Bernatz R. Fourier Series and Numerical Methods for Partial Differential Equations

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122 HEAT TRANSFER IN 1D

for which a solution for A(t) and B(t) is guaranteed if the determinant

α2&ι

—

a\b\c

—

a\b2

is not zero. Because the IBVP in question pertains to ID heat transfer, we know that

for many applications a\ = hi,

ct2

—«1,

i>i = h^ and 62 = «2» and each of these

constants are non-negative. The determinant is then given by

—/i2^i — h\h2C

—

/11K2

which is zero if both h\ and /12 are zero. This is the case when flux is prescribed at

both boundaries. See Exercise 5.5 for a possible way to overcome this shortcoming.

We will assume that A(t) and B(t) can be determined. What is left to do is solve

the semihomogeneous IBVP for U(x, t) outlined as

IBVP<

= kU

(x,t)+q*(x,t), 0<x<c (PDE)

U(x,0) = f*(x) (IC)

(5>28)

oi£/(0,

t) + a

(0, t) = 0 (BC1)

t/(c, t) + 6

(c, i) = 0 (BC2)

where q*(x,t) = q(x, t) - A'(t)x- B'(t) and/*(x) = u(x,0)-A(0)x-B(0). The

semihomogeneous IBVP (5.28) is solved using the variation of parameter methods

described in Section 5.2 to determine a series solution for U(x,t). This solution

is used to express the solution u(x,t) to the original IBVP (5.24). Section 5.3.1

provides an example of this solution technique.

5.3.1 Example: Nonhomogeneous Boundary Condition

The example presented in this section pertains to an infinite slab with faces at x = 0

and x = c. The face at x = 0 is insulated, while the temperature T

of the

surroundings is maintained at x = c. The resulting IBVP for this situation is

IBVP

} = xy

X) K

^> (5.29)

—

(x,t), 0 < x < c

u(x,0)

—

x(l

—

(0,i}=0

hu(c,t) 4- Ku

(c,t)

—

(PDE)

(IC)

(BC1)

(BC2)

The positive constant k is the thermal diffusivity, h is the heat exchange coefficient,

and K is a positive constant representing the conductivity of the material. Here,

q(x, t) = 0 and f(x) = x(l - x) in IBVP (5.24).

Using the methods outlined in Section 5.3, we assume a solution u(x, t) of form

u(x, t) = U{x, t) + xA(t) + B(t)

Substituting for u(x,t) in IBVP (5.29) results in the homogeneous IBVP for U as

shown below.

NONHOMOGENEOUS BOUNDARY CONDITIONS 123

(x,

t), 0 < x < c (PDE)

IBVP ^

' °)

) ~

Ts (IC)

(5.30)

(0,t)=0

(BC1)

/i£7(c,

t) + nU

(C,

£)

= 0 (BC2)

and the determination of A(t) and B(t) with

A(t) = 0 and B(t) = T

Consequently, the functions q*(x, t) and /*(x,

£)

are

g*(x,¿) = ^(χ,ί)-^

(ί)-

β/

(*) =0-χ·0-0 = 0

and

f*(x)

= x(l -x)- A(0) - B(0) =

x(l-x)-0-T

Because q(x, t) is identically zero and neither A nor B are dependent on t, q* in this

case is zero so that the PDE in the transformed IBVP is homogeneous. Consequently,

the form of U(x, t) is

U(x,t) = ^2a

(x)T

(t)

n=0

where a

has no t dependence as in the nonhomogeneous case.

In search of our

(x),

the following Strum-Liouville for X results.

X"{x)

+ XX (x) = 0 X'(0) = 0 -X(c) + X\c) = 0

The eigenvalues and eigenfunctions for

this

Sturm-Liouville case were determined

in Exercise 3.I.e. They are given below:

v2.

Λ.Λ- /

hc/κ + sin

= α

; Χ

{χ) = Λ

— —^^ cosa

h/κ,

tana

c= n

—

1,2,3,...

The ODE for T is

T\t) + k\

T(t) = 0 with λ

= α

(5.31)

Consequently, the solutions of Equation (5.31) are constant multiples of

(t) = e-**'*

Combining the results for X and T give the form of C/

(x,

¿).

That is,

2/i 2

(x,t)

= \l-—— costtnxe"^* n = 1,2,3,..,

he + sin a;

124 HEAT TRANSFER IN 1D

Applying the Generalized Linearity principle, we know the function U(x,t),

defined as the series

U(x,t) = ^2,a

{x,t)

n=l

satisfies the homogeneous PDE and BCs of IBVP (5.30).

What remains of the solution process is to determine a

so that U(x,t) satisfies

the nonhomogeneous initial condition of IBVP (5.30). That is, find a

such that

2h/n

;

sin

The orthonormal nature of the eigenfunctions assure us that when

U(x,0) = y^

-— —~ cosa

x = x(l

-x)-T

y hcjK 4- sin a

= / \ T—, —Ö cosa

x I (x(l - x) -

)dx

(5.32)

hc/K,

sin a

c I

the series converges to the value

f(x+) - T

+ f(x-) - T

for all x in (0,

c).

(Refer to Section 2.13.)

The solution to the original IBVP (5.29) is obtained by adding the term T

to the

result for U(x,t). That is,

u(x,t) =

U{x,t)+T

(5.33)

cos(a

x)e-^

+ T

(5.34)

_^. »\„-<*u

hc/κ -f- sin a

n=l V

where a

is given by Equation (5.32) and a

satisfies

tana

c = —

In order to visualize the solution, values will be provided for the constants in this

example. Let c

2, k = 0.01, κ = 0.1, and T

= 4. Calculation of u(x, t) is made

for two values of h to demonstrate the effect of the heat exchange coefficient of the

time-evolution of u. The first computation is made for h = 0.1. The temperature

surface u(x, t) is shown in Figure 5.3 for 0 < x < 2 and 0 < t < 100. The parabolic

shape of the initial temperature distribution is evident for t

—

0. As t increases, the

left-hand boundary temperature (at x — 2) increases to the limiting temperature of

= 4. The rate at which this temperature increases is controlled, in part, by the heat

exchange coefficient h. In this case, the temperature u{2,100) is « 2.5. When the

same calculation is done, but for h = 0.01, the left-hand boundary temperature at t

= 100 has reached an approximate value of 1.0, as evident in Figure 5.4.

NONHOMOGENEOUS BOUNDARY CONDITIONS 125

1 1.5 2

Figure 5.3 Plot of

the

material temperature u(x, t) for h

0.1.

u(xj)

Figure 5.4 Plot of

the

material temperature u(x, t) for h - 0.01.

5.3.2 Example: Time-Dependent Boundary Condition

In this section, we consider an example wherein the boundary condition on the left

face of our infinite slab is time-dependent. The details of the IB VP are

IBVP<

( u

= ku

(x,t), 0<#<c (PDE)

u(x, 0) = 0 (IC)

(5.35)

M0,t)=0 (BC1)

[ hu{c, t) +

(C,

t) = h(sint + 1) (BC2)

The IB VP (5.35) specifies there

no internal source of heat and the initial temperature

distribution is zero for all x. The face at x

0 of the infinitely long slab is insulated,

126 HEAT TRANSFER IN

while Newton's law

heat transfer governs the boundary condition

—

where

the temperature

the surrounding environment

given

sin

t + 1.

A solution

the form

u(x, t) = U(x, t) + A(t)x

B(t) is

sought. The semiho-

mogeneous IBVP

for U

results through

the

determination

of A(t) and B(t) by the

system

A(t)

= 0

h(A(t)c

+ B(t)) + KA(t) =

ft(sini

+ l)

that gives

B(t) =

sint

+ 1 and A(t) = 0.

With

A(t) and B(t)

determined,

the

functions q*(x,

and

f*(x, t) are

q*(x,t) =0-Ä(t)x-B'(t)

-cost

and

f*(x,t)

= 0- A(0)x - B(0) = -1

so that the semihomogeneous IBVP

{

(x,

t) -

cosi,

0 < x < c (PDE)

tf(s,0) =

-l (IC)

(536)

E/*(0,í)=0

(BC1)

hU(c, t) + KU

(C, t) = 0 (BC2)

From Section 5.2, we know

the

solution

for U(x, t) is

U{x,t)

= Y^B

(t)X

{x)

where

(x)

are the

appropriate eigenfunctions determined

the boundary condi-

tions

IBVP (5.36),

and

the time-dependent coefficients

(t)

are given

(t)

(T)e

^dT

+ J°

f*(x)X

(x)dx

where

Qn(t)=

[

q*(x,t)X

(x)dx

The boundary conditions

IBVP (5.36) give orthonormal eigenfunctions

{x)

= \

——.

r~9

cosa

y he/ κ

sin

with eigenvalues

given

= a

and

tan

c =

Oin

NONHOMOGENEOUS BOUNDARY CONDITIONS 127

Figure 5.5 Plot of

the

material temperature u(x, t).

Figure 5.5 shows the resulting temperature surface u(x, t) for the original IBVP

(5.35) for the case h = 0.1, κ = 0.01,

= 0.1, and c

100. The temperature at the

right boundary (x = 1) eventually oscillates between 0.2 and 1.8, an interval slightly

less in range than that of the surrounding temperature. However, the temperature at

the left boundary (x = 0) oscillates over a much smaller range of values (w 0.8 ...

1.2). Additionally, the oscillation at the left boundary lags that of the right boundary.

The periods of oscillation appear to be the same.

5.3.3 Laplace Transforms

An alternate method for solving heat transfer problems having time-dependent bound-

ary conditions of

limited type is by an application of Duhamel's theorem. One way

of establishing the desired result is through Laplace transforms. A review of Laplace

transforms is given in this section.

If a function u(t) is "reasonably" continuous and bounded for t > 0, then the

Laplace transform of u(t), denoted as C[u(t)}, exists and is given by

/»OO

C[u(t)] = / u(t)e~

U(s). (5.37)

The process defined above transform a function u of t into a unique function U of

The transformation is linear and one-to-one. Consequently, the inverse Laplace

transformation, denoted by

[U(s)]

—

u(t), is defined and linear as well.

Using the definition given by Equation (5.37), it is reasonable to define the Laplace

transform for i¿(x, t) as

/»OO

C[u(x,t)] = / u(x,t)e-

dt = U(x,s) (5.38)

128 HEAT TRANSFER IN 1D

The method of integration by parts can be used to show (see Exercise 5.10)

/»OO

C[u

(x,

t)}

= / u

(x, t)e~

dt = sU(x, s) - u(x, 0) (5.39)

Additionally,

—C[u{x,t)) = — / u{x,t)e~

= / u

(x,t)e~

= C[u

(x,t)]

(x,s)

(5.40)

provided the order of partial differentiation and integration may be interchanged.

Once the result in Equation (5.40) is established, it follows that

C[u

(x, t)] =

(x,

s) (5.41)

The convolution of two functions / and g is defined as

(f*9)(t)= [ f(r)g(t-r)dr (5.42)

provided the integral exists for t > 0. It can be shown that

I f(r)9(t - T)dr = I fit- r)g(r) (5.43)

JO JO

so that (/ * g)(t) = (g *

f)(t).

An additional property of convolutions is

C[(f * g)(t)} = C[f(t)}C[g(t)} (5.44)

Using the invertibility of the Laplace transform, it follows that

C-

[C[f{t)]C[g(t)}] = C-\C[U*g){t)\]

(/**)(*)

= / f(r)g(t - r)dt (5.45)

5.3.4 Duhamel's Theorem

Duhamel's theorem for IBVP (5.46) is developed in this section.

( u

= ku

0 < x < c (PDE)

u(x, 0) = 0 (IC)

IBVP<

u(0,t)=0 (BC1)

{ u(c,t)=g(t) (BC2)

(5.46)

NONHOMOGENEOUS BOUNDARY CONDITIONS

129

The time-dependent function

g(t) in

BC2

assumed

to be

continuous

and

differ-

entiable

for t > 0.

The process begins

finding

the

solution

to the

simpler IBVP

(5.47)

0 < x < c (PDE)

IBVP<

v(x,0)

= 0

v(cA)

= 1

(IC)

(BC1)

(BC2)

using Laplace transformation techniques. The PDE

transformed first.

=4>

C[v

]

C[ku

=> sV-v(x,0)

= kV

=^>

sV - kV

= 0

(5.47)

(5.48)

The Laplace transform process

this case changes the original PDE into

second-

order ordinary differential equation

in the

new variable

The general solution

the equation given

Equation (5.48)

V(x, s)

= Ae^

Be~^

(5-49)

The coefficients

A and B may be

determined using boundary conditions resulting

from

the

Laplace transformation

the boundary conditions given

IBVP (5.47).

A table giving Laplace transformations

basics functions

beneficial

for

this

objective. The transforms

are

v(0,t)

C[v(0,t)}

C[0]

V(0,s)=0

(5.50)

and

v(l,t)

= l

C[v(l,t)]=C[l]

V(c,s)

= -

Applying the transformed BC given

Equation (5.50) results

(5.51)

V(a:,

s) =

Csinh

( χ/

—x

(5.52)

Applying

the

transformed

given

Equation (5.51) allows

value

for C to be

determined

V(x,

s) = .

]

,__.

sinh

(JT_X)

(5.53)

ssinh

130 HEAT TRANSFER IN 1D

The last step in our solution process in to find the inverse transform of the expression

for V(x, s) given in Equation (5.53). For sake of reference, let v(x, t) be such that

v{x,t) = £

5sinh

(vl

)

sinh

(5.54)

The solution process for IBVP (5.46) follows that for the IBVP (5.47) up to the

point of determining the coefficient C shown in Equation (5.52). In the present case,

the coefficient C must be such that the Laplace transformed BC2 be satisfied as

x = c. That is, it must be

(5.55)

Csmh(

l-x)=C[g(t)]=G(s)

Here, the solution for u(x,t) may be expressed as

G(s)

i(x,t)

sinh

(\/f

)

= C

G(s)s—-

sinh

5sinh(yfc)

= C-

[G(s)sV(x,s)]

= C~

[G(s)(sV(x, s) - v(x, 0))]

= C-^GWVtfas))

= (g*v

)(t)

= / g(j)v

{x,t - r)dr (5.56)

Using integration by parts, the expression found in Equation (5.56) is equivalent to

u(x,t)= / g

(T)v(x,t-r)dT-g(t)v(x

0)+g(0)v(x,t) (5.57)

Finally, using the convolution property given in Equation (5.43) and the fact that

v(x, 0) = 0, the result is the solution for u(x, t) given by Duhamel's theorem.

u(x,t)= I g

(t-T)v(x,t)dr + g(0)v(x,t) (5.58)

Duhamel's theorem formulation was used to solve the IBVP (5.35) from Section

5.3.2. Figure 5.6 shows the temperature "surface" plotted for t from 0 to 30. The

result is almost identical to that shown in Figure 5.5, where the temperature surface

was constructed using methods outlined in Section 5.3.2.

SPHERICAL COORDINATE EXAMPLE

131

Figure 5.6 Plot of

the

material temperature u(x, t) found using Duhamel's theorem.

5.4 SPHERICAL COORDINATE EXAMPLE

There are occasions when the temperature

spherically shaped solid material is a

function of radius

only. In this case, the original Laplacian in spherical coordinates

τ-72

1/ x 1

V U = -{rU)rr +

. 2η

ΦΦ + ~2

r^ sin

r(sin0

reduces

—

-(ru)

An example

such

occurrence

is a

solid sphere

radius

whose surface tem-

perature is maintained at

temperature of

0°C.

The sphere has an initial temperature

distribution

of f(r).

The following IBVP results:

IBVP<

(r, t) =

£(ru)

0 < r < c (PDE)

u(r,0)

= /(r) (IC)

(5.59)

(Q,t)

= 0

u(c,t)

(BC1)

(BC2)

The boundary condition at

r = 0

is a Neumann condition based on the symmetry the

temperature profile

function

of r

only) will have at the center

the sphere.

The

PDE of

IBVP (5.59)

is not in a

separable form.

The

change

variable

v(r,

t) = ru(r, t) is

made

order to transform the PDE to

separable form. Then,

u(r,t)

-v(r,t)

(r,t)

-v

(r,t)

and

ru(r,t)

-v(r,t)

-{ru)

(r,t)

(5.60)

(5.61)