Introduction to Partial Differential Equations (MATH10100)
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A rigorous introduction covering the basics of elliptic, hyperbolic, parabolic and dispersive PDEs. This is a pure maths course.
From Newton's laws of motion, to Maxwell's equations of electrodynamics and Einstein's equations of relativity, Partial Differential Equations (PDEs) provide a mathematical language to describe the physical world. It is perhaps less known that PDEs have been the driving force behind a large part of Analysis. The theory of Fourier series was first developed in an attempt to solve the wave and heat equations. A large part of the modern theory of Integration was developed in order to make rigorous sense of the integrals that appear in the formulas defining the Fourier coefficients. More recently, a stunning success of geometric PDEs was Perelman's proof of the Poincare conjecture, a long-standing problem in Topology, using the Ricci flow.This course is a rigorous introduction to the wave, heat, and Laplace equations. These are the prototypes of hyperbolic, parabolic and elliptic equations, the three main types of PDEs. We'll investigate under what conditions solutions exist and whether or not they are unique. We'll also study some of the basic properties of solutions such as finite speed of propagation, the Huygens principle and conservation of energy. These properties originate in Physics but have powerful mathematical expressions that allow us to develop rigorously a large part of the theory of PDEs.
Written Exam 80%, Coursework 20%, Practical Exam 0%
Additional Assessment Information
Coursework: 20%Exam 80%
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