Chapter 12: Boundary-Value Problems in Rectangular Coordinates - - PowerPoint PPT Presentation

Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Chapter 12: Boundary-Value Problems in Rectangular Coordinates

王奕翔

Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw

December 18, 2013

1 / 38 王奕翔 DE Lecture 14

Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

In this lecture, we focus on solving some classical partial differential equations in boundary-value problems. Instead of solving the general solutions, we are only interested in finding useful particular solutions. We focus on linear second order PDE: (A, · · · , G: functions of x, y) Auxx + Buxy + Cuyy + Dux + Euy + Fu = G. notation: for example, uxy :=

∂2u ∂x ∂y.

Method: Separation of variables – convert a PDE into two ODE’s Types of Equations: Heat Equation Wave Equation Laplace Equation

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Classification of Linear Second Order PDE

Auxx + Buxy + Cuyy + Dux + Euy + Fu = G. notation: for example, uxy :=

∂2u ∂x ∂y. 1 Homogeneous vs. Nonhomogeneous

Homogeneous ⇐ ⇒ G = 0 Nonhomogeneous ⇐ ⇒ G ̸= 0.

2 Hyperbolic, Parabolic, and Elliptic: A, B, C, · · · , G: constants,

Hyperbolic ⇐ ⇒ B2 − 4AC > 0 Parabolic ⇐ ⇒ B2 − 4AC = 0 Elliptic ⇐ ⇒ B2 − 4AC < 0

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Superposition Principle

Theorem If u1(x, y), u2(x, y), . . . , uk(x, y) are solutions of a homogeneous linear PDE, then a linear combination u(x, y) :=

k

∑

n=1

cnun(x, y) is also a solution. Note: We shall assume without rigorous argument that the linear combination can be an infinite series u(x, y) :=

∞

∑

n=1

cnun(x, y)

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

1 Separation of Variables and Classical PDE’s 2 Wave Equation 3 Laplace’s Equation

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Separation of Variables

To find a particular solution of an PDE, one method is separation of variables, that is, assume that the solution u(x, y) takes the form of a product of a x-function and a y-function: u(x, y) = X(x)Y(y). Then, with the following, sometimes the PDE can be converted into an ODE of X and an ODE of Y: ux = dX dx Y = X ′Y, uy = XdY dy = XY ′ uxx = d2X dx2 Y = X ′′Y, uyy = Xd2Y dy2 = XY ′′, uxy = X ′Y ′ Note: Derivatives are with respect to different independent variables. For example, X ′ := dX dx .

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Convert a PDE into Two ODE’s

Example Use separation of variables to convert the PDE below into two ODE’s. x2uxx + (x + 1)uy + (x + xy)u = 0 With u(x, y) = X(x)Y(y), the original PDE becomes x2X ′′Y + (x + 1)XY ′ + (x + 1)yXY = 0 = ⇒ x2X ′′Y = −(x + 1)X(Y ′ + yY) = ⇒ x2X ′′ (x + 1)X = −Y ′ Y − y = λ separation constant Left-hand side is a function of x, independent of y; Right-hand side is a function of y, independent of x. Hence, the above is equal to something independent of x and y

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Convert a PDE into Two ODE’s

Example Use separation of variables to convert the PDE below into two ODE’s. x2uxx + (x + 1)uy + (x + xy)u = 0 With u(x, y) = X(x)Y(y), the original PDE becomes x2X ′′Y + (x + 1)XY ′ + (x + 1)yXY = 0 = ⇒ x2X ′′Y = −(x + 1)X(Y ′ + yY) = ⇒ x2X ′′ (x + 1)X = −Y ′ Y − y = λ separation constant = ⇒ { x2X ′′(x) − λ(x + 1)X(x) = 0 Y ′(y) + (y + λ)Y(y) = 0

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Some Remarks

1 The method of separation of variables can only solve for some linear

second order PDE’s, not all of them.

2 For the PDE’s considered in this lecture, the method works. 3 The method may work for both homogeneous (G = 0) and

nonhomogeneous (G ̸= 0) PDE’s Auxx + Buxy + Cuyy + Dux + Euy + Fu = G.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Three Classical PDE’s

In this lecture we focus on solving boundary-value problems of the following three classical PDE’s that arise frequently in physics, mechanics, and engineering:

1 (One-Dimensional) Heat Equation/Diffusion Equation

kuxx = ut, k > 0

2 (One-Dimensional) Wave Equation/Telegraph Equation

a2uxx = utt

3 (Two-Dimensional) Laplace Equation

uxx + uyy = 0

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Heat Transfer within a Thin Rod: Heat Equation

x

0 x x + ∆x L

Cross section of area A

Assumptions: Heat only flows in x-direction. No heat escapes from the surface. No heat is generated in the rod. Rod is homogeneous with density ρ. Let u(x, t) denote the temperature of the rod at location x at time t. Consider the quantity of heat with [x, x + dx]: (γ : 比熱) dQ = γ (ρAdx) u = ⇒ Qx = γρAu = ⇒ Qxt = γρAut Heat transfer rate through the cross section = −KAux, and hence the net heat rate inside [x, x + dx] is dQt = −KAux(x, t) − (−KAux(x + dx, t)) dQt = KAu (ux(x + dx, t) − ux(x, t)) = KAuxxdx = ⇒ Qtx = KAuxx

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Heat Transfer within a Thin Rod: Heat Equation

x

0 x x + ∆x L

Cross section of area A

Assumptions: Heat only flows in x-direction. No heat escapes from the surface. No heat is generated in the rod. Rod is homogeneous with density ρ. Let u(x, t) denote the temperature of the rod at location x at time t. Hence, { Qxt = γρAut Qtx = KAuxx = ⇒ γρ

• Aut = K
• Auxx =

⇒ ( K γρ ) uxx = ut = ⇒ kuxx = ut, k > 0

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Heat Equation: Initial and Boundary Conditions

x

0 x x + ∆x L

Cross section of area A

Initial Condition: Provides the spatial distribution of the temperature at time t = 0. u(x, 0) = f(x), 0 < x < L Boundary Conditions: At the end points x = 0 and x = L, give the constraints on u: (Dirchlet condition), for example, (u0: constant) u(L, t) = u0 Temperature at the right end is held at constant. ux: (Neumann condition), for example, ux(L, t) = 0 The right end is insulated. ux + hu: (Robin condition), for example, (h > 0, um: constants) ux(L, t) = −h {u(L, t) − um} Heat is lost from the right end.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Heat Equation: Boundary-Value Problems

x

0 x x + ∆x L

Cross section of area A

A problem involving both initial and boundary conditions is called a boundary-value problem At the two boundaries x − 0 and x = L,

• ne can use different kinds of conditions.

Examples: kuxx = ut, 0 < x < L, t > 0

Heat equation

u(0, t) = u0, ux(L, t) = −h {u(L, t) − um} , t > 0

Boundary condition

u(x, 0) = f(x), 0 < x < L

Initial condition

kuxx = ut, 0 < x < L, t > 0

Heat equation

ux(0, t) = 0, u(L, t) = 0, t > 0

Boundary condition

u(x, 0) = f(x), 0 < x < L

Initial condition

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Dynamics of a String Fixed at Two Ends: Wave Equation

u ∆s u(x, t) x x + ∆x L x x u ∆s x + ∆x x θ1 θ2 T1 T2 (a) Segment of string

Assumptions: No external force. Tension force is large compared to gravity and is the same at all points. Slope of the curve is very small at all points. Vertical displacement ≪ string length. String has mass per unit length ρ. Let u(x, t) denote the vertical position (displacement)

• f the string at location x at time t.

Consider the string in [x, x + dx]. Net vertical force is T (sin θ2 − sin θ1) ≈ T (tan θ2 − tan θ1) = T {ux(x + dx, t) − ux(x, t)} = Tuxxdx

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Dynamics of a String Fixed at Two Ends: Wave Equation

u ∆s u(x, t) x x + ∆x L x x u ∆s x + ∆x x θ1 θ2 T1 T2 (a) Segment of string

Assumptions: No external force. Tension force is large compared to gravity and is the same at all points. Slope of the curve is very small at all points. Vertical displacement ≪ string length. String has mass per unit length ρ. Let u(x, t) denote the vertical position (displacement)

• f the string at location x at time t.

Since the slope is small, the mass ≈ ρdx. Hence Tuxx✚

✚

dx = (ρ✚

✚

dx) utt = ⇒ T ρ uxx = utt = ⇒ a2uxx = utt

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Wave Equation: Initial and Boundary Conditions

u ∆s u(x, t) x x + ∆x L x x u ∆s x + ∆x x θ1 θ2 T1 T2 (a) Segment of string

Initial Conditions: Provide the initial displacement u and velocity ut at time t = 0. u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L Boundary Conditions: At the end points x = 0 and x = L, give the constraints

• n u, ux, or ux + hu. Usually in the scenario of strings,

the boundary conditions are u(0, t) = 0, u(0, L) = 0, t > 0 Both ends are fixed. ux(0, t) = 0, ux(0, L) = 0, t > 0 Free-ends condition

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Wave Equation: Boundary-Value Problems

u ∆s u(x, t) x x + ∆x L x x u ∆s x + ∆x x θ1 θ2 T1 T2 (a) Segment of string

Examples: Both ends are fixed: a2uxx = utt, 0 < x < L, t > 0

Wave equation

u(0, t) = 0, u(L, t) = 0, t > 0

Boundary condition

u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L

Initial condition

Free Ends: a2uxx = utt, 0 < x < L, t > 0

Wave equation

ux(0, t) = 0, ux(L, t) = 0, t > 0

Boundary condition

u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L

Initial condition

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Laplace’s Equation

(x, y) y Thermometer Temperature as a function of position

• n the hot plate

W O

x

H

Laplace’s equation usually occurs in time-independent problems involving potentials. Its solution can also be interpreted as a steady-state temperature distribution. Two-dimensional Laplace Equation ∇2u := uxx + uyy = 0 Three-dimensional Laplace Equation ∇2u := uxx + uyy + uzz = 0

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Laplace’s Equation: Boundary Conditions

(x, y) y Thermometer Temperature as a function of position

• n the hot plate

W O

x

H

Boundary Conditions: In the x-direction, at the end points x = 0 and x = a, give the constraints on u, ux, or ux + hu. In the y-direction, at the end points y = 0 and y = b, give the constraints on u, uy, or uy + hu. Examples: Both ends in x are insulated ux(0, y) = 0, ux(a, y) = 0 Temperatures of two ends in y are held at different distributions u(x, 0) = f(x), u(x, b) = g(x)

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Laplace’s Equation: Boundary-Value Problems

Example: uxx + uyy = 0, 0 < x < a, 0 < y < b

Laplace’s equation

ux(0, y) = 0, ux(a, y) = 0, 0 < y < b

Boundary condition

u(x, 0) = f(x), u(x, b) = g(x), 0 < x < a

Boundary condition

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Modifications of Heat and Wave Equations

In the derivation of the heat equation and the wave equation, we assume that there is no internal or external influences. For example, no heat escapes from the surface, no heat is generated in the rod, no external force act on the string, etc. Taking external and internal influences into account, more general forms

• f the heat equation and the wave equation are the following:

kuxx + G(x, t, u, ux) = ut Heat Equation a2uxx + F(x, t, u, ut) = utt Wave Equation Example: kuxx − h(u − um) = ut

heat transfers from the surface to an environment with constant temperature um

a2uxx + f(x, t) = utt

External force f acts on the string

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Homogeneous vs. Nonhomogeneous Boundary Conditions

Homogeneous Boundary Condition: ux(0, y) = 0, ux(a, y) = 0, u(x, 0) = 0, u(0, L) = 0 Nonhomogeneous Boundary Condition: ux(0, y) = f(y), ux(a, y) = g(y), u(x, L) = um Typically, when using separation of variables, start with the independent variable associated with homogeneous boundary conditions.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

1 Separation of Variables and Classical PDE’s 2 Wave Equation 3 Laplace’s Equation

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Wave Equation: a Boundary-Value Problem

Solve u(x, t) : auxx = utt, 0 < x < L, t > 0 subject to : u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L

We focus on solving the above BVP (both ends are fixed). Step 1: Separation of variables: Assume that the solution u(x, t) = X(x)T(t), X, T ̸= 0. Then, a2uxx = utt = ⇒ a2X ′′T = XT ′′ = ⇒ X ′′ X = T ′′ a2T = −λ = ⇒ { X ′′ + λX = 0 T ′′ + a2λT = 0 The 2 homogeneous boundary conditions become X(0) = X(L) = 0.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Solve in the x-Dimension and Find λ

Solve : X ′′ + λX = 0, T ′′ + a2λT = 0 subject to : X(0) = 0, X(L) = 0 X(x)T(0) = f(x), X(x)T ′(0) = g(x), 0 < x < L

Step 2: λ remains to be determined. What values should λ take?

1 λ = 0: X(x) = c1 + c2x. X(0) = X(L) = 0 =

⇒ c1 = c2 = 0.

2 λ = −α2 < 0: X(x) = c1e−αx + c2eαx.

Plug in X(0) = X(L) = 0, we get c1 = c2 = 0.

3 λ = α2 > 0: X(x) = c1 cos(αx) + c2 sin(αx).

Plug in X(0) = X(L) = 0, we get c1 = 0, and c2 sin(αL) = 0. Hence, c2 ̸= 0 only if αL = nπ. Since X ̸= 0, pick λ = n2π2 L2 , n = 1, 2, . . . = ⇒ X(x) = c2 sin nπ

L x.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Solve in t-Dimension and Superposition

Solve : X ′′ + λX = 0, T ′′ + a2λT = 0 subject to : X(0) = 0, X(L) = 0 X(x)T(0) = f(x), X(x)T ′(0) = g(x), 0 < x < L

Step 3: Once we fix λ = n2π2

L2 , n = 1, 2, . . ., we obtain

X(x) = c2 sin (nπ L x ) , T(t) = c3 cos (nπa L t ) + c4 sin (nπa L t ) = ⇒ un(x, t) = { An cos (nπa L t ) + Bn sin (nπa L t )} sin (nπ L x ) , (An := c2c3, Bn := c2c4) = ⇒ u(x, t) :=

∞

∑

n=1

un(x, t) is a solution, by the superposition principle.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Plug in Initial Condition, Revoke Fourier Series, and Done

Solve u(x, t) : auxx = utt, 0 < x < L, t > 0 subject to : u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L

Step 4: Plug in the initial conditions and find {An, Bn | n = 1, 2, . . .}. u(x, 0) = f(x), u(x, t) =

∞

∑

n=1

{ An cos (nπa L t ) + Bn sin (nπa L t )} sin (nπ L x ) = ⇒ f(x) =

∞

∑

n=1

An sin (nπ L x ) , 0 < x < L From the Fourier sine series expansion on (0, L), we get An = 2 L ∫ L f(x) sin (nπ L x ) dx.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Plug in Initial Condition, Revoke Fourier Series, and Done

Solve u(x, t) : auxx = utt, 0 < x < L, t > 0 subject to : u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L

Step 4: Plug in the initial conditions and find {An, Bn | n = 1, 2, . . .}. ut(x, 0) = g(x), u(x, t) =

∞

∑

n=1

{ An cos (nπa L t ) + Bn sin (nπa L t )} sin (nπ L x ) = ⇒ g(x) =

∞

∑

n=1

Bn nπa L sin (nπ L x ) , 0 < x < L From the Fourier sine series expansion on (0, L), we get Bn nπa L = 2 L ∫ L g(x) sin (nπ L x )

• dx. =

⇒ Bn = 2 nπa ∫ L g(x) sin (nπ L x ) dx.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Final Solution

Solve u(x, t) : auxx = utt, 0 < x < L, t > 0 subject to : u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = f(x), ut(x, 0) = g(x), 0 < x < L

Step 5: The final solution is u(x, t) =

∞

∑

n=1

{ An cos (nπa L t ) + Bn sin (nπa L t )} sin (nπ L x ) =

∞

∑

n=1

Cn sin (nπa L t + φn ) sin (nπ L x ) An = 2 L ∫ L f(x) sin (nπ L x ) dx, Bn = 2 nπa ∫ L g(x) sin (nπ L x ) dx Cn = √ A2

n + B2 n,

sin φn = An Cn , cos φn = Bn Cn

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Standing Waves

The final solution u(x, t) =

∞

∑

n=1

Cn sin (nπa L t + φn ) sin (nπ L x ) is a linear combination of standing waves or normal modes un(x, t) = Cn sin (nπa L t + φn ) sin (nπ L x ) , n = 1, 2, . . . For a normal mode n, at a fixed location x, the string moves with time-varying amplitude Cn sin ( nπ

L x

) frequency fn := nπa/L

2π

= na

2L

Fundamental Frequency: f1 := πa/L

2π

=

a 2L

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

L

(a) First standing wave (b) Second standing wave

Node L L 2 x x

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

1 Separation of Variables and Classical PDE’s 2 Wave Equation 3 Laplace’s Equation

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Laplace’s Equation: a Boundary-Value Problem

Solve u(x, y) : uxx + uyy = 0, 0 < x < a, 0 < y < b subject to : ux(0, y) = 0, ux(a, y) = 0, 0 < y < b u(x, 0) = 0, u(x, b) = f(x), 0 < x < a

We focus on solving the above BVP (both ends x = 0 and x = a are insulated). Step 1: Separation of variables: Assume that the solution u(x, y) = X(x)Y(y), X, Y ̸= 0. Then, uxx + uyy = 0 = ⇒ X ′′Y + XY ′′ = 0 = ⇒ X ′′ X = −Y ′′ Y = −λ = ⇒ { X ′′ + λX = 0 Y ′′ − λY = 0 The 3 homogeneous boundary conditions become X ′(0) = X ′(a) = Y(0) = 0.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Solve in the x-Dimension and Find λ

Solve : X ′′ + λX = 0, Y ′′ − λY = 0 subject to : X ′(0) = 0, X ′(a) = 0 Y(0) = 0, X(x)Y(b) = f(x), 0 < x < a

Step 2: λ remains to be determined. What values should λ take?

1 λ = 0: X(x) = c1 + c2x. X ′(0) = X ′(a) = 0 =

⇒ c2 = 0.

2 λ = −α2 < 0: X(x) = c1e−αx + c2eαx.

Plug in X ′(0) = X ′(a) = 0, we get c1 = c2 = 0.

3 λ = α2 > 0: X(x) = c1 cos(αx) + c2 sin(αx).

Plug in X ′(0) = X ′(a) = 0, we get c2 = 0, and c1α sin(αa) = 0. Hence, c1 ̸= 0 only if αa = nπ. Since X ̸= 0, pick λ = n2π2 a2 , n = 0, 1, 2, . . . = ⇒ X(x) = c1 cos ( nπ

a x

) .

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Solve in y-Dimension and Superposition

Solve : X ′′ + λX = 0, Y ′′ − λY = 0 subject to : X ′(0) = 0, X ′(a) = 0 Y(0) = 0, X(x)Y(b) = f(x), 0 < x < a

Step 3: Once we fix λ = n2π2

a2 , n = 0, 1, 2, . . ., we obtain X(x) = c1 cos

( nπ

a x

) Y(y) = {

✚ ✚

c3 + c4y, n = 0

✘✘✘✘✘ ✘

c3 cosh ( nπ

a y

) + c4 sinh ( nπ

a y

) , n ≥ 1 (Y(0) = 0 = ⇒ c3 = 0) = ⇒ un(x, y) = { A0y, n = 0 An cos ( nπ

a x

) sinh ( nπ

a y

) , n ≥ 1 , (An := c1c4) = ⇒ u(x, y) :=

∞

∑

n=0

un(x, y) is a solution, by the superposition principle.

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Plug in Initial Condition, Revoke Fourier Series, and Done

Solve u(x, y) : uxx + uyy = 0, 0 < x < a, 0 < y < b subject to : ux(0, y) = 0, ux(a, y) = 0, 0 < y < b u(x, 0) = 0, u(x, b) = f(x), 0 < x < a

Step 4: Plug in the initial conditions and find {An | n = 1, 2, . . .}. u(x, b) = f(x), u(x, y) = A0y +

∞

∑

n=1

An cos (nπ a x ) sinh (nπ a y ) = ⇒ f(x) = A0b +

∞

∑

n=1

An cos (nπ a x ) sinh (nπ a b ) , 0 < x < a From the Fourier cosine series expansion on (0, a), we get 2A0b = 2 a ∫ a f(x) dx, An sinh (nπ a b ) = 2 a ∫ a f(x) cos (nπ a x ) dx

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Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation

Final Solution

Solve u(x, y) : uxx + uyy = 0, 0 < x < a, 0 < y < b subject to : ux(0, y) = 0, ux(a, y) = 0, 0 < y < b u(x, 0) = 0, u(x, b) = f(x), 0 < x < a

Step 5: The final solution is u(x, y) = A0y +

∞

∑

n=1

An cos (nπ a x ) sinh (nπ a y ) A0 = 1 ab ∫ a f(x) dx An = 2 a sinh ( nπ

a b

) ∫ a f(x) cos (nπ a x ) dx, n ≥ 1

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