Question 9.CS.5: Linear algebraic equations can arise when modeling distribut...
Linear algebraic equations can arise when modeling distributed systems. For example, Fig. 9.7 shows a long, thin rod positioned between two walls that are held at constant temperatures. Heat flows through the rod as well as between the rod and the surrounding air. For the steady-state case, a differential equation based on heat conservation can be written for such a system as
\frac{d^2 T}{d x^2}+h^{\prime}\left(T_a-T\right)=0 (9.24)
where T = temperature (°C), x = distance along the rod (m), h′ = a heat transfer coefficient between the rod and the surrounding air (m−2), and T_a = the air temperature (°C).
Given values for the parameters, forcing functions, and boundary conditions, calculus can be used to develop an analytical solution. For example, if h′ = 0.01, T_a = 20, T(0) = 40, and T(10) = 200, the solution is
T=73.4523 e^{a .1 x}-53.4523 e^{-0.1 x}+20 (9.25)
Although it provided a solution here, calculus does not work for all such problems. In such instances, numerical methods provide a valuable alternative. In this case study, we will use finite differences to transform this differential equation into a tridiagonal system of linear algebraic equations which can be readily solved using the numerical methods described in this chapter.
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