\frac{d^{2}x}{dt^{2}} = −\frac{C}{x^{2}}

\begin{aligned}\frac{dv}{dt} & = −\frac{K}{x^{2}} \\ \frac{dx}{dt} & = v \end{aligned}

flow in from left = f(x_{i}) dy dz flow out to right = f(x_{i+1}) dy dz

f(x_{i+1}) dy dz = f(x_{i}) dy dz Δx ≡ x_{i+1} − x_{i}

f(x + Δx) = f(x)

\frac{df}{dx} = 0

f(x) = C (some constant value)

\frac{∂q}{∂x} = 0

q = −k \frac{∂T}{∂x}

\frac{∂^{2}T}{∂x^{2}} = 0

\frac{∂^{2}T}{∂x^{2}} = S(x)

forward difference D_{i}^{+} = \frac{T_{i+1} − T_{i}}{x_{i+1} − x_{i}}

backward difference D_{i}^{−} = \frac{T_{i} − T_{i−1}}{x_{i} − x_{i−1}}

centered second difference D_{i}^{2} = \frac{D_{i}^{+} − D_{i}^{−}}{\frac{1}{2}(x_{i+1} − x_{i−1})}

q = −kρC \frac{∂T}{∂x}

− \frac{∂q}{∂x} Δt = ρC ΔT

k \frac{∂^{2}T}{∂x^{2}} = \frac{∂T}{∂t}

T_{i}^{new} = T_{i}^{old} + k (\frac{∂^{2}T}{∂x^{2}} )_{i} dt