MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx

To avoid this divergence we terminate the series by restricting W & to integer values given by 1 - 2 2 ’ 5.25 Polynomial solutions obtained in this way can be expressed in terms of the

COSMOLOGY AND GEGENBAUER POLYNOMIALS 73

w and X are two separation constants For wave problems w corresponds to the angular frequency

Two linearly independent solutions of Equation (5.7) can be immediately written as

T ( t ) = eiwt and e-iwt, (5.10) while the Second Equation (5.8) is nothing but the differential equation [Eq

(2.182)] that the spherical harmonics satisfy with X and m given as

X = - 1 ( 1 + 1 ) , I=O,1,2 , , and m = O , f l , , f l (5.11) Before we try a series solution in Equation (5.9) we make the substitution

X ( X ) = Co sin' ~ ~ ( c o s x ) , (5.12) where

z = cosx, x E [-I, 11 (5.13) and obtain the following differential equation for C(x):

Substitution (5.12) is needed to ensure a two-term recursion relation with the Frobenius method This equation has two regular singular points a t the end points 2 = 3~1 We now try a series solution of the form

74 GEGENBAUER AND CHEBYSHEV POLYNOMIALS

Equation (5.16) cannot be satisfied for all x unless the coefficients of all the

powers of x are zero, that is

we see that both of these series diverge a t the end points, x = 3z1, as -

To avoid this divergence we terminate the series by restricting W & to integer values given by

1 - 2 2 ’

(5.25) Polynomial solutions obtained in this way can be expressed in terms of the Gegenbauer polynomials Note that these frequenck mean that one can only fit integer multiples of full wavelengths around the circumference, 2 ~ & , of the universe, that is,

( I f N ) X N = 2 T & , N = 0 , 1 , 2 , (5.26) Using the relation

W N =

we easily obtain the frequencies of Equation (5.25)

GEGENBAUER EQUATION AND 1TS SOLUTlONS 75 5.2 GEGENBAUER EQUATION AND ITS SOLUTIONS

The Gegenbauer equation is in general written as

(1 - x 2 ) L- (2X + 1 ) x Z + n(n + 2X)C;(z) = 0 (5.27) For X = 1/2, this equation reduces to the Legendre equation For the integer values of n, its solutions reduce to the Gegenbauer or the Legendre polyno- mials as:

(5.28)

5.2.1

The orthogonality relation of the Gegenbauer polynomials is given as

Orthogonality and the Generating Function

The generating function of the Gegenbauer polynomials is defined as

(5.31)

+(t,x,O,4) = (cleiWNt + C Z e - i u N t )(sin' x ) C ~ ~ , ( c o s x ) ~ m ( 8 , 4)

5.3 CHEBYSHEV EQUATION AND POLYNOMIALS

5.3.1

Polynomials defined as

Chebyshev Polynomials of the First Kind

Tn(cosx) = cos(nx), n = 0,1,2 (5.32) are called the Chebyshev polynomials of first kind, and they satisfy the Cheby- shev equation

where we have defined

x = cosx

(5.33)

(5.34)

76 GEGENBAUER AND CHEBYSHEV POLYNOMIALS

5.3.2

The Chebyshev equation after (1 + 1)-fold differentiation yields

Relation of Chebyshev and Gegenbauer Polynomials

5.3.3

Chebyshev polynomials of the second kind are defined as

Chebyshev Polynomials of the Second Kind

Un(x) = sin(nX), n = O,1,2 , (5.39) where x = cosx Chebyshev polynomials of the first and second kinds are linearly independent, and they both satisfy the Chebyshev Equation (5.33)

In terms of x the Chebyshev polynomials are written as

and

CHEBYSHEV EQUATION AND POLYNOMIALS 77

For some n values Chebyshev polynomials are given as

Chebyshev Polynomials of the First Kind

78 GEGENBAUER AND CHEBYSHEV POLYNOMIALS

5.3.4 Orthogonality and the Generating Function of Chebyshev

Polynomials

The generating functions of the Chebyshev polynomials are given as

and

(5.45) Their orthogonality relations are

80 GEGENBAUER AND CHEBYSHEV POLYNOMIALS

gives a three-term recursion relation and then drive the transformation

which gives a differential equation for C ( c 0 s x ) with a two-term recursion relation

5.2 Using the line element

ds2 = c2dt2 - &(t)2[dX2 + sin2 xdB2 + sin2 Xsin2 13d4~],

find the spatial volume of a closed universe What is the circumference?

5.3 Show that the solutions of

5.4 Show the orthogonality relation of the Gegenbauer polynomials:

5.5 Show that the generating function

PROBLEMS 81

5.7 Show the following special values:

and

5.8 Show the relations

5.9 Using the generating function

show that

5.10 Show that Tn(z) and Un(z) satisfy the recursion relations

(1 -z2)T;(2) = -nzTn(z) +nTn_l(z)

and

(1 - 22)u;(z) = -nzU,(z) + nUn-l(.)

5.11 Using the generating function

82 GEGENBAUER AND CHEBYSHEV POLYNOMIALS

Un+1 (z) - 2zUn(z) + un- 1 (z) = 0

5.15

Show the relations

Chebyshev polynomials Tn(z) and Un(z) can be related to each other

BESSEL FUNCTIONS

The important role that trigonometric and hyperbolic functions play in the study of oscillations is well known The equation of motion of a uniform rigid rod of length 21 suspended from one end and oscillating freely in a plane is given as

In this equation I is the moment of inertia, m is the mass of the rod, g is the acceleration of gravity, and 6 is the angular displacement of the rod from its

equilibrium position For small oscillations we can approximate sin B with 6;

thus the general solution is given in terms of trigonometric functions as

6(t) = Acoswot + Bsinwot , (wi = mgl/I) (6.2) Suppose the rod is oscillating inside a viscous fluid exerting a drag force proportional to 8 Now the equation of motion will be given as

13 = ke - mglo, (6.3)

where k is the drag coefficient For low viscosity the general solution is still expressed in terms of trigonometric functions albeit an exponentially decaying amplitude However, for high viscosity, (k/21)2 > wg, we need the hyperbolic functions, where the general solution is now given as

q t ) = e - ( k / 2 W [Acosh qot + B sinh got] , (9: = ( l ~ / 2 1 ) ~ - wi) (6.4)

83

84 BESSEL FUNCTIONS

Fig 6.1 Flexible chain

We now consider small oscillations of a flexible chain with uniform density

po(g/cm) and length 2 We assume that the loops are very small compared

t o the length of the chain We show the distance measured upwards from the free end of the chain with x and use y(z,t) to represent the displacement of the chain from its equilibrium position (Fig 6.1) For small oscillations we

assume that the change in y with x is small; hence ay/ax << 1 We can write the y-component of the tension along the chain as Ty(x) = pogx(ay/dx) This gives the restoring force on a mass element of length Ax as

We can now write the equation of motion of a mass element of length Ax as

Since Ax is small but finite, we obtain the differential equation to be solved for y(x,t) as

We separate the variables as

- + + w2u(z) = 0

(6.10)

(6.11)

(6.12)

To express the solutions of this equation we need a new type of function

called the Bessel function This problem was first studied by Bernoulli in

1732, however, he did not recognize the general nature of these functions As

we shall see, this equation is a special case of Bessel’s equation

86 BESSEL FUNCTIONS

(6.17) Solutions of the first two equations can be written immediately as

and

@ ( 4 ) = cleimb + cae-im@ (6.19) The remaining Equation (6.17) is known as the Bessel equation and, with the definitions

z = k p and R(p) = Jm(z), (6.20) can be written as

6.2 SOLUTIONS OF BESSEL’S EQUATION

6.2.1 Bessel Functions J i r n ( z ) , N r n ( z ) , and H 2 y 2 ) ( z )

Series solution of Bessel’s equation is given as

L m ( z ) = (-l)mJrn(z) (6.24) When m takes integer values, the second and linearly independent solution can be taken as

(6.25)

SOLUTIONS O f BESSEL’S EQUATION 87

which is called the Neumann function or the Bessel function of the second kind Note that N,(z) and Jm(z) are linearly independent even for the integer values of m Hence it is common practice to take N,(z) and Jm(z)

as the two linearly independent solutions for all n

Other linearly independent solutions of Bessel’s equation are given as the Hankel functions:

All the other functions diverge as

where y = 0.5772 In the limit as z + co functions J,(z), N,(z), H:)(z),

88 BESSEL FUNC JIONS

x-0 2”r(m + 1) ’

and

(6.39)

6.2.3

Spherical Bessel functions jl(x), ni(x), and h1(”2)(x) are defined as

Spherical Bessel Functions j , ( z), nl( z), and ( Z)

(6.40)

(6.41)

OTHER DEFINITIONS OF THE BESSEL FUNCTIONS 89

Bessel functions with half integer indices, J1++(x) and N,++(x), satisfy the differential equation

while the spherical Bessel functions, jl(x), nl(x), and hi1'2)(x) satisfy

Spherical Bessel functions can also be defined as

1

j[(X) = (-.)"'A) -, sin x

x d x x

n1(.) = (-x), (;g( -) cos x Asymptotic forms of the spherical Bessel functions are given as

X1 2 2 +.-), x < 1 , jdx) + (21 + l)!! (1 - 2(2l+ 1)

(21 - l)!! 2 2 f - x l + l (1 - 2(1- 21) + -), x < 1 ,

90 BESSEL FUNCTIONS

6.3.2 Integral Definitions

Bessel function J,(x) also has the following integral definitions:

(6.48) and

1

2 (1 - t2)n-6 cosztdt, ( n > ) (6.49)

Jn(IL.1 =

6.4 RECURSION RELATIONS OF THE BESSEL FUNCTIONS

Using the series definitions of the Bessel functions we can obtain the following recursion relations

From the asymptotic form [Eq (6.30)j of the Bessel function it is clear that

it has infinitely many roots:

J,(Z,~) = 0, 1 = 1,2,3, (6.54)

BOUNDARY CONDITIONS FOR THE BESSEL FUNCTIONS 91

% , stands for the Ith root of the nth order Bessel function When n takes integer values the first three roots are given as

6.6 BOUNDARY CONDITIONS FOR THE BESSEL FUNCTIONS For the roots given in Equation (6.55) we have used the Dirichlet boundary

condition, that is,

as

r

a

Example 6.1 Flexible chain problem: We now return to the flexible

chain problem, where the equation of motion was written as

d2u l d u

dz2 z d z

General solution of this equation is given as

Since No(wz) diverges at the origin, we choose a1 as zero and obtain the displacement of the chain from its equilibrium position as (Fig 6.2)

y(x, t ) = aoJ0(2w&)cos(wt - 6) (6.72)

BOUNDARY CONDITIONS FOR THE BESSEL FUNCTIONS 93

Fig 6.2 Jo and No functions

If we impose the condition

Example 6.2 Tsunamis and wave motion in a channel: Theequation

of motion for one dimensional waves in a channel with breadth b(z) and depth h(z) is given as

(6.75)

94 BESSEL FUNCTIONS

where ~ ( x , t ) is the displacement of the water surface from its equilib- rium position and g is the acceleration of gravity If the depth of the channel varies uniformly from the end of the channel, x = 0, to the mouth (x = u ) as h(x) = b z / u we can try a separable solution of the

(6.80) With an appropriate normalization a snapshot of this wave is shown

in Fig 6.3 Note how the amplitude increases and the wavelength d e

creases as shallow waters is reached If hb is constant or at least a slow varying function of position, we can take it outside the brackets

in Equation (6.75), thus obtaining the wave velocity as & This is characteristic of tsunamis, which are wave trains caused by sudden dis- placement of large amounts of water by earthquakes, volcanos, meteors, etc Tsunamis have wavelengths in excess of 100 km and their period

is around one hour In the Pacific Ocean, where typical water depth is

4000 m, tsunamis travel with velocities over 700 km/h Since the en- ergy loss of a wave is inversely proportional to its wavelength, tsunamis could travel transoceanic distances with little energy loss Because of their huge wavelengths they are imperceptible in deep waters; however,

in reaching shallow waters they compress and slow down Thus to con- serve energy their amplitude increases to several or tens of meters in height as they reach the shore

When both the breadth and the depth vary as b(x) = box/u and h(z) =

hox/u, respectively, the differential equation to be solved for A ( z ) b e

comes

& A dA

dx2 dx

WRONSKIANS OF PAIRS OF SOLUTIONS 95

I Y

Fig 6.3 Channel waves

where k = w2a/gho as before The solution is now obtained as

- - - - ) c o s ( w ~ + ~ ) , (6.82)

kX k2X2

( 1 2 ) + ( 1 2 ) (2.4) which is

(6.83)

6.7 WRONSKIANS OF PAIRS OF SOLUTIONS

The Wronskian of a pair of solutions of a second-order linear differential equa- tion is defined by the determinant

(6.86) (6.87)

We multiply the second equation by u 1 and subtract the result from the first equation multiplied by u 2 to get

C

w [u1 ( X I , u2(x)1 = > ;

(6.90) (6.91) (6.92) Since C is independent of x it can be calculated by using the asymptotic forms

of these functions in the limit x -+ 0 as

0, 1,2, ) in terms of JO and 51 Show that for m = 0 the second equation is

replaced by

6.2 Write the wave equation

in spherical polar coordinates Using the method of separation of variables show that the solutions for the radial part are given in terms of the spherical Bessel functions

6.3

antenna Use the result in Problem 6.2 to find the solutions for On the surface, T = a, take the solution as a spherically split

and assume that in the limit as T -+ 00 solution behaves as

6.4 Solve the wave equation

- = 0, k = - = wave number, for the oscillations of a circular membrane with radius a and clamped at the boundary What boundary conditions did you use? What are the lowest three modes?

6.5 Verify the following Wronskians:

n

98 BESSEL FUNCTIONS

6.6 Find the constant C in the Wronskian

6.7 Show that the stationary distribution of temperature, T ( p , z), in a cylin- der of length 2 and radius a with one end held a t temperature TO while the rest of the cylinder is held a t zero is given as

Hint: Use cylindrical coordinates and solve the Laplace equation,

a‘”(p, 2 ) = 0,

by the method of separation of variables

6.8

temperature f ( p ) Solve the heat transfer equation

Consider the cooling of an infinitely long cylinder heated to an initial

with the boundary condition

and the initial condition

then find C, so that the initial condition T(p,O) = f ( p ) is satisfied Where

does z, come from?

The hypergeometric equation has three regular singular points at z = 0 , l and

00; hence we can find a series solution about the origin by using the Frobenius method Substituting the series

03

y = CarxJ+T, a0 # 0, r=O

00

x u , (s + T ) ( s + T - 1) zs+-l r=O

03

-Cur ( s + ?-) ( s + T - 1) 2S+r

r = O

03 + c C u , ( s + T ) zs+-

HYPERGEOMETRIC SERIES 101

and the recursion relation

( s + r - l + a ) ( s + r - 1 + b )

a, = (s + r ) (s + r - 1 + c) a,-l, r 2 1 (7.8) Roots of the indicia1 equation are

C (a + 1) ( b + 1)

( c + 1)2 ( a + 2) ( b + 2)

s = l - c ,

(7.12)

(7.13)

(7.14)

z = -1 one needs c > a + b - 1 Similarly the second solution, y2 (z) , can be expressed in term of the hypergeometric function as

9 2 (x) = ~ ~( a - ~c + ~1, b - 8 c + 1,2 ' - C ; Z) , c # 2,3,4, (7.20) Thus the general solution of the hypergeometric equation is

(z) = A F ( a , b,c;z) + Bz'-"F(a - c + I, b - C + 1,2 - C; z) (7.21) Hypergeometric functions are also written as 2F1 (a, b, C; z)