Differential equations

Table of contents:

INTRODUCTION

1. Introduction to differential equations

   • Notation

   • Classification

   • Solution

2. Linear Differential Equations

   • Linear ODEs

   • Linear PDEs

PART I: ORDINARY DIFFERENTIAL EQUATIONS

1. First-order ODEs

   • Direction field

   • First order linear differential equation

2. Higher order Linear ODEs

   • Wronskian

   • Second-order linear ODE

   • General n-th order linear ODE

   • Linear ODEs with Constant Coefficients

3. Series ODE

   • Series solutions of ODEs

   • series_solution

   • Frobenius method

PART II: PARTIAL DIFFERENTIAL EQUATIONS

1. Introductory definitions and concepts

   • Partial differentiation

   • Auxiliary conditions

   • Well-posedness

   • Elliptic, parabolic and hyperbolic PDEs

2. First-order linear PDEs

   • Method of characteristics

3. Canonical form of second-order linear PDEs

   • Hyperbolic case

   • Parabolic case

   • Elliptic case

4. Poisson’s and Laplace’s equation

   • Laplace operator

   • Physical and geometric meaning

5. Heat (diffusion) equation

   • Separation of variables

   • Transforming inhomogeneous BCs to homogeneous

6. Solving PDEs with fourier methods

   • Diffusion equation in 1D

   • Example - \delta function

7. Wave equation

   • Physical interpretation

   • D’Alembert’s formula

8. Self-similar solutions

Further reading and references

• Prof Velisa Vesovic’s lecture notes on Maths Methods 2 (available internally on ESESIS)

• Prof Matthew Piggott’s lecture notes on Mathematics for Scientists and Engineers (available internally on ESESIS)

• Prof Alan Heavens’s lecture notes on Differenential Equations for second year physics

• Boyce, W.E., DiPrima, R.C. and Villagómez Velázquez, H., 2004. Elementary differential equations and boundary value problems.

• Kreyszig, E., 2009. Advanced Engineering Mathematics, 10th Eddition.

• Riley, K.F., Hobson, M.P. and Bence, S.J., 1999. Mathematical methods for physics and engineering.

• Needham, T., 1998. Visual complex analysis. Oxford University Press.

• Farlow, S.J., 1993. Partial differential equations for scientists and engineers. Courier Corporation.