Computational Physics - Department of Physics

304 10 Partial Differential Equations

the diffusion equation, or heat equation.
If we specialize to the heat equation, we assume that the diffusion of heat through some
material is proportional with the temperature gradientT(x,t)and using conservation of en-
ergy we arrive at the diffusion equation

κ

Cρ

∇^2 T(x,t) =

∂T(x,t)

∂t

whereCis the specific heat andρthe density of the material. Here we let the density be rep-
resented by a constant, but there is no problem introducing an explicit spatial dependence,
viz.,
κ
Cρ(x,t)
∇^2 T(x,t) =
∂T(x,t)
∂t

Setting all constants equal to the diffusion constantD, i.e.,

D=

Cρ

κ,

we arrive at
∇^2 T(x,t) =D∂T(x,t)
∂t
Specializing to the 1 + 1 -dimensional case we have

∂^2 T(x,t)

∂x^2 =D

∂T(x,t)

∂t.

We note that the dimension ofDis time/length^2. Introducing the dimensionless variables
αxˆ=xwe get
∂^2 T(x,t)
α^2 ∂xˆ^2

=D

∂T(x,t)

∂t

,

and sinceαis just a constant we could defineα^2 D= 1 or use the last expression to define a
dimensionless time-variableˆt. This yields a simplified diffusion equation

∂^2 T(xˆ,ˆt)

∂xˆ^2

=∂T(xˆ,

tˆ)

∂ˆt

It is now a partial differential equation in terms of dimensionless variables. In the discussion
below, we will however, for the sake of notational simplicity replacexˆ→xandtˆ→t. Moreover,
the solution to the 1 + 1 -dimensional partial differential equation is replaced byT(xˆ,tˆ)→u(x,t).

10.2.1Explicit Scheme

In one dimension we have the following equation

∇^2 u(x,t) =∂u(x,t)

∂t

,

or
uxx=ut,

with initial conditions, i.e., the conditions att= 0 ,

u(x, 0 ) =g(x) 0 <x<L