- The $(2)$'nd case ($A \ne 0 \:,\: B \ne 0 \:,\: C = 0$) corresponds with the
governing equation:
$$
A \frac{d^2T}{dx^2} + B \frac{dT}{dx} = 0 \slechts
- \frac{d^2T}{dx^2} + P \frac{dT}{dx} = 0
$$
The differential equation for Convection and Diffusion is recognized.
From $(P^2 - Q^2 = C = 0)$ it follows that: $|P| = |Q|$. But then, from "The
Hyperbolic Connection", we know:
$$
a + b = 1 \slechts | P | = | Q |
$$
The case ($a + b = 1$) has been covered at length in "Some Stable Solutions"
and a solution of the ODE for Convection and Diffusion has been found there
also.
- The $(3)$'rd case is $(A \ne 0 \: , \: B = 0 \: , \: C \ne 0)$. The governing equation is: $$ A \frac{d^2T}{dx^2} + C.T = 0 \slechts \frac{d^2T}{dx^2} - \kwart Q^2 T = 0 $$ Meaning that $P = 0$, hence $a = b$. And the discriminant is $Q^2 = - C/A$. A safe solution of the tri-diagonal system can only exists if the latter is a positive real number. Again, oscillatory and vibrating solutions cannot be (safely) obtained with a tri-diagonal system of linear equations.
- The $(4)$'th case is $(A \ne 0 \: , \: B = 0 \: , \: C = 0)$. The governing equation is: $$ \frac{d^2T}{dx^2} = 0 $$ Meaning that $P = 0$, hence $a = b$. And $P^2 = Q^2$, hence $Q = 0$. Therefore $a.b = 1/4$. This is only possible if: $a = b = 1/2$. The tri-diagonal system corresponds with the Finite Difference scheme for Pure 1-D Diffusion: $$ - \half T_{i-1} + T_i - \half T_{i+1} = 0 $$ It is recognized that this is equivalent with one of the special cases, as noted in "Persistent Schemes": \begin{eqnarray*} a = b = \half & \quad \mbox{Symmetric matrix} \end{eqnarray*} The solution of both the governing equation and the tri-diagonal system is a straight line between the boundaries, as has been established in the preamble of "Some Stable Solutions".
- The $(5)$'th case is $(A = 0 \: , \: B \ne 0 \: , \: C \ne 0)$. The governing
equation is:
$$
B \frac{dT}{dx} + C.T = 0
$$
It was proved in "L.U. Decomposition" that the governing equations of the Upper
and Lower matrices are of the same type, namely:
$$
\frac{dT}{dx} - \gamma . T = 0
$$
Details are handled at length in the abovementioned chapter.
Here, of course: $\gamma = - C/B$.
- The $(6)$'th case is $(A = 0 \: , \: B \ne 0 \: , \: C = 0)$. The governing equation is: $$ \frac{dT}{dx} = 0 $$ This case is recognized as a further specialization of $(5)$, though it was already mentioned as such in "Persistent Schemes": \begin{eqnarray*} a = 0 , b = 1 & \quad \mbox{Lower diagonal matrix} \\ a = 1 , b = 0 & \quad \mbox{Upper diagonal matrix} \end{eqnarray*} - The $(7)$'th case is $(A = 0 \: , \: B = 0 \: , \: C \ne 0)$. The governing equation is: $$ T = 0 $$ Equivalent again with one of the special cases in "Persistent Schemes": \begin{eqnarray*} a = b = 0 & \quad \mbox{Identity matrix} \end{eqnarray*} - The $(8)$'h case is $(A = 0 \:,\: B = 0 \:,\: C = 0)$. The ultimate degenerate case. There is no governing equation and no tri-diagonal system of equations at all.