Chapter 2

Solve the heat equation \(\partial u / \partial t=k \partial^{2} u / \partial x^{2}, 00\), subject to $$ \begin{array}{ll} \frac{\partial u}{\partial x}(0, t)=0 & t>0 \\ \frac{\partial u}{\partial x}(L, t)=0 & t>0 . \end{array} $$ (a) \(u(x, 0)= \begin{cases}0 & xL / 2\end{cases}\) (b) \(u(x, 0)=6+4 \cos \frac{3 \pi x}{L}\) (c) \(u(x, 0)=-2 \sin \frac{\pi x}{L}\) (d) \(u(x, 0)=-3 \cos \frac{8 \pi x}{L}\)

For the following partial differential equations, what ordinary differential equations are implied by the method of separation of variables? *(a) \(\frac{\partial u}{\partial t}=\frac{k}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)\) (b) \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}-v_{0} \frac{\partial u}{\partial x}\) *(c) \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\) (d) \(\frac{\partial u}{\partial t}=\frac{k}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial u}{\partial r}\right)\) *(e) \(\frac{\partial u}{\partial t}=k \frac{\partial^{4} u}{\partial x^{4}}\) *(f) \(\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}\)

Show that any linear combination of linear operators is a linear operator.

Consider \(u(x, y)\) satisfying Laplace's equation inside a rectangle \((0

Solve $$ \begin{aligned} \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} \quad \text { with } \quad \frac{\partial u}{\partial x}(0, t) &=0 \\ u(L, t) &=0 \\ u(x, 0) &=f(x) \end{aligned} $$ For this problem you may assume that no solutions of the heat equation exponentially grow in time. You may also guess appropriate orthogonality conditions for the eigenfunctions.

Consider the differential equation $$ \frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0 $$ Determine the eigenvalues \(\lambda\) (and corresponding eigenfunctions) if \(\phi\) satisfies the following boundary conditions. Analyze three cases \((\lambda>0, \lambda=0, \lambda<0)\). You may assume that the eigenvalues are real. (a) \(\phi(0)=0 \quad\) and \(\phi(\pi)=0\) *(b) \(\phi(0)=0 \quad\) and \(\phi(1)=0\) (c) \(\frac{d \phi}{d x}(0)=0\) and \(\frac{d \phi}{d x}(L)=0 \quad\) (If necessary, see Sec. 2.4.1.) *(d) \(\phi(0)=0 \quad\) and \(\frac{d \phi}{d x}(L)=0\) (e) \(\frac{d \phi}{d x}(0)=0\) and \(\phi(L)=0\) *(f) \(\phi(a)=0\) and \(\phi(b)=0\) (You may assume that \(\lambda>0\).) (g) \(\phi(0)=0 \quad\) and \(\frac{d \phi}{d x}(L)+\phi(L)=0 \quad\) (If necessary, see Sec. 5.8.)

(a) Show that \(L(u)=\frac{\partial}{\partial x}\left[K_{0}(x) \frac{\partial u}{\partial x}\right]\) is a linear operator. (b) Show that usually \(L(u)=\frac{\partial}{\partial x}\left[K_{0}(x, u) \frac{\partial u}{\partial x}\right]\) is not a linear operator.

Solve Laplace's equation outside a circular disk \((r \geqslant a)\) subject to the boundary condition: (a) \(u(a, \theta)=\ln 2+4 \cos 3 \theta\) (b) \(u(a, \theta)=f(\theta)\) You may assume that \(u(r, \theta)\) remains finite as \(r \rightarrow \infty\).

Solve the eigenvalue problem $$ \frac{d^{2} \phi}{d x^{2}}=-\lambda \phi $$ subject to $$ \phi(0)=\phi(2 \pi) \quad \text { and } \quad \frac{d \phi}{d x}(0)=\frac{d \phi}{d x}(2 \pi) $$

Consider the heat equation $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} $$ subject to the boundary conditions $$ \begin{aligned} &u(0, t)=0 \\ &u(L, t)=0 \end{aligned} $$ Solve the initial value problem if the temperature is initially (a) \(u(x, 0)=6 \sin \frac{9 \pi x}{L}\) (b) \(u(x, 0)=3 \sin \frac{\pi x}{L}-\sin \frac{3 \pi x}{L}\) *(c) \(u(x, 0)=2 \cos \frac{3 \pi x}{L}\) (d) \(u(x, 0)= \begin{cases}1 & 0

Show that \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}+Q(u, x, t)\) is linear if \(Q=\alpha(x, t) u+\beta(x, t)\) and in addition homogeneous if \(\beta(x, t)=0\).

For Laplace's equation inside a circular disk \((r \leqslant a)\), using \((2.5 .28 a)\) and \((2.5 .29)\), show that $$ u(r, \theta)=\frac{1}{\pi} \int_{-\pi}^{\pi} f(\bar{\theta})\left[-\frac{1}{2}+\sum_{n=0}^{\infty}\left(\frac{r}{a}\right)^{n} \cos n(\theta-\bar{\theta})\right] d \bar{\theta} $$ Using \(\cos z=\operatorname{Re}\left[e^{i z}\right]\), sum the resulting geometric series to obtain Poisson's integral formula.

Explicitly show there are no negative eigenvalues for \(\frac{d^{2} \phi}{d x^{2}}=-\lambda \phi \quad\) subject to \(\quad \frac{d \phi}{d x}(0)=0 \quad\) and \(\quad \frac{d \phi}{d x}(L)=0\).

Consider $$ \frac{\partial u}{\partial t}-k \frac{\partial^{2} u}{\partial x^{2}} $$ subject to \(u(0, t)=0, u(L, t)=0\), and \(u(x, 0)=f(x)\).

In this exercise we derive superposition principles for nonhomogeneous problems. (a) Consider \(L(u)=f\). If \(u_{p}\) is a particular solution, \(L\left(u_{p}\right)=f\), and if \(u_{1}\) and \(u_{2}\) are homogencous solutions, \(L\left(u_{i}\right)=0\), show that \(u=u_{p}+c_{1} u_{1}+c_{2} u_{2}\) is another particular solution. (b) If \(L(u)=f_{1}+f_{2}\), where \(u_{p}\), is a particular solution corresponding to \(f_{i}\), what is a particular solution for \(u\) ?

Solve Laplace's equation inside the quarter-circle of radius \(1(0 \leqslant \theta \leqslant \pi / 2\), \(0 \leqslant r \leqslant 1\) ) subject to the boundary conditions: *(a) \(\frac{\partial u}{\partial \theta}(r, 0)=0, \quad u(r, \pi / 2)=0, \quad u(1, \theta)=f(\theta)\) (b) \(\frac{\partial u}{\partial \theta}(r, 0)=0, \quad \frac{\partial u}{\partial \theta}(r, \pi / 2)=0, \quad u(1, \theta)=f(\theta)\) *(c) \(u(r, 0)=0, \quad u(r, \pi / 2)=0, \quad \frac{\partial u}{\partial r}(1, \theta)=f(\theta)\) (d) \(\frac{\partial u}{\partial \theta}(r, 0)=0, \quad \frac{\partial u}{\partial \theta}(r, \pi / 2)=0, \quad \frac{\partial u}{\partial r}(1, \theta)-g(\theta)\) Show that the solution [part (d)] exists only if \(\int_{0}^{\pi / 2} g(\theta) d \theta=0\). Explain this condition physically.

This problem presents an alternative derivation of the heat equation for a thin wire. The equation for a circular wire of finite thickness is the two- dimensional heat equation (in polar coordinates). Show that this reduces to \((2.4 .19)\) if the temperature does not depend on \(r\) and if the wire is very thin.

Evaluate (be careful if \(n=m\) ) $$ \int_{0}^{L} \sin \frac{n \pi x}{L} \sin \frac{m \pi x}{L} d x \quad \text { for } n>0, m>0 . $$ Use the trigonometric identity $$ \sin a \sin b=\frac{1}{2}[\cos (a-b)-\cos (a+b)] . $$

If \(L\) is a linear operator, show that \(L\left(\sum_{n=1}^{M} c_{n} u_{n}\right)=\sum_{n=1}^{M} c_{n} L\left(u_{n}\right) .\) Use this result to show that the principle of superposition may be extended to any finite number of homogeneous solutions.

Solve Laplace's equation inside a semicircle of radius \(a(0

Evaluate $$ \int_{0}^{L} \cos \frac{n \pi x}{L} \cos \frac{m \pi x}{L} d x \quad \text { for } n \geqslant 0, m \geqslant 0 \text {. } $$ Use the trigonometric identity $$ \cos a \cos b=\frac{1}{2}[\cos (a+b)+\cos (a-b)] . $$ (Be careful if \(a-b=0\) or \(a+b=0 .\) )

Solve I aplace's equation inside a \(60^{\circ}\) wedge of radius \(a\) subject to the boundary conditions: (a) \(u(r, 0)=0, \quad u\left(r, \frac{\pi}{3}\right)=0, \quad u(a, \theta)=f(\theta)\) "(b) \(\frac{\partial u}{\partial \theta}(r, 0)=0, \quad \frac{\partial u}{\partial \theta}\left(r, \frac{\pi}{3}\right)=0, \quad u(a, \theta)=f(\theta)\)

Solve Laplace's equation inside a circle of radius \(a\), $$ \nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0, $$ subject to the boundary condition $$ u(a, \theta)=f(\theta) . $$

Solve Laplace's equation inside a circular annulus \((a

Consider $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}-\alpha u . $$ This corresponds to a one-dimensional rod either with heat loss through the lateral sides with outside temperature \(0^{\circ}(\alpha>0\), see Exercise 1.2.4) or with insulated lateral sides with a heat source proportional to the temperature. Suppose that the boundary conditions are $$ u(0, t)=0 \quad \text { and } \quad u(L, t)=0 \text {. } $$ (a) What are the possible equilibrium temperature distributions if \(\alpha>0\) ? (b) Solve the time-dependent problem \([u(x, 0)=f(x)]\) if \(\alpha>0\). Analyze the temperature for large time \((t \rightarrow \infty)\) and compare to part (a).

Solve Laplace's equation inside a \(90^{\circ}\) sector of a circular annulus \((a

Using the muximum principles for Laplace's equation, prove that the solution of Poisson's equation, \(\nabla^{2} u=g(\mathbf{x})\), subject to \(u=f(\mathbf{x})\) on the boundary, is unique.

Solve Laplace's equation inside a rectangle: $$ \nabla^{2} u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$ subject to the boundary conditions $$ \begin{aligned} u(0, y) &=g(y) & & u(x, 0)=0 \\ u(L, y) &=0 & & u(x, H)=0 \end{aligned} $$

(a) Using the divergence theorem, determine an alternative expression for \(\iiint u \nabla^{2} u d x d y d z\). b) Using part (a), prove that the solution of Laplace's equation \(\nabla^{2} u=0\) (with \(u\) given on the boundary) is unique. (c) Modify part (b) if \(\nabla u \bullet \hat{\mathbf{n}}=0\) on the boundary. d) Modify part (b) if \(\nabla u \cdot \hat{\mathbf{n}}+h u=0\) on the boundary. Show that Newton's law of cooling corresponds to \(h<0\).

Prove that the temperature satisfying L.aplace's equation cannot attain its minimum in the interior.

Show that the "backwards" heat equation $$ \frac{\partial u}{\partial t}=-k \frac{\partial^{2} u}{\partial x^{2}} $$ suhject to \(u(0, t)=u(L, t)=0\) and \(u(x, 0)=f(x)\), is not well posed. [Hint: Show that if the data are changed an arbitrarily small amount, for example $$ f(x) \longrightarrow f(x)+\frac{1}{n} \sin \frac{n \pi x}{L} $$ for large \(n\), then the solution \(u(x, t)\) changes by a large amount.]

Solve Laplace's equation inside a semi-infinite strip \((0

Consider Laplace's equation inside a rectangle \(0 \leqslant x \leqslant L, 0 \leqslant y \leqslant H\), with the boundary conditions $$ \frac{\partial u}{\partial x}(0, y)=0, \quad \frac{\partial u}{\partial x}(L, y)=g(y), \quad \frac{\partial u}{\partial y}(x, 0)=0, \quad \frac{\partial u}{\partial y}(x, H)=f(x) . $$ (a) What is the solvability condition and its physical interpretation? (b) Show that \(u(x, y)=A\left(x^{2}-y^{2}\right)\) is a solution if \(f(x)\) and \(g(y)\) are constants [under the conditions of part (a)]. (c) Under the conditions of part (a), solve the general case [nonconstant \(f(x)\) and \(g(y)\) ]. [Hints: Use part (b) and the fact that \(f(x)=f_{a v}+\left[f(x)-f_{a v}\right]\), where \(\left.f_{a v}=\frac{1}{L} \int_{0}^{L} f(x) d x .\right]\)