Equilibrium Heat Distribution
In the next chapter, we will see the Ascoli-Arzela and Schauder theories used once again, to demonstrate the existence of solutions to a type of problem in the theory of ordinary differential equations that is quite different from what we encountered in studying the forced pendulum. The purpose of the present chapter is to present an illustration of how problems like those discussed in the next chapter come up. Although a single application is hardly sufficient to justify the entire theoretical apparatus, I hope that you will be willing to take my word for it that there are more where this one came from. Furthermore, by following the steps from the mathematical model of the physical problem to the abstract mathematics, you will be able to see why the differential equation problem has the particular form that it does. If I didn’t show you something of the sort, you would have every right to suspect that the somewhat complicated hypotheses of the mathematical problem were constructed just so that the abstract theory we’ve been developing could be applied to it.
KeywordsOrdinary Differential Equation Heating Process Abstract Theory Temperature Function Previous Chapter
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