Courses Detail Information

MATH6002J – Methods of Applied Mathematics II

Instructors:

Credits: 3 Credits

Pre-requisites: Graduate standing or Vv216/Vv256/Vv286.

Description:

The course revolves around solving differential equations through methods inspired by the treatment of point sources (charges, masses, forces, etc.). Examples from mechanical as well as electrical engineering will be used throughout.

Our initial motivation is the desire to understand the treatment of point sources. Starting from the Dirac delta function as a formal symbol to denote a point source, we begin a formal treatment of generalized functions (distributions), including principal value integrals, notions of convergence and delta families, the distributional Fourier transform and solutions of distributional equations.

We will then apply the theory of distributions to ordinary differential equations (ODEs). Strong, weak and distributional solutions are introduced and general solution formulas obtained. The main focus is then on obtaining Green’s functions for boundary value problems (BVPs) for ODEs, leading to a brief discussion of solvability and modified Green’s functions for ODEs.

The final part of the course extends the ODE methods to PDEs. Green’s formulas for boundary value problems of the first, second and third kind are derived. Subsequently, methods for finding Green’s functions are explored, including that of full and partial eigenfunction expansions and the method of images. To round off the topic, a short introduction to the use of Green’s functions for the Laplace equation in the boundary element method (BEM) is presented.

Course Topics:

Introduction
Test Functions
Distributions
Families of Distributions
The Classical Fourier TransformTempered Distributions and the Fourier TransformDifferential Operators and Types of SolutionsInitial Value Problems
Second-Order Boundary Value Problems
Adjoint BVPs and Higher-Order EquationsSolvability Conditions and Modified Green’s FunctionsBoundary Value Problems for PDEsEigenfunction ExpansionsEigenfunction Expansions
The Method of lmages
The Boundary Element Method