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Boundary Value Problems in Cylindrical

Coordinates

Y. K. Goh

2009

Y. K. Goh

Boundary Value Problems in Cylindrical Coordinates
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Outline

Differential Operators in Various Coordinate Systems

Laplace Equation in Cylindrical Coordinates Systems

Bessel Functions

Wave Equation the Vibrating Drumhead

Heat Flow in the Infinite Cylinder

Heat Flow in the Finite Cylinder

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Differential Operators in Various Coordinates

Differential Operators in VariousCoordinates

Y. K. Goh

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Differential Operators in Polar Coordinates

Polar Coordinates x= r cos , y=r sin r2 =x2 +y2, tan = yx

Differential relations r

x = xr , x = yr2 , and 2r

x2 = y2

r3 , 2

x2 = 2xyr4 ry =

yr ,

y =

xr2

, and 2r

y2 = x

2

r3,

2y2

= 2xyr4

2

x2+

2

y2 = 0, and

x

r

x+

y

r

y = 0

Differential operators ux = cos

ur

sin r

u , and

uy = sin

ur +

cos r

u

2u=2u

x2+

2u

y2 =

2u

r2 +

1

r

u

r +

1

r22u

2

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Laplacian in Various Coordinates

Polar Coordinates (r, ) : x=r cos , y=r sin

2u=2u

r2 +

1

r

u

r +

1

r22u

2

Cylindrical Coordinates (,,z) : x= cos , y= sin , z =z

2u=1

u

+

1

22u

2+

2u

z2

Spherical Coordinates (r,,) : x=r cos sin , y =r sin sin , z=r cos 2u=

1

r2

r

r2

u

r

+

1

r2 sin

sin

u

+

1

r2 sin2

2u

2

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Dirichlet Problems on a Disk or inside Inf Cylinder

A Dirichlets problem inside a Disk or Infinite Cylinder

Variables u(,,z) =u(r, ).

PDE

2u=2u

r2+

1

r

u

r+

1

r22u

2 = 0, 0< r < L,0

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Separation of variables

u(r, ) =R(r)() = R

R +

1

r

R

R =

=.

Sturm-Liouville Eq. associated to is Harmonic Eq.

+ = 0, 0

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ODE associated to R is a Euler Equation

r2R +rR R= 0, 0< r < L.

R(r) = C0+D0ln r for= 0;Cnrn +Dnrn, forn= 0, n= 1, 2, . . . .

Regularity condition: |u| is finite in 0< r < L requiresthatD0=Dn= 0, because ln r andr

n as r 0.

A general solution 0< r < Land 0

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Coefficients

a0= 1

2

20

f() d

anLn =

1

2 2

0

f()cos nd

bnLn =

1

2

20

f()sin nd

A complex form of the general solution is

u(r, ) =

n=

cnr|n|ein, 0< r < L, 0

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Neumann Problems on a Disk or inside Inf Cylinder

A Neumanns problem on a Disk or inside Infinite Cylinder


PDE

2u=2u

r2+

1

r

u

r+

1

r22u

2 = 0, 0< r < L,0 L, 0

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Neumann Problems on a Disk or inside Inf Cylinder

General solution

u(r, ) =a0

2 +

n=0

(anrn cos n+bnr

n sin n)

Boundary condition

ur

r=L

=n=1

annLn1 cos n+bnnLn1 sin n

= f()

Coefficients

annLn1 =

1

2 2

0f()cos nd

bnnLn1 =

1

2

2

0f()sin nd

Note that the B.C. must satisfies

20

f() d= 0

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Dirichlet Problems outside a Disk or Inf. Cylinder

A Dirichlets problem outside a Disk or Infinite Cylinder


PDE

2u= 2u

r2 +1

rur

+ 1r2

2u2

= 0, r > L, 0 L, 0

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Neumann Problems outside a Disk or Inf Cylinder

A Neumanns problem outside a Disk or Infinite Cylinder


PDE

2u= 2u

r2 +1

rur

+ 1r2

2u2

= 0, r > L, 0 L, 0

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Robins Problem on a Wedge

A Robins problem on a wedge


PDE+ B.C..

2

u=

2u

r2+

1

r

u

r +

1

r22u

2 = 0, 0< r < L, 0 < . u(r, 0) =u(r, ) = 0, 0< r < L;

r u(L, ) = u(L, ) , 0 < .

Solution u(r, ) =

n=1

bn

rn

sinn

where

bn= 22(1)n

nL(+n)

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Bessel Functions

Bessels functions have wide application in mathematics,especially when dealing with cylindrical symmetry or cylindricalcoordinate systems.

Definition (Bessels functions & Bessels equation)Bessels functions J orYare solutions to the Besselsequation of order

x2y +xy + (x2 2)y= 0. (1)

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Bessel Functions

If we seek a series solution about x= 0for the Besselsequation, then

y(x) =

n=

anxn+r, an= 0forn

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Bessel Functions

If2 is not an integer,

two linearly independent solutions for Eq.(1)are

y1(x) =x

1 x

2

2(2+2) + x4

2(4)(2+2)(4+2) +. . . y2(x) =x

1 x

2

2(22) + x4

2(4)(22)(42) +. . .

After normalizing the solutions, we obtain the general solution to the Bessels equation

y(x) =c1J(x) +c2J(x), where

J(x) y1(x); and J(x) y2(x); and

J(x) is called the Bessels functions of the first kind.

Y. K. GohBoundary Value Problems in Cylindrical Coordinates
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Bessel Functions

Definition (Bessels functions of the first kind)The Bessels function of the first kind of order is defined as

J(x) =

n=0

(1)n

x2

+2n

n!(n+)!

= 1

!(

1

2x)

1

(x2

)2

+ 1+

(x2

)4

2(+ 1)(+ 2)+. . .

.

Definition (Gamma/factorial function)For is not an integer, ! is defined as

! (+ 1)

0

tet dt.

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Bessel Functions

If2 is an integer, and =N+ 1

2, for some integer N 0,

the resulting functions are called spherical Besselsfunctions

jN(x) = (/2x)1/2JN+1/2(x) (We will come back to speherical Bessels function later)

=N, for some integer N 0, JN(x) = (1)

NJN(x) is linearly dependent to JN(x);

a linearly independent second solution is YN(x) and it iscalled the Bessels function of the second kind; now the general solution of Eq.(1) is

y(x) =c1JN(x) +c2YN(x).

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Bessel Functions

Definition (Bessels functions of the second kind)The Bessels function of the second kind of order is definedas

Y(x) = J(x) cos() J(x)

sin() .

If=N, for some integer N 0, then

YN(x) = limNY(x).

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Bessel Functions

A variation of Bessels equation of order is of the form,

x2y +xy + (2x2 2)y = 0 (2)

Eq.(2) can be converted to the standard BesselsEquation by defining new variable t=x.

The general solution is

y(x) =c1J(x) +c2Y(x).

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Bessel Functions

A modified Bessels Equation is of the formx2y +xy (x2 +2)y= 0. (3)

The general solution to modified Bessels equation is

y(x) = c1J(ix) +c2Y(ix); or

y(x) = c1I(x) +c2K(x).

Definition (Modified Bessels functions)

First kind I(x) =iJ(ix) =

n=0

(x

2

)+2n

n!(n+)! ;

Second kind K(x) = limr

Ir(x)cos r Ir(x)

sin r .

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Bessel Functions

A variation of modified Bessels Equation is of the form

x

2

y

+xy

(

2

x

2

+

2

)y= 0, (4)

with the general solution

y(x) =c1I(x) +c2K(x).

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Bessel Functions

Figure: Bessels functions of thefirst kind, Jn(x).

Figure: Bessels functions of thesecond kind, Yn(x).

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Bessel Functions

Figure: Modified Besselsfunctions of the first kind, In(x).

Figure: Modified Besselsfunctions of the second kind,Kn(x).

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Bessel Functions

Another representation of the Bessels functions for integervalues of=N is in the integral form

JN(x) =

1

0 cos(N x sin ) d

YN(x) = 1

0

sin(x sin N ) d

1

0

e

N

+ (1)

N

e

Ne

x sinh

d

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Bessel Functions

Also the Bessels functions could be generated from thegenerating function

e(x/2)(t1/t) =

n=

Jn(x)tn

eiz cos =

n=

inJn(z)ein

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Bessel Functions

Recursion relations of Bessels functions

d

dx[xJ(x)] =x

J1(x)

ddx [xJ(x)] = xJ+1(x)

J1(x) +J+1(x) =2

xJ(x)

J1(x) J+1(x) = 2J(x)

J(x) = xJ(x) +J1(x) =

xJ(x) J+1(x)

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Bessel Functions

Orthogonality properties of Bessels functions: If and are two roots ofJ, then

1

0

xJ(x)J(x) dx = ,

2

J2+1()

= ,

2 J2()

Thus, the generalized Fourier series in terms ofJ is

f(x) =

n=1

cnJ(nx), and where the coefficient is given

bycn=R1

0 xf(x)J(nx) dxR10 xJ2 (nx) dx

= 2J2+1

(n)

1

0 xf(x)J(nx) dx.

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The Vibrating Drumhead (Radial Symmetric I.C.)

The vibrating of a thin circular membrane with uniform densitywith radial symmetric initial conditions is given by

PDE: 2u

t2 =c22u

r2 +

1

r

u

r , 0< r < , t >0.

B.C.: u(, t) = 0, t 0;

I.C.: u(r, 0) =f(r), u

r(r, 0) =g(r), 0< r < .

The initial conditions are said to be radial symmetric asf andg are depend only on r but not .

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Separation of variables:

u(r, t) =R(r)T(t) = T

c2T =

1

R(R +

1

rR) = 2.

The ODEs and corresponding B.C.

rR +R +2rR = 0, R() = 0 (5)

T +c22T = 0 (6)

Solutions to ODEs R(r) =c1J0(r) +Y0(r)

Since u is bounded near r= 0, this gives c2= 0; B.C.R() = 0 = J0() = 0; Let n, n= 1, 2, . . . be the roots ofJ0, then

=n= n , n= 1, 2, . . . .

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Non-trivial solutions to the ODEsRn(r) = J0(nr) =J0

n

r

, n= 1, 2, . . . ,

Tn(t) = Ancos cnt+Bnsin cnt

General solution and coefficients

u(r, t) =n=1

(Ancos cnt+Bnsin cnt)J0(nr) ,

An =

2

2J21 (n) 0 f(r)J0(nr)r dr,

cnBn = 2

2J21 (n)

0

g(r)J0(nr)r dr.

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The Vibrating Drumhead (Asymmetric I.C.)

The vibrating of a thin circular membrane with uniform densitywith asymmetric initial conditions is given by

PDE: 2u

t2 =c2

2u

r2 +

1

r

u

r +

1

r22u

2 ,0< r < ,0 0.

B.C.: u(,,t) = 0, 0

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Separation of variables: u(r,,t) =R(r)()T(t) =Tc2T

= R

R + R

rR+

r2= 2; and

r2R+rR+r2RR

=

=2.


+2

= 0, (0) = (2),

(0) =

(2)r2R +rR + (2r2 2)R= 0, R() = 0

T +c22T = 0

Solutions to ODEsmn= mn

andmn isn

th root ofJm.

() = m() =Amcos m+Bmsin m, m= 0, 1, 2, . . .

R(r) =Rmn(r) =Jm(mnr), m= 0, 1, 2, . . . , n= 1, 2, . . .

T(t) =Tmn(t) =Amncos cmnt+Bmnsin cmnt.Y. K. Goh

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General solution u(r,,t) and coefficients

u =

m=0

n=1

Jm(mnr)(amncos m+bmnsin m)cos cmnt

+

m=0

n=1

Jm(mnr)(amncos m+b

mnsin m)sin cmnt

a0n = 1

2J21 (0n)

0

20

f(r, )J0(0nr) rdrd

amn = 22J2m+1(mn)

0

20

f(r, )cos mJm(mnr) rdrd

bmn = 2

2J2m+1(mn)

0 2

0

f(r, ) sin mJm(mnr) r dr d

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a0n = 1

2

J

2

1 (0n)

0

2

0

g(r, )J0(0nr) rdrd

amn = 2

2J2m+1(mn)

0

20

g(r, )cos mJm(mnr) rdrd

bmn = 2

2J2m+1(mn)

0

2

0

g(r, )sin mJm(mnr) rdrd

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Heat Flow in the Infinite Cylinder (Radial Sym.)

The radial flow of heat in a infinite cylinder or on a disk withlateral surface is kept at zero temperature is given by

PDE:

u

t =c

22ur2 +

1

r

u

r

, 0< r < , t >0. B.C.: u(, t) = 0, t 0;

I.C.: u(r, 0) =f(r), 0< r < .

The initial conditions are said to be radial symmetric as

f is depend only on r.

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Separation of variables:u(r, t) =R(r)T(t) =

T

c2T =

1

R(R +

1

rR) = 2.


rR

+R

+2

rR = 0, R() = 0T +c22T = 0.

Solutions to ODEs

Rn(r) = AnJ0(nr)Tn(t) = Cne

c22nt

n = n/, n= 1, 2, . . . , n are roots ofJ0.

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General solution and coefficients

u(r, t) =

n=1

anJ0(nr)ec2

2

nt,

an = 2

2J21 (n)

0

f(r)J0(nr)r dr.

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Heat Flow in a Finite Cylinder

The heat flow in a finite cylinder of radius and height 2hwith lateral surface and bases are kept at zero temperature isgiven by

PDE: ut

=c2

2ur2

+1r

ur

+ 1r2

2u2

+2u

z2

,

0< r < ,0 0.

B.C.: u(,,z,t) =u(r,,h, t) =u(r,,h,t) = 0,

I.C.: u(r,,z, 0) =f(r,,z).

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Heat Flow in a Finite Cylinder

Separation of variables: u(r,,z,t) =R(r)()Z(z)T(t)

The general solution

u(r,,z,t) =

m=0

n=1

k=1

{ ec2(2mn+

2k)tJm(mnr)sin k(z+h)

(Amnkcos m+Bmnksin m) }

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