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Chapter 10: Problem 1
\(f(x)=x^{2}, \quad 0
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Let \(\left\\{f_{n}(x)\right\\}\) be an orthogonal set of functions on the interval \([a, b]\) with respect to the weight function \(w(x)\) . Show that they satisfy the Pythagorean property $$\left\|f_{m}+f_{n}\right\|^{2}=\left\|f_{m}\right\|^{2}+\left\|f_{n}\right\|^{2}$$ if $$m \neq n$$
Find a solution to the mixed boundary value problem \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0, \quad 1< r <3, \quad-\pi \leq \theta \leq \pi,\) \(u(1, \theta)=f(\theta), \quad-\pi \leq \theta \leq \pi,\) \(\frac{\partial u}{\partial r}(3, \theta)=g(\theta), \quad-\pi \leq \theta \leq \pi.\)
Chemical Diusion. Chemical diusion through a thin layer is governed by the equation $$\frac{\partial C}{\partial t}=k \frac{\partial^{2} C}{\partial x^{2}}-L C$$ where \(C(x, t)\) is the concentration in moles/cm \(^{3},\) the diffusivity \(k\) is a positive constant with units \(\mathrm{cm}^{2} / \mathrm{sec}\) and \(L>0\) is a consumption rate with units \(\sec ^{-1}\). Assume the boundary conditions are $$C(0, t)=C(a, t)=0, \quad t>0$$ and the initial concentration is given by $$C(x, 0)=f(x), 0< x < a$$ Use the method of separation of variables to solve formally for the concentration \(C(x, t)\). What happens to the concentration as \(t \rightarrow+\infty ?\)
\(f(x)=\pi-x, \quad 0
$$ f(x)=x^{2}, \quad g(x)=0 $$
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