Courses Detail Information
VV557 – Methods of Applied Mathematics II
Credits: 3 Credits
Pre-requisites: Vv256 Applied Calculus IV, Vv286 Honors Mathematics IV, or consent of instructor
The course consists of three parts, each dealing with certain mathematical techniques useful for solving differential equations. 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 inte- grals, notions of convergence and delta families, the distributional Fourier transform and solutions of distributional equations.
The second part of the course applies 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 third 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, the method of images and (if time per- mits) conformal mappings. Finally, a short introduction to the use of Green’s functions for the Laplace equation in the boundary element method (BEM) is presented.