Partial Differential Equations and Integral Transforms
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Module Aims
Aim 1
The aim of this course is to study fields, particularly subfields of the complex numbers, and their subfield structure by using Galois groups.
Module Content
Partial Differential Equations
Characteristics & classification of PDEs (parabolic, elliptic, hyperbolic)
First order PDEs: Diffusion equation.
Second order PDEs: The Heat Equation. Method of Separation of Variables in Cartesian and polar co-ordinates. The Wave Equation, Vibrations of an elastic string. Laplaces Equation, Dirichlet & Neumann problems, Legendre Polynomials & their properties including orthogonality & generating functions.
Integral Transforms
Laplace Transforms: Definition; transforms of standard functions. First Shift Theorem, multiplication and division by t. Inverse transforms. Partial fractions. Transforms of derivatives. Solution of first and second order differential equations by Laplace Transforms. Solutions of simultaneous differential equations. Heaviside unit step function and its Laplace Transform. Second Shift Theorem. Use of Laplace Transforms to solve PDE’s: Heat Equation, Wave Equation and Laplace’s Equation.
Fourier Transforms: Transforms for derivatives for general, even and odd functions. Use of Fourier Transforms to solve PDE’s.
Learning Outcomes
On successful completion of this module, a student will be able to:
Teaching Methods
Classes consist of formal lectures and tutorials. Lectures introduce the theory with some proof, and provide illustrative examples. Tutorial sheets containing practice questions will be provided for the students to attempt and these will be discussed in the tutorials.
The module will be assessed principally by examination. However, questions from the tutorial sheets will be assessed to gauge student understanding and engagement throughout the year.
Assessment Methods
The module is assessed through Worksheets and a Written examination.
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Optional
