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

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132 HEAT TRANSFER IN 1D

so the PDE in v is

1 k

-v

(r, t) = -v

(r, t)

(r, t) = kv

(r, t) (5.62)

r r

The IC under the change of variable becomes

-v(r, t) = /(r) => v(r, t) = rf(r) (5.63)

The boundary condition at r = 0 for v is determined by first finding a formula for

(x,t) in terms of f

(r, t).

9 1 rv

- υ

—-ü(r,t) = 2

/ N i/ i , , rv

— v ,„ ^

r(r,t) = is^M) = ^— (5.64)

Consequently, for u

(0,£) = 0, it must follow that v(0,t) = 0. The boundary

condition for r

—

c becomes

u(c, t) = 0

-v(c, t) = 0 => v(c, t) = 0 (5.65)

The Sturm-Liouville problem that results from separation of variables on the PDE

and BCs for v(r, t) is

Ä"(r) +

(

) = 0 Ä(0) = 0 and Ä(c) - 0 (5.66)

so the eigenvalues and eigenfunctions are

2 ηπ

-sin(a

r) a

= —, n = 1,2,3,... (5.67)

c c

and the general solution for v(r,

£)

;(r,i) = X;6„J?

(r)e-

(5.68)

n=l

The coefficients b

are determined by matching the initial condition

rf(r),

so that

= Γ rf(r)R

(r)dr (5.69)

The last step is transforming the solution given in Equation (5.69) to one for u(r, t),

which gives

u(r, t) = - Σ b

Rn(r)e-

(5.70)

n=l

Figure 5.7 shows the temperature result for 0 < t < 10 for a sphere of radius

thermal diffusivity A: = 0.1, and an initial temperature distribution given by

f(r) = r(2

—

r). Observe that the temperature at r = 0 increases from the initial

value of zero for t from zero until t « 2. Thereafter, the temperature decreases in an

exponential fashion for all r.

EXERCISES 133

u(rjt)

Figure 5.7 Temperature of

the

sphere of

radius

2 over for 0 < t < 10.

EXERCISES

In the exercises that follow, as in other locations in the text, the function H^

^ (x)

a < x < b

0 otherwise

is defined as:

[a,b](

)

5.1 Suppose that in the separation process in the solution to IBVP (5.2), the sepa-

ration process resulted in the equation

T'(t)

X»{x)

T(t) X(x)

As in the example, set each side equal to — λ, and show that the same eigenfunctions

result as before. (The purpose of

this

exercise is to demonstrate either setup gives the

same result for eigenfunctions. However, one setup results in simpler manipulations.)

5.2 Solve the following homogeneous IBVPs by constructing a 20-term approxi-

mating series for u{x,t). In each case, (i) specify the appropriate eigenvalues and

orthonormal eigenfunctions, (ii) clearly identify the integral formula for any Fourier

series coefficients that need to be determined, (iii) provide a 3D plot of the solution

u(x, t) Gver an appropriate domain, (iv) give the value of u(c/2,1).

IBVP<

0.05u

u(x,0) = 1

—

0<x< 1

u(0,t

(PDE)

(IC)

(BC1)

(BC2)

(5.71)

IBVP<

0.1u

U{X,0) =

#[0.8,1.2]

0*0

u(0,t) =0

u(2,t) =0

0 <x < 2

(PDE)

(IC)

(BC1)

(BC2)

(5.72)

134 HEAT TRANSFER IN 1D

IBVP<

( ut=

0.1u

0 < x < 2 (PDE)

U(X,0)=F

.8,1.2](X) (IQ

(5.73)

u(0,t) = 0

(2,t)

= 0

(BC1)

(BC2)

IBVP

ut =

0.1u

0 < x < 2 (PDE)

u(x

0) = 2i7

.4,0.6] (x) + #[1.4,1.6] (IC)

u(0,£) = 0

0.2u(2,t)+0.3u

(2,i) =0

(5.74)

(BC1)

(BC2)

5.3 Solve the following semihomogeneous IBVPs by constructing a 20-term ap-

proximating series for

u(x,t).

In each case, (i) specify the appropriate eigenvalues

and orthonormal eigenfunctions, (ii) clearly identify the integral formula for any

Fourier series coefficients that need to be determined, (iii) provide a 3D plot of the

solution u(x, t) over an appropriate domain, (iv) give the value of u(c/2,1).

[ Ut

—

§.\u

+ siri7nr, 0 < x < 1 (PDE)

u(x, 0) = 0 (IC)

IBVP< (5.75)

u{l,t) =0

(BC1)

(BC2)

IBVP

—

0.1u

4- cos7nr, 0 < x < 1 (PDE)

lfc(x,0) =

#[0.4,0.6]

0*0

(IC)

(5.76)

(0,t)

{l,t)=0

(BC1)

(BC2)

[ u

=0.1u

+ e-*#

,o.6], 0<x<l (PDE)

IBVP i

(

i ^)

= s

n πχ

ÖQ

(5.77)

(0,£) = 0

I u(l,t) = 0

(BC1)

(BC2)

( ut = (Un** + x(2 - x)e~%, 0 < x < 2 (PDE)

IBVP ¿

(

> °) = ° (

)

(5.78)

(0,f) = 0

0.2u(2,t)

+ 0Mu

(2,t) =0

(BC1)

(BC2)

EXERCISES 135

IBVP<

ut = 0.1

u(x,0) =

w(0,t) =

0.2u(l,i

^Xj

~T~

= X

■2Íf

) + 0,01u

(l,/

4,0.6]

) = 0

x<l

(PDE)

(IC)

(BC1)

(BC2)

(5.79)

5.4 Solve the following nonhomogeneous IB VPs using methods outlined in Section

5.3,

where the substitution u(x, t) — U(x, t) + A(t)x + B(t) is employed. Construct

an approximating 20-term series for U(x,t). In each case, (i) specify formulas

for the nonhomogeneous term g* of the PDE for U(x,t), the initial condition /*

of the semihomogeneous IBVP in U(x, t), and formulas for A(t) and B(t), (ii) the

appropriate eigenvalues and eigenfunctions, and (iii) provide a 3D plot of

the

solution

u(x, t) over an appropriate domain, and (iv) give the value ofu(c/2,1).

IBVP<

[ u

0.1u

+ e 2 sin7T£,

J i¿(#,0) = 1

( u(l,t) = cosí

f u

= 0Au

+ e"*^/f[0.4,0.6]

1 u(x,0) = 0

u(0,i) = l

I 0.2w(l,i) +

0.4^(1,t)

= 2

r ^ =0Au

+x(l + t)-

)

M(X,0)

= x — x

u(0,i) = l

l ^(Ι,^ίίΗί)-

0 <x < 1

(PDE)

(IC)

(BC1)

(BC2)

(PDE)

(IC)

(BC1)

(BC2)

(PDE)

(IC)

(BC1)

(BC2)

(5.80)

(5.81)

(5.82)

( u

=0.lu

+x(l + t)-

, 0<x<l (PDE)

IBVP<

u(x, 0) = x

—

«x(0,t) = l

«(1,ί) = ί(1+*)

(IC)

(BCD

(BC2)

(5.83)

5.5 In Section 5.3, it was pointed out that if Neumann conditions are prescribed

at both endpoints in the ID heat transfer case, the determination of A(t) and B(t)

136 HEAT TRANSFER IN 1D

such that u(x, t) = [/(#, t) + A{t)x -f -B(i) solves the nonhomogeneous may not be

possible. For the IB VPs given below, use a function of the form

u{x, t) = U(x, t) 4- A(t)x

+ B(t)x

in an effort to transform the nonhomogeneous IBVP to a semihomogeneous IBVP. If

possible, solve the original IBVP. In each case, (i) specify formulas for the nonhomo-

geneous term

of the PDE for U(x, t), the initial condition /* of the semihomoge-

neous IBVP in U(x, t), and formulas for A(t) and B(t), (ii) specify the appropriate

eigenvalues and orthonormal eigenfunctions, (iii) provide a clear integral formula for

any Fourier series coefficients that need to be determined, and (iv) create a 3D plot

of the temperature u(x, t) over an appropriate domain.

' ut = (Un** 0 < x < 1 (PDE)

IBVP<

u(x,0) = 2x

(IC)

(5.84)

IBVP¿

(0,t)

= -2 (BCl)

(l,t)=A (BC2)

' u

0.05u

0 < x < 1 (PDE)

U(Z,0) = #[0.5,1] (Z) (IC)

*(0,*) = 1 (BCl)

*(!,«) = e~* (BC2)

(5.85)

5.6 Suppose a sphere of radius 4 is submerged in a fluid of temperature 20°C and

the initial temperature of the sphere is f(r) = 10/(1 + r) °C. Take the boundary

condition at r = 4 to be Dirichlet, u(4) = 20. Use the methods outlined in Section

5.4 to determine the temperature of the sphere as a function of r and t. Generate a

plot of the temperature over the interval 0 < r < 2 for 0 < t < 10. Use κ

—

0.1.

5.7 A solid sphere of radius c

surrounded by a

fluid

kept at

constant temperature

of 40°C. Applying Newton's law of cooling and Fourier's law of heat transfer, the

boundary condition at r = c is

hu(c, t) + KU

(c, t) = hT

where n is the thermal conductivity of the material and h is the surface conductance.

Suppose a sphere of radius 2 is submerged in a fluid of temperature 40 °C and the

initial temperature of the sphere is 10°C. Use the methods outlined in Section 5.4

to determine the temperature of the sphere as a function of r and t. Generate a plot

of the temperature over the interval 0<r<2for0<¿<10. Let κ — 0.1 and

= 0.1.

5.8 In Exercise 4.13, the PDE for ID heat transfer was derived for the case of

surface heat transfer along the later length of

slender wire. It is the PDE associated

EXERCISES 137

with the IBVP shown below. Find the solution u(x, t) and use technology to plot

u(x, t) for 0 < x < c and 0 < t < 5.

IBVP<

( Uf=0.1u

+ b[l-u(x,t)], 0<x<l (PDE)

u(x,0)

=-H

[0m5tl]

(x)

(IC)

(5.86)

(0,t)

= l

(l,t)

=e *

(BC1)

(BC2)

5.9 Show that the formula for u(x, t) of Equation (5.56) found by Duhamel's

theorem reduces to the solution v(x, t) of IBVP (5.47) in the event g(r) = 1.

5.10 Provide the details showing how integration by parts is used to establish the

result given in Equation (5.39).

5.11 For the given IBVP, (i) find a solution using Duhamel's theorem as formu-

lated by Equation (5.58), (ii) demonstrate your solution satisfies the time-dependent

boundary condition by generating a plot of the solution at the appropriate boundary

over a suitable time interval, and (iii) calculate u(c/2,1).

IBVP<

IBVP

ut =

0.1^

w(x,0) = 0

u(0,t) = 0

w(l,i) =

-t

sin£

ut = Ο.ΟΙη^,

u(x,0) =0

u(0,t) = 0

u(l,t) =ί(1+ί)"

= 0.01i¿

u(x,0) =0

u(0,t) =0

u(l,t) = t(l + *)-

0<x < 1

(PDE)

(IC)

(BC1)

(BC2)

0 < x < 1 (PDE)

(IC)

(BC1)

(BC2)

0 < x < 1 (PDE)

(IC)

(BCD

(BC2)

IBVP<

= 0Au

u(x,0) =0

u(0,t) = sini

uh,t) =0

0 < x < 1 (PDE)

(IC)

(BCD

(BC2)

(5.87)

(5.88)

(5.89)

(5.90)

138 HEAT TRANSFER IN 1D

IBVP<

f u

= OAu

, 0 < x < 1 (PDE)

u{x, 0) = 0 (IC)

IBVP<

tx*(0,i) = 0

[ u(l,t) = sint

( u

0.05u

u(x,0) = 1

u(0,t) =0

I 0.1u

(l,í) + 0.2u(l,£) = 0.2sin(7rt)

(BCl)

(BC2)

0 < x < 1 (PDE)

(IC)

(BCl)

(BC2)

IBVP<

( u

= O.luss, 0 < x < 1 (PDE)

u(x, 0) = 0 (IC)

u(0,t) = sin¿

[ u(l,t) = cotí

(BCl)

(BC2)

(5.91)

(5.92)

(5.93)

CHAPTER 6

HEAT TRANSFER IN 2D AND 3D

In this chapter, our attention is focused on methods for solving heat transfer problems

in the event temperature is a function of two spacial dimensions. The development

process is very similar to that for the ID case. We begin by considering the homo-

geneous IBVP, then the semihomogeneous problem (a nonhomogeneous PDE and

homogeneous boundary conditions), and then the case for IBVPs with nonhomoge-

neous boundary conditions.

6.1 HOMOGENEOUS 2D IBVP

We consider the case of the homogeneous 2D heat equation with homogeneous

boundary conditions as indicated in the IBVP below.

IBVP<

ut(x, y,t)=k [u

(x, y, t) +

(x,

y, t)}

u(x,y,0) = f(x,y)

a\u{x, 0, t) +

<i2U

(x,

0,£) = 0 0 < x < c

biu(c, y, t) +

(c,y,

t) = 0 0 < y < d

c\u{x, d, t) + C2U

(x,

<i,

t) = 0 0 < x < c

dmiO, y, t) + d

(0, y,t)=0 0<y<d

Fourier Series and Numerical Methods for Partial Differential Equations,

First Edition. By Richard Bernatz

(PDE)

(IC)

(BC1) (

·!)

(BC2)

(BC3)

(BC4)

139

140 HEAT TRANSFER IN 2D AND 3D

Figure 6.1 shows the problem domain for this IBVP.

2/f

Figure

6.1

Problem domain for IBVP 6.19.

We solve the 2D heat transfer IBVP using separation of variables much like the

case for ID heat transfer. That is, we assume the solution u(x,y,t) has the separable

form

u(x,y,t)=X(x)Y(y)T(t) (6.2)

so,

the PDE of IBVP (6.19) becomes

X(x)Y(y)T'(t) = k [X"{x)Y{y)T{t) = X(x)Y"(y)T(t)\ (6.3)

Dividing all terms of Equation (6.3) by X(x)Y(y)T(t) and simplifying gives

T'(t) _ X"(x) Y"(y)

kT(t) X(x)

Y(y)

(6.4)

A separation constant — λ is introduced so that Equation (6.4) may be written as

separate equations

x"(x)

r^)

and

X(x) Y(y)

T'{t)

kT(t)

Equation (6.5) is separated further by letting

-λ

(6.5)

(6.6)

so that

and

—

X"{x)

+ μΧ{χ) = 0

Y" (y) + vY{y) = 0

(6.7)

(6.8)

HOMOGENEOUS 2D IBVP 141

Substituting the separated form of u(x, y, t) into BCl - BC4 of IBVP (6.19) gives

ajFfO) + a

Y'(0) = 0 BCl

X(c) + b

X'(c)=0 BC2

rfi

Y(d) + c

Y'(d) = 0 BC3 ^'

diX(O) + d

X'(0) = 0 BC4

Combining BC2 and BC4 of (6.9) with the differential equation (6.7) results in

the regular Sturm-Liouville problem

X"{x)

+ μΧ(χ) = 0

X(0) + d

X'(0) = 0 (6.10)

6iX(c) + b

X'(c) = 0

for X(x). Similarly, combining BCl and BC3 of (6.9) with the differential equation

(6.8) gives the regular Sturm-Liouville problem

(

/) + i/y(y) = o

a{Y(0) + a

(0) = 0 (6.11)

ciY(d) + c

F'(d) - 0

The regularity of the two Sturm-Liouville problems (6.10) and (6.11), as well as

the possible values of the coefficients ai, a

, &i,..., di, d

assure us that Properties

1-5 of Chapter 3 hold. Consequently, the eigenvalues μ and ^ are discrete non-

negative real numbers. Let μ

— α^ and Vm — β^- Additionally, the associated

eigenfunctions X

(x) and Y

(y), respectively, are unique and orthogonal relative to

the set to which they belong.

Next, we turn our attention to the differential equation for T(t). Knowing the

form of λ, we may write Equation (6.6) as

{t)

= -k{a

+ß

)T{t) (6.12)

with solutions of

(í) = e-

/c(a

+/3

)í

(6.13)

forn = 0,1,2... .

Summarizing the solution process thus far, we know the functions of the form

Xn^YMe-^+tf^

n = 0,1,2,... m = 0,1,2,... (6.14)

solve the PDE and BCs of the IBVP (6.19). Applying the general principle of

superposition, the function defined by the double infinite series

oo oo

u(x,

j/,*)

= Σ Σ A

(x)Ym(y)e-

k{a

+ß

(6.15)

n=0

ra=0

where A

are arbitrary constants, satisfies the PDE and BCs of the IBVP as well.

The function u(x, y, t) will satisfy the IBVP (6.19) provided values of

nrn

may be

determined such that

oo oo

u(x, y^) = J2^2 A

(x)Y

(y) = /(x, y) (6.16)

n=0 m=0