Chapter 12 Boundary-Value Problem in

1
Chapter 12 Boundary-Value Problem in
Rectangular Coordinates

• Role of Chapter 12
• Discuss the boundary-value problem for the
  case of two independent variables.

(??????? Chapter 13 ????? ?????????????)
(x-y ??)
Use the methods of (1) separation of variables or
(2) the Fourier transform to solve the problem.
Chapter 12
Section 14.4
2
?????
12.1 ???? Separation of variables (???)
12.2 ?????
12.4 Wave equation
???12.1 ???? (????????)
12.5 Laplaces equation
???? (1) ?? separation of variables ? PDE
??? (2) ?????
3
?? boundary value problem (BVP)
initial value problem (IVP)
? BVP
IVP partial differential equation
(PDE) ordinary differential equation
(ODE)
4
Section 12.1 Separable Partial Differential
Equations
12.1.1 Section 12.1 ??
(1) linear second order partial differential
equation for two independent

variables
7 terms
B2 - 4AC gt 0 hyperbolic, B2 - 4AC 0
parabolic B2 - 4AC lt 0 elliptic
(2) Partial differential equation (PDE) ??????
Separation of variables (see pages 724-726).
?? real separation constant (page 724)
5
12.1.2 Linear Second Order Partial Differential
Equation
independent variables x, y
dependent variables u(x, y), ??? u
homogeneous G(x, y) 0, nonhomogeneous
G(x, y) ? 0
6
particular solution, general solution ???????
?Theorem 12.1.1? Superposition Principle
If u1, u2, ., uk are solutions of a homogeneous
linear partial differential equation, then
is also a solution of the homogeneous linear
partial differential equation.
7
12.1.3 Method of Separation of Variables
? PDE with BVP (or IVP) ???
(1) method of separation of variables
? PDE ???? x ?? y ????,
???? u(x, y) X(x)Y(y)
(2) using the Fourier transform (or Fourier
cosine transform, Fourier sine transform) (see
Section 14.4)
????? PDE ODE
Note Laplace transform can also be used for
solving the PDE (Section 14-2,??????)
8
Method of Separation of Variables ???
(Step 1) ???? u(x, y) X(x)Y(y)
????
(Step 2) ? u(x, y) X(x)Y(y) ?? PDE,? PDE ??
function of X function of
Y ?? ???
? ??? real separation constant
9
Steps 3, 4, 5 ?????? Cases ??
??trivial ????,????? cases ????
(Step 3) ? function of X ?? ????,?? X(x)
?(a) ??????? boundary (initial)
conditions, ????????
(?? pages 754, 770
???) (b) ??,?? Y(y) ?????
(? boundary (initial) conditions ??)
(c) ?????,????,??? ? ???
(Step 4) ? function of Y ?? ????,?? Y(y)
??????? Step 3 ??
(Step 5) u(x, y) X(x)Y(y)
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(Step 6) ????????????
(Step 7) ???? boundary (initial) conditions ?
coefficients ?? ? ???????? Fourier series,
Fourier cosine series ?
Fourier sine series
? ??? boundary (initial) conditions,Steps 6, 7
????
Rules
x ? BVP (IVP) ?? ?? X(x)
y ? BVP (IVP) ?? ?? Y(y) ?? BVP
(IVP) ?? X(x) ? Y(y) ??
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?? X(x)
?? Y(y)
12
Note Separation of variables ??????????? PDE
???? ?????? X(x)Y(y) ???
Separation of variables ?????????????
13
Example 2 (text page 462)
Step 1 ???? u(x, y) X(x)Y(y)
(????)
Step 2 ? u(x, y) X(x)Y(y) ??
real separation constant
?
(????)
14
Case 1 for Steps 3, 4, 5
Step 3-1
roots 0, 0
auxiliary function
Step 4-1
Step 5-1
15
Case 2 for Steps 3, 4, 5
??????,?
roots of the auxiliary function 2?, ?2?
Step 3-2
???????
Step 4-2
Step 5-2
16
Case 3 for Step 3
??????,?
roots of the auxiliary function j2?, ?j2?
Step 3-3
Step 4-3
Step 5-3
???? boundary conditions,????? general
solution,????????????
Step 6
? ?????
(?nonseparable ????????)
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12.1.4 Classification
The PDE is said to be hyperbolic (???)
The PDE is said to be parabolic (???)
The PDE is said to be elliptic (???)
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??????,??? 2 ????? x-y ??????
?
?
y
y
x
x
19
?
y
x
??????????????,??????
???,???????
20
Example 3 (text page 463)
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12.1.5 ?????????
(1) ????????,?????? classification of equations
? ? method of separation of variables. (2)
??, method of separation of variables ?????,????
?,???? (Sections 12-4, 12-5 ???????)
??????? u(x, y) X(x)Y(y)
??? function of X function of Y ?? (3)
Method of separation of variables ????,??????
cases. (4) Separation of variables?? BVP ?
IVP?,????? cases ? ???????? (Step 6)
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(5) ?????? BVP ? IVP,??? ??? (6)
Hyperbolic, parabolic, elliptic ???,?????
special cases ?? (7) ??? ? BVP ?IVP ????
Steps 3, 4 ???? (?? pages 754, 770 ???)
???? ? BVP ?IVP ?? Step 7 ????
23
Section 12.2 Classical PDEs and Boundary-
Value Problems
12.2.1 ????
(1) one-dimensional heat equation (???? heat
equation)
k gt 0
(2) one-dimensional wave equation (???? wave
equation)
(3) two-dimensional form of Laplaces equation
(???? Laplaces equation )
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?? heat equation, (page 741)
wave equation, (page 743) Laplaces
equation, (page 746) Laplacian,
(page 747) Dirichlet condition,
(page 750) Neumann condition,
(page 750) Robin condition (page 750)
????????????,?????????
25
12.2.2 One-Dimensional Heat Equation
???????? (??????? text page 466)
u(x, t) temperature, t time, x location
Fig. 12.2.1
heat equation ??diffusion equation
26
Example
u(x, t) temperature, t time, x location
u(x, t0)
x0
x-axis
x0 ??????
27
12.2.3 One-Dimensional Wave Equation
????????? (??????? text page 468)
u(x, t) height, t time, x location
(1) ????
f(x)
u(x, t) 0 when x 0
u(x, t) 0 when x L
Fig. 12.2.4
wave equation ?? telegraph equation
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(2) ?????????
??
Fig. 12.2.2
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? Wave equation ????? Theory of
high-frequency transmission line Fluid
mechanics (????) Acoustics (??) Elasticity
(???) Microwave engineering (????)
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12.2.4 Two-Dimensional Form of Laplaces
Equation
??? Fig. 12.2.3,????????????
u(x, y) temperature, x, y location
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Laplaces Equation ??? Laplacian ??, ?2u(x, y) 0
Laplacian ?2
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? Laplaces Equation ????? Static displacement
of membranes Edge detection (????)
Microwave engineering (????)
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12.2.5 Modification
????,?????????
?heat equation ? modification
?wave equation ? modification
34
12.2.6 Boundary Conditions ? Initial Conditions
Dirichlet condition
u .. (???)
Neumann condition
(???)
Robin condition
(??)
h is a constant
35
Section 12.4 Wave Equation
12.4.1 ????
?????? (one-dimensional wave equation)
0 lt x lt L
t gt 0
BVP and IVP
for t gt 0
for 0 lt x lt L
??? page 753-761??? page 762
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???,Sections 12.4 ? 12.5 ???? Section 12.1 ?
method of separation of variables ???? (???
method of separation of variables ????)
??
standing waves (page 763) normal
modes (page 763) first standing wave
(page 764) fundamental frequency (page
764) nodes (page 766)
overtones (page 766)
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12.4.2 Solutions for Wave Equations (???????)
0 lt x lt L
t gt 0
for t gt 0
????
for 0 lt x lt L
?? (??method of separation of variables)
Step 1
???? u(x, t) X(x)T(t)
Step 2
? u(x, y) X(x)T(t) ??
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?
??2 ? ODEs
Steps 3, 4, 5 ????
(1) ?? x ? boundary condition ???,???? X(x)
(2) ?? ? 0, ? lt 0 , ? gt 0 ?? cases (3) ??
u(0, t) X(0)T(t) 0
for all t gt 0
T(t) ??? 0 (?? u(x, t) X(x)T(t) 0 for any x,
t)
?? X(0) 0
??,?
?????? X(L) 0
and
subject to
X(0) 0
X(L) 0
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subject to
X(0) 0
and
X(L) 0
Case 1 for Steps 3, 4, 5 ? 0
Step 3-1
?? boundary conditions
?? case ?? trivial solution u(x, t) X(x)T(t)
0 ?????
? 0 ???
???? Step 4-1, Step 5-1
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Case 2 of Steps 3, 4, 5 ? lt 0
? ? -?2
Step 3-2
???? boundary conditions
Solution
????
?? boundary conditions X(0) 0 and X(L) 0
?? case ?? trivial solution u(x, t) X(x)T(t)
0 ?????
? lt 0 ???
???? Step 4-2, Step 5-2
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Case 3 of Steps 3, 4, 5 ? gt 0
? ? ?2
Step 3-3
Solution
?? boundary conditions X(0) 0 and X(L) 0
n ?????
???? ?????
???

?????????? ?, ?????????
42
n ??????, c2 ????
Step 4-3
Solution
n ?????
Step 5-3
n ?????
An c2c3, Bn c2c4,
43
??
???????,?? n ?????
Step 6
44
???? n ?????,???? n ??1 ?? ?, ??? ? ? ?? ? ?
??
???
45
Step 7
? initial conditions
????,An ? f(x) ? Fourier sine series,
? g(x) ? Fourier sine series
46
12.4.3 ????
??
??
???
f(x)
u(x, t) 0 when x 0
u(x, t) 0 when x L
Fig. 12.2.4
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12.4.4 ??
??
un(x, t) ??? standing waves (??) ? normal modes
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n 1 ?, u1(x, t) ??? first standing wave ?
first normal mode ? fundamental mode of vibration
?? t ??,??
?? 1/??
??? fundamental frequency (??) ? first harmonic
49
????, u2(x, t) ??? second standing wave
u3(x, t) ??? third standing wave
L
0
First standing wave
L
0
Second standing wave
L
0
Third standing wave
Fig. 12.4.2
50
?,?? t ????,
? nth standing wave ? node (??)
un(x, t) ??? 1/??
??? overtones (??)
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Section 12.5 Laplaces Equation
12.5.1 Section 12.5 ??
0 lt x lt a,
0 lt y lt b,
(??method of separation of variables ??)
for 0 lt y lt b,
??? 1?
for 0 lt x lt a
??? 2?
for 0 lt y lt b,
for 0 lt x lt a
??? 3?
for 0 lt x lt a
for 0 lt y lt b,
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? ???? superposition principle
Sections 12.4, 12.5 ???,???? (1) Method of
separation of variables ??? (a) Wave
equation ? (b) Laplaces equation (2)
Superposition principle
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12.5.2 Solutions for Laplaces Equations
(???????)
0 lt y lt b,
0 lt x lt a,
for 0 lt y lt b,
for 0 lt x lt a
Step 1
???? u(x, y) X(x)Y(y)
Step 2
?? ??
?
??2 ? ODEs
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Steps 3, 4, 5 ????
(1) ?? x ? boundary condition ???,???? X(x) (2)
?? ? 0, ? lt 0 , ? gt 0 ?? cases (3) ??
for all 0 lt y lt b,
Y(y) ??? 0 (?? u(x, y) X(x)Y(y) 0)
??
??,?
??,?
55
Case 1 of Steps 3, 4, 5 ? 0
Step 3-1
solution
? boundary conditions
Y(0) 0
Step 4-1
solution
?? boundary condition Y(0) 0, c3 0
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Step 5-1
Case 2 of Steps 3, 4, 5 ? lt 0
? ? -?2
Step 3-2
solution
????
? boundary conditions
??
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??, case 2 ?? trivial solution u(x, y) X(x)Y(y)
0 ?????
? lt 0 ???
(???? Steps 4-2, 5-2)
Case 3 of Steps 3, 4, 5 ? gt 0
? ? ?2
Step 3-3
solution
? boundary conditions
n ?????
58
??????????? c1 0 and c2 0 ?????????? ?,
?????????
n ?????
Step 4-3
since
Y(0) 0
solution
?????
?? boundary condition Y(0) 0
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Step 5-3
n ?????
Step 6
???????,?????
??? n ?? 1 ?? ?,??? ?? ?? ?? ????? page 760
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Step 7
nonzero boundary condition
????,2A0b ? (n 1, 2,
., ?) ? f(x) ? Fourier cosine series ?
coefficients
remember Section 11-3 ? Fourier cosine series ?
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12.5.3 Laplaces Equations with Dirichlet
Problem
0 lt x lt a,
0 lt y lt b,
0 lt y lt b,
0 lt x lt a,
? method of separation of variables,??????
????????
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12.5.4 Superposition Principle
Dirichlet Problem ?????????
0 lt x lt a,
0 lt y lt b,
for 0 lt y lt b,
for 0 lt x lt a,
??????????,????? separation of variable ??????
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??? 1
0 lt x lt a,
0 lt y lt b,
for 0 lt y lt b,
for 0 lt x lt a,
??? 2
0 lt x lt a,
0 lt y lt b,
for 0 lt y lt b,
for 0 lt x lt a,
?? u1(x, y), u2(x, y) ?????? 1, ??? 2 ?? ? u (x,
y) u1(x, y) u2(x, y) ???????
(??? Section 4-1 ? superposition principle)
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? u (x, y) u1(x, y) u2(x, y)
u (0, y) u1(0, y) u2(0, y) 0 F(y) F(y)
u (a, y) u1(a, y) u2(a, y) 0 G(y) G(y)
u (x, 0) u1(x, 0) u2(x, 0) f(x) 0 f(x)
u (x, b) u1(x, b) u2(x, b) g(x) 0 g(x)


Fig. 12.5.3
66
??? 1 ??
??? 2 ??
??????
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12.5.5 Sections 12.4 ? 12.5 ???????
(1) Method of separation of variables ? PDE
??????,??????? pages 724-726 ? 7 ?
steps,?????,??????
(?????????????????)
(2) ??????,??????????? ????????????????????
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(3) ??????, ? boundary conditions ?? u(0,
y) 0, u(L, y) 0, ??????? sine ??
? boundary conditions ?? ???????
cosine ? constant ??
??? 2L/n
or
???? 2L/n
(4) ?????,?? u(x, y) ? boundary conditions
?? u(a, y) 0 ??? X(a) 0 , ??
u(x, b) 0 ??? Y(b) 0 ? ??
??? X'(a) 0, ??
??? Y'(b) 0
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(5) ?? wave equations ??, X(x) ? T(t) ????????
?? X(x) ? sine cosine, T(t) ??
sine cosine ?? Laplaces equations??,
X(x) ? Y(y) ?????? ??X(x) ? sine
cosine, Y(y) ? sinh cosh
(6) ??? cosh(x), sinh(x) ???
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(7) Method of separation of variables ???????????
(? ?? pages 753-761 wave equations ??)
(a) (b) Steps 3, 4, 5 ????? cases (c)
????? c1 0 ?
?? c1 c2 0 ?? ? ??? ?n/L, ???
page 757 ?? (d) ? Step 6,???????????,?? u(x,
t) ???? ??? page 759 ??
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Exercise for Practice
Section 12-1 3, 6, 9, 10, 12, 14, 16, 18, 22,
23, 27, 30, 32 Section 12-2 3, 4, 8, 10,
11 Section 12-4 1, 4, 7, 9, 11, 15, 17, 19,
20 Section 12-5 2, 5, 6, 9, 10, 11, 12, 14, 16,
17, 18 Review 12 1, 2, 5, 13, 14
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Happy New Year!
????????,????!