Further Maths 2C1 Lecture Notes

79 Pages·2002·0.782 MB·English

by  Walet N.

2
2C1:
Further Mathematical Methods
Niels Walet, Fall 2002
Version: October 23, 2002
Copyright by Niels R. Walet, Dept of Physics, UMIST
Copying for personal or study use permitted
Copying for commercial use or profit disallowed
4
Contents
1 Introduction 7
1.1 Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 PDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Classification of partial differential equations. 11
3 Boundary and Initial Conditions 15
3.1 Explicit boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Dirichlet boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 von Neumann boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.3 Mixed (Robin’s) boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Implicit boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 A slightly more realistic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 A string with fixed endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.2 A string with freely floating endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.3 A string with endpoints fixed to strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Fourier Series 21
4.1 Taylor series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Introduction to Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Orthogonality and normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.5 When is it a Fourier series? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.6 Fourier series for even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.7 Convergence of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Separation of variables on rectangular domains 29
5.1 Cookbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 parabolic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 hyperbolic equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.4 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.5 More complex initial/boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.6 Inhomogeneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 D’Alembert’s solution to the wave equation 39
7 Polar and spherical coordinate systems 45
7.1 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2 spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8 Separation of variables in polar coordinates 51
5
6 CONTENTS
9 Series solutions of O.D.E. 55
9.1 Singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.2 *Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.2.1 Two equal roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.2.2 Two roots differing by an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.2.3 Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
9.2.4 Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10 Bessel functions 61
10.1 Temperature on a disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.2 Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
10.3 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
10.4 Bessel functions of general order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10.5 Properties of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
10.6 Sturm-Liouville theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
10.7 Our initial problem and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10.8 Fourier-Bessel series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.9 Back to our initial problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
11 Separation of variables in three dimensions 73
11.1 Modelling the eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
11.2 Properties of Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
11.2.1 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
11.2.2 Rodrigues’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
11.2.3 A table of properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
11.3 Fourier-Legendre series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
11.4 Modelling the eye–revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chapter 1
Introduction
In this course we shall consider so-called linear Partial Differential Equations (P.D.E.’s). This chapter is
intendedtogiveashortdefinitionofsuchequations,andafewoftheirproperties. However,beforeintroducing
a new set of definitions, let me remind you of the so-called ordinary differential equations (O.D.E.’s) you have
encountered in many physical problems.
1.1 Ordinary differential equations
ODE’sareequationsinvolvinganunknownfunctionanditsderivatives,wherethefunctiondependsonasingle
variable, e.g., the equation for a particle moving at constant velocity,
d
x(t)=v, (1.1)
dt
which has the well known solution
x(t)=vt+x . (1.2)
0
The unknown constant x is called an integration constant, and can be determined if we know where the
0
particleislocatedattimet=0. Ifwegotoasecondorderequation(i.e., onecontainingthesecondderivative
of the unknown function), we find more integration constants: the harmonic oscillator equation
d2
x(t)= ω2x(t) (1.3)
dt2 −
has as solution
x=Acosωt+Bsinωt, (1.4)
which contains two constants.
As we can see from the simple examples, and as you well know from experience, these equations are
relatively straightforward to solve in general form. We need to know only the coordinate and position at one
time to fix all constants.
1.2 PDE’s
Rather than giving a strict mathematical definition, let us look at an example of a PDE, the heat equation in
1 space dimension
∂2u(x,t) 1∂u(x,t)
= . (1.5)
∂x2 k ∂t
It is a PDE since partial derivatives are involved.
•
7
8 CHAPTER 1. INTRODUCTION
To remind you of what that means: ∂u(x,t) denotes the differentiation of u(x,t) w.r.t. x
∂x
keeping t fixed,
∂(x2t+xt2)
=2xt+t2. (1.6)
∂x
It is called linear since u and its derivatives appear linearly, i.e., once per term. No functions of u are
•
allowed. Terms like u2, sin(u), u∂u, etc., break this rule, and lead to non-linear equations. These are
∂x
interesting and important in their own right, but outside the scope of this course.
Equation (1.5) above is also homogeneous (which just means that every term involves either u or one of
•
its derivatives, there is no term that does not contain u). The equation
∂2u(x,t) 1∂u(x,t)
= +sin(x) (1.7)
∂x2 k ∂t
is called inhomogeneous, due to the sin(x) term on the right, that is independent of u.
Why is all that so important? A linear homogeneous equation allows superposition of
solutions. If u and u are both solutions to the heat equation,
1 2
∂2u (x,t) 1∂u (x,t) ∂2u (x,t) 1∂u (x,t)
1 1 = 2 2 =0, (1.8)
∂x2 − k ∂t ∂x2 − k ∂t
any combination is also a solution,
∂2[au (x,t)+bu (x,t)] 1∂[au (x,t)+bu (x,t)]
1 2 1 2 =0. (1.9)
∂x2 − k ∂t
For a linear inhomogeneous equation this gets somewhat modified. Let v be any solution
to the heat equation with a sin(x) inhomogeneity,
∂2v(x,t) 1∂v(x,t)
=sin(x). (1.10)
∂x2 − k ∂t
Inthatcase v+au , with u asolution tothehomogeneousequation, seeEq.(1.8), isalso
1 1
a solution,
∂2[v(x,t)+au (x,t)] 1∂[v(x,t)+au (x,t)]
1 1 =
∂x2 − k ∂t
∂2v(x,t) 1∂v(x,t) ∂2u (x,t) 1∂u (x,t)
+a 1 1 = sin(x). (1.11)
∂x2 − k ∂t ∂x2 − k ∂t
(cid:146) (cid:147)
FinallywewouldliketodefinetheorderofaPDEasthepowerinthehighestderivative,evenitisamixed
derivative (w.r.t. more than one variable).
Quiz Which of these equations is linear? and which is homogeneous?
∂2u ∂u
a) +x2 =x2+y2. (1.12)
∂x2 ∂y
∂2u ∂u ∂2u
b) y2 +u +x2 =0. (1.13)
∂x2 ∂x ∂y2
1.2. PDE’S 9
∂u 2 ∂2u
c) + =0. (1.14)
∂x ∂y2
(cid:146) (cid:147)
What is the order of the following equations?
∂3u ∂2u
a) + =0. (1.15)
∂x3 ∂y2
∂2u ∂4u ∂2u
b) 2 + =0. (1.16)
∂x2 − ∂x3∂y ∂y2
10 CHAPTER 1. INTRODUCTION