Laplace transform isotherm .pdf - University of Hertfordshire ...

This is the most general form of the heat equation.
In our work, we consider the case where K is constant and ρ and c are
independent of time and we use the following form of the heat equation,
which is the most convenient for our purposes:
where α = K
pc
∇ 2 u = 1 ∂u
α ∂t
is the thermal diffusivity.
(2.2)
The partial differential equation for diffusion problems is set up in a
similar manner (Crank 1979) and its form is
∂2C 1 ∂C
=
∂x2 D ∂t
C is the concentration of the diffusing substance and D is the diffusion
coefficient and the law equivalent to Fourier’s law in equation (2.1) is Fick’s
law
q = −DgradC
The heat equation is a partial differential equation and in the next section
we look at the classification of partial differential equations, because the
nature of the solution depends on the classification of the equation.
2.1.1 Classification of partial differential equations
In order to classify the equations, we will consider the linear partial differ-
ential operator L in two independent variables for simplicity, which can be
extended to three or more variables (Weinberger 1965). We define the linear
partial differential operator as
L [u] ≡ A(x,y) ∂2u ∂y2 +B (x,y) ∂2u ∂x∂y +C (x,y) ∂2u ∂u ∂u
∂x2+D (x,y) +E (x,y) +F (x,y)u
∂y ∂x
and if
L [u] = G
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