Applied Mathematics — Partial Differential Equation. Wu-ting Tsai. 11.1. Basic
Concepts. Partial differential equation ⇐⇒ Ordinary differential equation. .
APPLIED MATHEMATICS
Part 5: Partial Differential Equations
Wu-ting Tsai
Contents
11 Partial Differential Equations
2
11.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . .
3
11.2 Modeling: Vibrating String. Wave Equation . . . . . . . . .
5
11.3 Separation of Variables. Use of Fourier Series . . . . . . . .
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11.4 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . .
15
11.5 Laplace Equation in a Rectangular Domain . . . . . . . . .
23
11.6 Two-Dimensional Wave Equation. Use of Double Fourier Series 27 11.7 Heat Equation: Use of Fourier Integral . . . . . . . . . . . .
34
11.8 Heat Equation: Use of Fourier Transform . . . . . . . . . .
38
11.9 Heat Equation. Use of Fourier Cosine and Sine Transforms .
41
11.10 Wave Equation. Use of Fourier Transform . . . . . . . . . .
43
1
Chapter 11 Partial Differential Equations
2
Applied Mathematics — Partial Differential Equation
11.1
Wu-ting Tsai
Basic Concepts
Partial differential equation ⇐⇒ Ordinary differential equation
order of differential equation linear ⇐⇒ nonlinear homogeneous ⇐⇒ nonromogeneous For examples: 2 ∂ 2u 2∂ u =c ∂t2 ∂x2
one-dimensional wave equation
2 ∂u 2∂ u =c ∂t ∂x2
one-dimensional heat equation
∂ 2u ∂ 2u + =0 ∂x2 ∂y 2
two-dimensional Laplace equation
∂ 2u ∂ 2u + = f (x, y) ∂x2 ∂y 2
two-dimensional Poisson’s equation
Note: u(x, y) = x2 − y 2, ex cos y and ln(x2 + y) all satisfy the two-dimensional Laplace equation ⇒ “boundary condition” makes the solution unique (or “initial condition” when t is one of the variable).
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Applied Mathematics — Partial Differential Equation
Wu-ting Tsai
Theorem : Superposition of solutions If u1 and u2 are any solutions of a linear and homogeneous partial differential equation in R ⇒ u = c1u1 + c2u2, where c1 and c2 are constant, is also solution of the equation in R ✷
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Applied Mathematics — Partial Differential Equation
11.2
Wu-ting Tsai
Modeling: Vibrating String. Wave Equation
Consider a vibrating, elastic string with length L and deformation u(x, t).
Assumptions: 1. 2. 3. 4. 5.
homogeneous string (i.e. constant density) perfectly elastic and cannot sustain bending neglect gravitation force (i.e. tension gravitational force) small deformation (i.e. du/dx is small) u is in a plane only
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Applied Mathematics — Partial Differential Equation
Wu-ting Tsai
Since du/dx (slope) is small ⇒ no horizontal motion ⇒ constant horizontal tension T1 cos α = T2 cos β = T in vertical direction:
T2 sin β − T1 sin α = ρ∆x
2
∂ u ÷T 2 ∂t
T2 sin β T1 sin α ρ∆x ∂ 2u ⇒ − = · 2 T2 cos β T1 cos α T ∂t ρ∆x ∂ 2u ⇒ tan β − tan α = · 2 T ∂t
∂u ∂u + ∆x − ρ ∂ 2u ∂x ∂x x x = · 2 ⇒ ∆x T ∂t ∆x → 0 ∂ 2u 1 ∂ 2u ⇒ = ∂x2 c2 ∂t2
where c2 ≡
6
T ρ
✷
Applied Mathematics — Partial Differential Equation
11.3
Wu-ting Tsai
Separation of Variables. Use of Fourier Series
One-dimensional wave equation: 2 ∂ 2u 2∂ u =c ——— (1) ∂t2 ∂x2
Boundary conditions: u(0, t) = 0 ——— (2) u(L, t) = 0 ——— (3) Initial conditions: u(x, 0) = f (x) ——— (4)
∂u = g(x) ——— (5) ∂t t=0 Method of separation of variables (product method): u(x, t) = F (x)G(t)
⇒
∂ 2u d2 G ¨ = F 2 ≡ FG 2 ∂t dt ∂ 2 u d2 F = G ≡ F
G 2 2 ∂x dx
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Applied Mathematics — Partial Differential Equation
¨ = c2F
G (1) ⇒ F G ¨ G F
⇒ = =k 2G c F function of t function of x where k is constant to be determined.
F
− kF = 0 ——— (6)
¨ − c2kG = 0 ——— (7) G
⇒ F (x)
F
− kF = 0 • If k = 0
F (0) = F (L) = 0
⇒ F (x) = ax + b
F (0) = b = 0 F (L) = aL = 0 ⇒a=b=0 ⇒ F (x) = 0 • If k = µ2 > 0
⇒ u(x, t) = 0
×
⇒ F (x) = Aeµx + Be−µx
F (0) = A + B = 0 F (L) = AeµL + Be−µL = 0 ⇒A=B=0 ⇒ F (x) = 0
⇒ u(x, t) = 0
× 8
Wu-ting Tsai
Applied Mathematics — Partial Differential Equation • • •
k = −p2 < 0
Wu-ting Tsai
⇒ F (x) = A cos px + B sin px
F (0) = A = 0 F (L) = B sin pL = 0 ⇒ pL = nπ
nπ , L
⇒p=
⇒ F (x) ≡ Fn(x) = sin
n = 1, 2, 3 . . .
nπ x, L
n = 1, 2, 3 . . .
nπ 2 and k = −p = − L 2
G(t)
2
¨ −c G
nπ 2 − G=0 L
¨ + λ2 G = 0, ⇒G n
λn =
cnπ L
G(t) ≡ Gn(t) = Bn cos λnt + Bn∗ sin λn t
• • •
un(x, t) = Fn (x)Gn(t) = (Bn cos λnt + Bn∗ sin λnt) sin
nπ x L
(n = 1, 2, 3 . . .)
So far, un(x, t) satisfies the differential equation (1), and the boundary conditions (2) and (3). un (x, t) is called eigenfunction or characteristic function. λn is called eigenvalue or characteristic value. 9
Applied Mathematics — Partial Differential Equation
Wu-ting Tsai
A single solution un (x, t) does not satisfy initial condition. Since the differential equation (1) is linear and homogeneous, we therefore assume the solution of (1) is: ∞
u(x, t) = =
n=1 ∞ n=1
un (x, t)
(Bn cos λnt + Bn∗ sin λnt) sin
nπ x L
Initial conditions:
u(x, 0) = f (x) ——— (4) ∂u = g(x) ——— (5) ∂t t=0 (4) ⇒ u(x, 0) = ⇒ Bn =
n=1
Bn sin
nπ x = f (x) (Fourier sine series of f (x)) L
2 L nπx dx f (x) sin L 0 L
n = 1, 2, 3 . . .
∞ ∂u nπx ∗ (5) ⇒ = (B λ cos λ t) sin n n t=0 ∂t t=0 n=1 n L
= ⇒
Bn∗ λn
⇒
Bn∗
∞ n=1
Bn∗ λn sin
nπx = g(x) (Fourier sine series of g(x)) L
2 L nπx = g(x) sin dx L 0 L
2 L nπx = g(x) sin dx λn L 0 L
10
n = 1, 2, 3 . . .
✷
Applied Mathematics — Partial Differential Equation
Wu-ting Tsai
Discussion of eigenfunctions and eigenvalues: (1) The eigenfunction un (x, t) = (Bn cos λnt + Bn∗ sin λnt) sin
nπ x L
λn cn represents a harmonic motion having frequency (cycles/unit time) = . 2π 2L This motion is called the normal mode.
(2) The frequency (eigenvalue) wn =
cn , 2L
c2 =
T ρ
where T = tension, ρ = density ⇒ increase tension or reduce density will increase the frequency.
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Applied Mathematics — Partial Differential Equation
Wu-ting Tsai
Discussion of the solution: u(x, t) = = λn =
∞ n=1 ∞ n=1
un(x, t) (Bn cos λnt + Bn∗ sin λnt) sin
nπ x L
cnπ L
2 L nπx Bn = f (x) sin dx, L 0 L
u(x, 0) = f (x)
2 L nπx ∗ g(x) sin Bn = dx, λn L 0 L
∂u = g(x) ∂t t=0
Consider the special case in which the string starts from rest, i.e. g(x) = 0, and Bn∗ = 0, then ∞ cnπ nπ u(x, t) = Bn cos t sin x L L n=1 ∞ nπ nπ 1 Bn sin (x − ct) + sin (x + ct) = 2 n=1 L L ∞ ∞ 1 nπ 1 nπ (x − ct) + (x + ct) = Bn sin Bn sin 2 n=1 L 2 L n=1 Fourier sine series of f ∗ (x−ct)
=
Fourier sine series of f ∗ (x+ct)
1 ∗ [f (x − ct) + f ∗(x + ct)] 2
where f ∗ (•) is an odd periodic function, and f (•) = f ∗ (•) for • ∈ [0, 2L].
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Applied Mathematics — Partial Differential Equation
1 Physical meaning of u(x, t) = [f ∗ (x − ct) + f ∗ (x + ct)] : 2
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Wu-ting Tsai
Applied Mathematics — Partial Differential Equation
Wu-ting Tsai
Ex : g(x) = 0
f (x) =
2k x, L
0