APPLIED MATHEMATICS

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 . . . . . . . .

7

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).

3


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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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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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

✷


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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¨ = 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


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

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n = 1, 2, 3 . . .

✷


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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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].

12


1 Physical meaning of u(x, t) = [f ∗ (x − ct) + f ∗ (x + ct)] : 2

13

Wu-ting Tsai


Wu-ting Tsai

Ex : g(x) = 0           

f (x) =          

2k x, L

0