\magnification=\magstep2

$$
{ {d^2 x}\over{dt^2} } \quad = \quad - {{C}\over{x^2}}
$$

$$
\eqalign{
 { {dv}\over{dt} } \quad &= \quad - {{ K}\over{x^2}}  \cr
 { {dx}\over{dt} } \quad &= \quad  {v} \cr
}
$$

$$
{\rm flow\ in\ from\ left\ } \quad = \quad
          f(x_{i}) \ dy \ dz 
$$

$$
{\rm flow\ out\ to\ right\ } \quad = \quad
          f(x_{i+1}) \ dy \  dz 
$$

$$
 f(x_{i+1})  \ dy \ dz \quad = \quad f(x_i)  \ dy \  dz 
$$

$$
\Delta x \quad \equiv \quad x_{i+1} - x_i 
$$

$$
f(x + \Delta x) \quad = \quad f(x)
$$

$$
{ {df}\over{dx} } \quad = \quad 0
$$

$$
f(x) \quad = \quad C \qquad ({\rm some\ constant\ value})
$$

$$
{ {\partial q }\over{\partial x} } \quad = \quad 0
$$

$$
{q} \quad = \quad - k {{\partial T} \over{\partial x}}
$$

$$
{{\partial^2 T}\over{\partial x^2}} \quad = \quad 0
$$

$$
{{\partial^2 T}\over{\partial x^2}} \quad = \quad S(x)
$$

$$
{\rm forward\ difference \ } D^+_i \quad = \quad  
       {{T_{i+1} - T_{i}} \over {x_{i+1} - x_{i} } }
$$

$$
{\rm backward\ difference \ } D^-_i \quad = \quad  
       {{T_{i} - T_{i-1}} \over {x_{i} - x_{i-1} } }
$$

$$
{\rm centered\ second\ difference \ } D^2_{i} \quad = \quad  
       {{D^+_{i} - D^-_{i}} \over {{{1}\over{2}} (x_{i+1} - x_{i-1}) } }
$$


$$
q \quad = \quad -k \rho C \thinspace {{\partial T}\over{\partial x}}
$$

$$
- \thinspace {{\partial q}\over{\partial{x}}}  \ \Delta t \quad = 
       \quad \rho C \thinspace \Delta T
$$


$$
k \thinspace {{\partial^2 T}\over{\partial x^2}} \quad = \quad
              {{\partial T}\over{\partial t}}
$$

$$
T_i^{\rm new} \quad = \quad T_i^{\rm old} \ + \ 
        k \thinspace \left( {{\partial^2 T}\over{\partial x^2}} \right)_i \thinspace dt 
$$

\bye



