conc_02_ibvps

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Concepts Review:

Basic guide to Boundary Conditions and IBVPs

1 Equations

We will concentrate on simplest linear PDEs with one space variable ( x) and one time variable

(t). The dependent variable is u(x, t).

Partial derivatives are denoted by subscripts.

We consider the following equations.

1. Homogeneous heat/diffusion equation

ut = a2uxx, a = const; (1)

2. Non-homogeneous heat/diffusion equation

ut = a2uxx + f(x, t), a = const (2)

(Here f(x, t) is a source term);

3. Homogeneous wave equation

utt = a2uxx, a = const; (3)

4. Non-homogeneous wave equation

utt = a2uxx + f(x, t), a = const (4)

(Here f(x, t) is a forcing term).

Equations (1), (2), as the name suggests, arise in heat conduction and diffusion problems,

as well as other numerous applications.

Equations (3), (4) describe small oscillations of a string (unloaded and loaded), lengthwise

oscillations of elastic rods and springs, and also arises in many other applications.

2 Initial and boundary conditions

In order to have a unique solution, the PDE has to be appended by an appropriate number of

initial and boundary conditions. Given a physical situation, it is most important to be able tocorrectly formulate such an Initial-Boundary Value Problem (IBVP).

The number of initial conditions (IC) is equal to the number of time derivatives (this is a

direct analogy with ODEs!) If one IC is required, it is usually u(0, x); if two are required, one

specifies u(0, x) and initial rate of change of u: ut(0, x). Note that initial conditions do depend

on x, i.e. are different for different x (0, L).

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For 2nd-order PDEs with one space variable, defined for an interval 0 < x < L, the number

of boundary conditions two. Boundary conditions (BC) are given on two boundaries of the

domain 0 < x < L, i.e. at x = 0 and x = L. BCs are not artificially made - we choose them

from a set of possible BCs, according to the physics of the problem.

For 2nd-order PDEs in multiple space dimensions, one boundary condition must be given in

each point of the domain boundary.

The solution of an IBVP we wish to find is u(x, t) for 0 < x < L, and all times t > 0.

Three main types of boundary conditions for PDE problems with one space and one time

variable are listed in the following table. Other types also occur in applications, but are out of

scope of the course.

In the table, I only gave BCs at x = 0. They have the corresponding form at x = L.

Table 1: Types of linear BCs

Name Type BC A physical application

Dirichlet Homogeneous u(0, t) = 0 Heat/diffusion: zero temperature/concentration is kep

(Type I) at x = 0 for all times.Wave: the left end of the string (or rod) is attached

at x = 0 and does not move for all times.

Non-homogeneous u(0, t) = (t) Heat/diffusion: temperature/concentration at x = 0

changes in time, according to the law (t).

Wave: the left end of the string (elastic rod) is moved

up/down (left/right) according to (t).

Neumann Homogeneous ux(0, t) = 0 Heat/diffusion: left end is insulated (flux q = kux

(Type II) vanishes at x = 0 for all times.)

Wave: the left end of the rod is free

for all times.

Non-homogeneous ux(0, t) = (t) Heat/diffusion: heat/substance flux through the end

x = 0 is (0, t) = K10

(t).

Wave: For longitudinal oscillations of an elastic rod

or spring, this BC means that to the left end a force

|F(t)| = k(t) is applied.

Mixed Homogeneous u(0, t) + ux(0, t) = 0 Heat: left end exchanges heat with the

(Type III) a, b = const. environment (which has zero temperature) according t

Newtons law of cooling: K0ux(0, t) = h(0 u(0, t)).

Wave: the left end of oscillating rod or springis attached elastically.

Non-homogeneous u(0, t) + ux(0, t) = Heat: left end exchanges heat with the

= (t) environment (which has temperature (t)) according

to Newtons law of cooling:

K0ux(0, t) = h((t) u(0, t)).

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3 Examples of IBVPs

Example 1. Describe small oscillations of a string of length L, with fixed ends, no displace-

ment, and initial velocity given by v0(x) = x(L x)2.

IBVP:

utt = a2

uxx, 0 < x < L, 0 < t;u(x, 0) = 0 (IC 1)

ut(x, 0) = x(L x)2 (IC 2)

u(0, t) = 0 (BC 1)

u(L, t) = 0 (BC 2)

Example 2. Describe small longitudinal oscillations of an elastic rod of length L, initially

undisturbed and at rest, with left end externally driven (position given by (t) = sin(2t)) and

a free right end.

IBVP:

utt = a2uxx, 0 < x < L, 0 < t;

u(x, 0) = 0 (IC 1)

ut(x, 0) = 0 (IC 2)

u(0, t) = sin(2t) (BC 1)

ux(L, t) = 0 (BC 2)

Example 3. Describe heat conduction in a metal rod of length L, with initial temperature

100K, with insulated left end, and the right end exchanging heat with environment whose

temperature is = 200K.

IBVP:

ut = a2uxx, 0 < x < L, 0 < t;

u(x, 0) = 100 (IC 1)

ux(0, t) = 0 (BC 1)

K0ux(L, t) = h(200 u(L, t)) (BC 2)

(Here K0, h , a are constant coefficients that can be found in a table for each material.)

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