(\epsilon\) is a small parameter, and \(A, B, \alpha, \beta\) are constants.

  3. In the third step of the project, students will focus on learning and applying two advanced mathematical techniques, namely regular perturbation and boundary layer methods, to address the problem identified in Step 2. For the regular perturbation aspect, students will derive the series expansion of the solution \(y(x)\) as a power series in \(\epsilon\). This involves obtaining approximations for both leading-order and next-order terms to understand how the solution behaves as \(\epsilon\) changes. On the other hand, for the boundary layer method, the first task is to identify the boundary layer in the given problem, followed by deriving inner and outer solutions within this layer. Subsequently, these inner and outer solutions are matched to synthesize an approximate global solution for the problem. Through these exercises, students will gain hands-on experience in employing regular perturbation and boundary layer methods to solve complex mathematical problems, thereby deepening their understanding of these techniques.

  4. In the fourth step of the project, students will conduct an in-depth error analysis to compare the perturbative solution with the numerical solution. A focal point of this analysis will be to discuss the range of \(\epsilon\) values for which the perturbative approach yields accurate results. The expected outcomes of this step include gaining a nuanced understanding of the concept of singular perturbations and their relevance in the problem at hand. Students will also acquire practical experience in applying perturbation methods to tackle a singular perturbation problem, along with insights into the accuracy and limitations of these methods. The evaluation criteria will focus on the quality of mathematical derivations, the accuracy and efficiency of the numerical implementations, the thoroughness of the error analysis, and an overarching understanding of the role and utility of perturbation methods in singular perturbations. Through this comprehensive task, students will cultivate both theoretical and practical skills in dealing with singular perturbation problems.