A schematic of the problem geometry is shown in the first snapshot.

For computational convenience we introduce the following dimensionless coordinates

**:**Thus the thermal conduction equations for the inner core and cladding are:

The quantity

is a dimensionless heat generation term defined as:

,

while

is the dimensionless absorption coefficient for the light source entering the core region.

The boundary conditions for the core region are:

BC1:

,

BC2:

,

0<ξ<ξ_{1}, at

,

BC3:

,

Here

is a Biot number and

is a ratio of thermal conductivities for the cladding and core regions. Typically the conductivity of the cladding is much smaller than that of the core so that

.

At the core/cladding interface we must have continuity of heat flux and temperature. These conditions are:

BC4:

,

BC5:

,

For the cladding region, we have Newton's law of cooling at all exposed surfaces:

BC7:

,

ξ_{1}<ξ<1, at

,

BC8:

,

ξ_{1}<ξ<1, at

.

**Finite difference formulation**We consider a uniform grid in the

and

directions. Let

be the number of grid points in the

direction,

be the number grid points in the

direction for the core, and

be the number of grid points in the

direction for the cladding region.

We discretize the domain into equal segments and use central differences for the Laplacian operator. The result is:

.

However at

there is a singularity. Using L'Hopital's rule we have at

:

.

Thus the Laplacian at

(i.e.

) is given by

,

where

is a fictitious node. Then noting that at

,

(because of symmetry). A central difference approximation of the derivative

at

gives

, so on eliminating the fictitious node we have:

.

As a rule we use central differences for derivatives at the boundary.

Consider BCs in

direction:

For BC 1 we require that:

BC1:

BC2:

for

.

BC3:

for

.

Similar finite difference formulas can be derived for the other BCs. However, for the interface conditions we use first-order differences. This allows us to readily couple the two grid boundaries.