Page 13 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

PARTIAL DIFFERENTIAL EQUATIONS IN ENGINEERING 3
The first equation is linear and nonhomogeneous. The third term is a mixed partial derivative.
Since it is of second order in x two boundary conditions are necessary on x. It is first order
in y, so that only one boundary condition is required on y. The second equation is linear and
homogeneous and is of second order in all three variables. The third equation is nonlinear
because the first term is not of first degree in u and u x .Itisoforder 1in x and order 2 in y.
In this book we consider only linear equations. We will now derive the partial differential
equations that describe some of the physical phenomena that are common in engineering
science.

Problems
Tell whether the following are linear or nonlinear and tell the order in each of the independent
variables:

2
u + xu + u = 0

tan(y)u y + u yy = 0
tan(u)u y + 3u = 0
u yyy + u yx + u = 0

1.3 THE HEAT CONDUCTION (OR DIFFUSION) EQUATION
1.3.1 Rectangular Cartesian Coordinates
The conduction of heat is only one example of the diffusion equation. There are many other
important problems involving the diffusion of one substance in another. One example is the
diffusion of one gas into another if both gases are motionless on the macroscopic level (no
convection). The diffusion of heat in a motionless material is governed by Fourier’s law which
states that heat is conducted per unit area in the negative direction of the temperature gradient
in the (vector) direction n in the amount ∂u/∂n,thatis

n
q =−k∂u/∂n (1.7)
n
where q denotes the heat flux in the n direction (not the nth power). In this equation u is the
local temperature and k is the thermal conductivity of the material. Alternatively u could be the
partial fraction of a diffusing material in a host material and k the diffusivity of the diffusing
material relative to the host material.
Consider the diffusion of heat in two dimensions in rectangular Cartesian coordinates.
Fig. 1.1 shows an element of the material of dimension x by y by z. The material has a
specific heat c and a density ρ. Heat is generated in the material at a rate q per unit volume.
Performing a heat balance on the element, the time (t) rate of change of thermal energy
within the element, ρc x y z∂u/∂t is equal to the rate of heat generated within the element