VV556/MATH6001J Methods of Applied Mathematics

VV556/MATH6001J Methods of Applied Mathematics

Number of Credits

3

Teaching Hours

48

Offering School

UM-SJTU Joint Institute

Course Teacher

Horst Hohberger

Course Level

Postgraduate Level

Language of Instruction

English

First Day of Class

Monday, 13rd February, 2023

Last Day of Class

Wednesday, 19th April, 2023

Course Component

Lecture

Mode of Teaching

Synchronous

Meeting Time

Week 1-10:

Mondays: 12:55 p.m.-15:30 p.m. + Wednesdays: 12:10 p.m.-13:50 p.m.

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Time Zone

Beijing Time(UTC+8)

Course-specific Restrictions (e.g. Prerequisites / Major / Year of Study)

Students must have previously taken courses on multivariable calculus, linear algebra and ordinary differential equations. Previous knowledge of partial differential equations is desirable but not necessary. The course is aimed at advanced undergraduate students as well as postgraduate students.

Course Website

English

Course 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.

Syllabus

English