Page 400 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

with the boundary equations Notes
u 0,t 0
u , L t 0
u x ,0 0
u
g x
t t 0

23.2 Boundary Value Problems (Heat Conduction or Diffusion)

Derivation of the Equation of Heat Conduction

In applied mathematics the partial differential equation
V 2 2
h V
t

2
where h is a constant and 2 is the Laplacian operator governs the temperature distribution V
in homogeneous solids.
To prove this, we know that the role of flow of heat in a homogeneous solid across the surface
V
is K per unit area, where V is the temperature and K a constant called the thermal
n

conductivity, denotes the differentiation along the normal. Taking an element of the solid
n
at the point P (x, y, z) as a rectangular parallelepiped with P centre and edges parallel to the co-
ordinate axes, of lengths dx, dy and dz, we find that the rate of flow of heat into the element is

K 2 Vdxdydz

But the element is gaining heat at the rate
V
C dxdydz
t
where is the density and C the specific heat. Thus, if there is no gain of heat in the element other
than by conduction, we have
V 2 2
C V
t

K
2
where C . ...(i)
C
If heat is being produced at ( , , )x y z in any other way, a term must be added to the right hand side
of (i).

23.2.1 Variable Heat Flow in One Dimension

If we consider the heat flow in a long thin bar or wire of constant cross-section and homogeneous
material which is along x-axis and is perfectly insulated, so that the heat flows in the x-
direction only, V depends only on x and t and therefore the heat equation becomes.

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