Introduction of Partial Differential

Equations and Boundary Value Problems

Y. K. Goh

2009

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Outline

I Definition

I Classification

I Where PDEs come from?

I Well-posed problem, solutions

I Initial Conditions and Boundary Conditions

I Strategies to solve PDEs

I Simple PDEs and their general solutions

I Introduction to 2nd order linear PDEs

I Classification of 2nd order linear PDEs

I Change of variables and canonical forms

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Definition

Definition (Partial Differential Equation)A partial differential equation (PDE) for a function of nvariables u(x1, . . . , xn) is an equation of the form

F (∂ku

∂xk1, · · · , ∂

ku

∂xkn, · · · , ∂u

∂x1

, · · · ∂u∂xn

, u, x1, · · · , xn) = 0 (1)

where F is a given function of the independent variables x1,..., xn and of the unknown function u = u(x1, · · · , xn) and ofa finite number of its partial derivatives.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Examples

In simple words, PDE is an equation that contains an unknownfunctions of several variables and its partial derivatives.One example is the two dimensional Laplace equation,

∇2u(x, y) =∂2u

∂x2+∂2u

∂y2= 0

or n-dimensional Laplace equation,

∇2u(x) =∂2u

∂x21

+∂2u

∂x22

+ · · ·+ ∂2u

∂x2n

= 0

or 3-dimensional Laplace equation in Spherical coordinates

∇2u =1

r2

∂

∂r

(r2∂u

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂u

∂θ

)+

1

r2 sin2 θ

∂2u

∂φ2= 0

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Linear PDEs

Some PDEs are linear

I Heat/diffusion eq.: ∂u∂t− k∇2u = 0, k > 0.

I Wave equation: ∂2u∂t2− k∇2u = 0, k > 0.

I 2D Laplace equation: ∇2u(x, y) = ∂2u∂x2 + ∂2u

∂y2= 0.

I Poisson equation: ∇2u = f(x), f(x) 6= 0.

I Helmholtz equation: ∇2u = −λu, λ > 0.

I Telegraph equation: utt + dut − uxx = 0.

I Schrodinger equation:i~ψt(x, t) = − ~2

2m∇2ψ(x, t) + V (x)ψ(x, t).

I Fokker-Planck eq.:ut −

∑ni,j=1(aiju)xixj

−∑n

i=1(biu)xi= 0.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Non-linear PDEs

Some PDEs are non-linear

I Nonlinear Poisson eq.: ∇2u = −f(u), f nonlinear.

I p-Laplacian equation: ∇ · (|∇u|p−2∇u) = 0.

I Minimal surface eq.: ∇ ·

(∇u√

1 + |∇u|2

)= 0.

I Hamilton-Jacobi eq.: ut +H(Du, x) = 0.

I Scalar conservation law: ut −∇ · F(u) = 0.

I Scalar reaction-diffusion eq.: ut −∇2u = f(u).

I Porous medium eq.: ut −∇2(uγ) = 0.

I Nonlinear wave eq.: utt −∇ · a(Du) = f(u).

I Korteweg-de Vries (KdV) eq.: ut + uux + uxxx = 0.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Homogeneous and Non-homogeneous PDEs

Some PDEs are homogeneous

I Heat/diffusion eq.: ∂u∂t− k∇2u = 0, k > 0.

I Wave equation: ∂2u∂t2− k∇2u = 0, k > 0.

I 2D Laplace equation: ∇2u(x, y) = ∂2u∂x2 + ∂2u

∂y2= 0.

and some are non-homogeneous

I Poisson equation: ∇2u = f(x), f(x) 6= 0.

I Helmholtz equation: ∇2u = −λu, λ > 0.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Classification of PDEs

For the ease of study, we often clasify PDE according to

I Linearity: linear, semi-linear, quasi-linear, non-linear;

I Homogeneity: homogeneous and non-homogeneousPDEs;

I Order: the order of the highest-order derivative presentin the PDE.

However, what is even more important is we need to

understand why we want to study PDEs!

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Importance Aspects of PDEs

PDEs arise from the modeling of numerous phenomena inphysical sciences and engineering: quantum mechanics,thermodynamics, electromagnetism, stock markets,automobile, aerodynamics, . . . .

Different from purely abstract mathematical problems, usuallythese PDEs represent the variation of physical quantities inspace and time. As the quantities are physical, we demandthat the mathematical problem must be a well-posedproblem (i.e. with a unique solution), and very often thesephysical quantities are subjected to certain constraints.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Importance Aspects of PDEs

Another important aspects of PDEs is the techniques ofsolving the PDEs.

In the next few slides, we will go through these importantaspects of PDEs:

I Modeling;

I Problems and constraints;

I Strategies of solving PDEs.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Modeling traveling wave on a vibrating string

When ∆x→ 0I Horizontal component: τ cosα ≈ τ cos βI Vertical component: −τ sinα + τ sin β = maI Divide by τ cosα: − tanα + tan β = ρ∆x

τ cosα∂2u∂t2

I 1D wave equation:∂2u

∂t2= c2∂

2u

∂x2, c2 = τ cosα

ρ∆x

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Modeling heat conduction across a bar

thermal energy in slice,

dQ = s(x)u(x, t)ρ(x)A∆x

I Energy Conservation:∂

∂t[s(x)u(x, t)ρ(x)A∆x] = φ(x, t)− φ(x+ ∆x, t)

I Fourier’s law of heat conduction: φ(x, t) = −K0∂u

∂x

I Heat equation:∂u

∂t=K0

sρ

∂2u

∂x2

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Well-posed Problem

Definition (Well-posed Problem)We say that a given problem for a PDE is well-posed if

I the problem in fact has a solution;

I the solution is unique;

I the solution depends continuously on the data given inthe problem.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Solution of PDE

Definition (Solution)A solution of a partial differential equation is any function thatsatisfies the equation.

For example if we could verify that ψ(x, y) = cxy is a solution

to the Laplace equation, ∂2ψ∂x2 + ∂2ψ

∂y2= 0, by differentiating and

substituting ψ(x, y) = cxy back into the Laplace equation.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Uniqueness of a Solution

However, more often uniqueness of a solution is not to beexpected. Unless certain conditions are given, we could havemore than one family of solution to a PDE, for example, thefollowing all are solutions to the Laplace equations (in fact,there are many more!)

ψ(x, y) = cxy

ψ(x, y) = c(x2 − y2)

ψ(x, y) =cx

x2 + y2

ψ(x, y) = cex cos y

ψ(x, y) = c ln(x2 + y2)

ψ(x, y) = c tan−1(y/x).

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

In order to identify a unique solution, we will need to havemore information, they are:

I Initial Conditions (the shape of the string, or its velocityat t = 0.)

I Boundary Condtions (the end points of the vibrated stringis fixed.)

The PDE, together with the initial conditions and boundaryconditions are called Boundary Value Problems.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Boundary conditions

We are familiar with the initial (Cauchy) conditions in theOrdinary Differential Equations, now we look at the boundaryconditions.The three common types of boundary conditions are:

1. Type 1 BC (Dirichlet condition): u = g;

2. Type 2 BC (Newmann or flux condition): ∂u∂n

= g;

3. Type 3 BC (Mixed or Robin or radiation condition):αu+ β ∂u

∂n= g.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Same strategies for solving PDEsWell-Posed Problem =

PDE + BC [+ IC]

HomogeneousBC?

Transform BC toHomogeneous BC

HomogeneousPDE?

Yes

No

RectangularBC?

Yes

CylindricalBC?

Yes

No

SphericalBC?

No

Other BC?

No

Order of PDE?

= 2

= 1 Transform PDEto ODE

Consider the homogeneouspart of the PDE and

calculate the general solution.Apply the variat ion of parameter

on the general solution.

X ’ ’+kX=0,etc.

No

Bessel Eq,etc.

Legendre Eq,Spherical

Bessel Eq, etc.

Yes

Yes

X’ ’+p(x)X ’+q(x)=0,etc.

SL Problem

aX(A)+bX’(A)=0cX(B)+dX’(B)=0

A < = r < = BC < = p < = DE < = z < = F

A < = r < = BC < = p < = DE < = q < = F

Yes aX(A)+bX’(A)=0cX(B)+dX’(B)=0

+

+

+

+

sin, cos, exp,etc.

sin, cos, J_v, Y_v, etc.

sin, cos, Y_nm, etc.

SpecialFunctions

Eigen-Functions

General Solution is aCombination of Fourier

Series or Fourier

Principle ofSuperpositionX’ ’+p(x)X’+q(x)=f_n(x)

Variation of Parameters

Specific Solution

Initial Conditions

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Solving Simple PDEs

Let consider simple PDEs for an unknown functions u(x, y).

I∂u

∂x= 0 =⇒ u(x, y) = f(y).

I∂u

∂x= f(x) =⇒ u(x, y) =

∫f(x) dx+ g(y)

I∂u

∂x= f(y) =⇒ u(x, y) = xf(y) + g(y)

I∂u

∂x= f(x, y) =⇒ u(x, y) =

∫f(x, y) dx+ g(y)

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Solving Simple PDEs

Increase the difficulties (if any at all!) one notch:

I∂2u

∂x2= 0 =⇒ u(x, y) = xf(y) + g(y).

I∂2u

∂x ∂y= 0 =⇒ u(x, y) = f(x) + g(y)

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Cauchy Problems

In order to pick up a particular solution we need to supply withinitial conditions, the resulting is a Cauchy (initial value)problem.

I∂u

∂x= y,when x = 1, u(1, y) = 2y =⇒ u(x, y) = xy+y.

I∂2u

∂x2= 0,when x = 0, u = y2,

∂u

∂x= 2y − sin 2y =⇒

u(x, y) = 2xy − x sin 2y + y2.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Linear 2nd Order PDEs

A general linear 2nd order can be written as

a∂2u

∂x2+ b

∂2u

∂x ∂y+ c

∂2u

∂y2+ d

∂u

∂x+ e

∂u

∂y+ fu+ g = h(x, y).

I if h(x, y) = 0 then it is a homogeneous PDE.

I the quatity ∆ = b2 − 4ac is called the discriminant.

I according to ∆, linear 2nd order PDE can be classified asI if ∆ > 0, then it is a hyperbolic PDEI if ∆ = 0, then it is a parabolic PDEI if ∆ < 0, then it is a elliptic PDE

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Classification of 3 common PDEs

Three PDEs that are the main focus of this course are waveequation, heat equation and Laplace equation. These PDEsare linear 2nd PDEs, and their are classify as

I Wave equation :∂2u

∂t2− c2∂

2u

∂x2= 0 (hyperbolic);

I Heat equation :∂u

∂t− c2∂

2u

∂x2= 0 (parabolic);

I Laplace equation :∂2u

∂x2+∂2u

∂y2= 0 (elliptic).

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

The advantage of the classification is, depending on ∆, wecould making a change of variables to convert the PDEs intocanonical forms.

Sometimes – no always – these canonical forms can be solvedeasily, expect solving for elliptic equation is a bit subtle andnot straight forward.

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Canonical Forms

By making a change in variables, s = ax+ by, t = cx+ dy,linear PDE could convert to its canonical form with properchoice of a, b, c and d.

I Hyperbolic equation:∂2u

∂s ∂t= F (s, t, u, us, ut);

I Parabolic equation:∂2u

∂s2= F (s, t, u, us, ut)

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems

Example

Consider the PDE : 6∂2u

∂x2+ 5

∂2u

∂x ∂y+∂2u

∂y2= 0 when

y = 0, u = cosx+ sinx, and ∂u∂y

= cosx+ 2 sinx.

I ∆ > 0 =⇒ hyperbolic;

I By making a change of variables : s = x− 2y, t = x− 3y;

I The PDE will convert to its canonical form:∂2u

∂s ∂t= 0;

I Which has a general solution,u = f(s) + g(t) = f(x− 2y) + g(x− 3y), and

I Particular solutionu = cos(x− 2y) + 4 sin(x− 2y)− 3 sin(x− 3y).

Y. K. Goh

Introduction of Partial Differential Equations and Boundary Value Problems