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PH2130: Mathematical Methods

This is an intermediate level course on Mathematical Methods in the Physical Sciences. The course builds on the material learned by students in their first year. An important feature of this course is the integration of the learning of Mathematics with the practical use of the Mathematica software package. The intention is that students will appreaciate the utility of Mathematica both as a help in learning Mathematics and in the practical solution of physical problems. Lecture sessions are combined with Laboratory computer classes.

Course Information

Lecture notes and other material

Lectures

Introduction to the course
- Course Strategy
- Book list
Equation spotting – classification: order, degree, homogeneity
- First and Second Order Linear Ordinary Differential Equations with Constant Coefficients
- Diffusion equation & wave equation, the Laplacian
  - Diffusion and Wave Equations
- Separation of variables, 1d wave equation
- 2d wave equation, drumhead, J₀ Bessel function
  - Motion of a circular drum head
- Orthogonal functions
  - Orthogonal Functions I
  - Fourier Series Mathematica Notebook
- Orthogonal function expansions
  - Orthogonal Functions II Fourier-Bessel Series
  - Fourier-Bessel Series Mathematica Notebook
- Series solution of the Simple Harmonic Oscillator equation, sines and cosines
  - (re-)discovering sines and cosines
- Series solutions: J₀ Bessel function and Legendre polynomials and functions
  - discovering Bessel and Legendre functions
  - Legendre's equation Mathematica Notebook
- Series solutions: Frobenius method
  - Extending series solutions - The Frobenius Method
- Factorials and Euler’s Gamma function
  - Factorials and Euler's Gamma function
- Schrodinger equation I Separation of the time variable
  - Eigenfunctions and Eigenvalues
- Schrodinger equation II Harmonic Oscillator, Hermite polynomials
  - The Simple Harmonic Oscillator
- Laplacian in different coordinate systems
  - Laplacian in different coordinate systems
- Solutions of Laplace equation in rectangular, polar and spherical coordinates
  - Solutions of Laplace equation in 3d
- Fourier transforms
  - Fourier transforms – Introduction
  - Properties of the delta function
  - Applications of Fourier transforms
- Orthogonal function solution of ODEs
  - Matrices and differential equations

Mathematica Laboratory Assignments

Lab 1 : Week 1 - optional
Lab 2 : Weeks 1 and 2
Lab 3 : Weeks 3 and 4
Lab 4 : Weeks 5 and 6
Lab 5 : Week 7
Lab 6 : Week 8
Lab 7 : Week 9 and 10

PH2130 Problem Sheets

Sheet 1 : To be handed in Friday 26th. October
Sheet 2 : To be handed in Friday 9th. November
Sheet 3 : To be handed in Friday 23rd. November
Sheet 4 : To be handed in Friday 7th. December
Sheet 5 : To be handed in Friday

Past Examination Papers

Brian Cowan - E-mail: B.Cowan@rhbnc.ac.uk