Question

12.38 Linear algebraic equations can arise in the solution of differential equations. For example, the following differential equa- tion derives from a heat balance for a long, thin rod (Fig. P12.38): $\frac{d^2T}{dx^2} + h'(T_a - T) = 0$ (P12.38.1) where $T_i$ designates the temperature at node $i$. This approximation can be substituted into Eq. (P12.38.1) to give $-T_{i-1} + (2 + h' \Delta x^2)T_i - T_{i+1} = h' \Delta x^2 T_a$ This equation can be written for each of the interior nodes of the rod resulting in a tridiagonal system of equations. The first and last nodes at the rod's ends are fixed by boundary conditions. (a) Develop an analytical solution for Eq. (P12.38.1) for a 10-m rod with $T_a = 20$, $T(x = 0) = 40$, $T(x = 10) = 200$, and $h' = 0.02$. (b) Develop a numerical solution for the same parameter values employed in (a) using a finite-difference solution with four in- terior nodes as shown in Fig. P12.38 ($\Delta x = 2$ m).

          12.38 Linear algebraic equations can arise in the solution of
differential equations. For example, the following differential equa-
tion derives from a heat balance for a long, thin rod (Fig. P12.38):
$\frac{d^2T}{dx^2} + h'(T_a - T) = 0$
(P12.38.1)
where $T_i$ designates the temperature at node $i$. This approximation
can be substituted into Eq. (P12.38.1) to give
$-T_{i-1} + (2 + h' \Delta x^2)T_i - T_{i+1} = h' \Delta x^2 T_a$
This equation can be written for each of the interior nodes of the
rod resulting in a tridiagonal system of equations. The first and last
nodes at the rod's ends are fixed by boundary conditions.
(a) Develop an analytical solution for Eq. (P12.38.1) for a
10-m rod with $T_a = 20$, $T(x = 0) = 40$, $T(x = 10) = 200$,
and $h' = 0.02$.
(b) Develop a numerical solution for the same parameter values
employed in (a) using a finite-difference solution with four in-
terior nodes as shown in Fig. P12.38 ($\Delta x = 2$ m).

12.38 Linear algebraic equations can arise in the solution of
differential equations. For example, the following differential equa-
tion derives from a heat balance for a long, thin rod (Fig. P12.38):
(d^2T)/(dx^2) + h'(Ta - T) = 0
(P12.38.1)
where Ti designates the temperature at node i. This approximation
can be substituted into Eq. (P12.38.1) to give
-Ti-1 + (2 + h' Δ x^2)Ti - Ti+1 = h' Δ x^2 Ta
This equation can be written for each of the interior nodes of the
rod resulting in a tridiagonal system of equations. The first and last
nodes at the rod's ends are fixed by boundary conditions.
(a) Develop an analytical solution for Eq. (P12.38.1) for a
10-m rod with Ta = 20, T(x = 0) = 40, T(x = 10) = 200,
and h' = 0.02.
(b) Develop a numerical solution for the same parameter values
employed in (a) using a finite-difference solution with four in-
terior nodes as shown in Fig. P12.38 (Δ x = 2 m).

Added by Jose Carlos L.

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