Chapter 5

Use the Rayleigh quotient to obtain a (reasonably accurate) upper bound for the lowest eigenvalue of (a) \(\frac{d^{2} \phi}{d x^{2}}+\left(\lambda-x^{2}\right) \phi=0 \quad\) with \(\frac{d \phi}{d x}(0)=0 \quad\) and \(\quad \phi(1)=0\) (b) \(\frac{d^{2} \phi}{d x^{2}}+(\lambda-x) \phi=0\) with \(\frac{d \phi}{d x}(0)=0\) and \(\frac{d \phi}{d x}(1)+2 \phi(1)=0\) *(c) \(\frac{d^{2} \phi}{d x^{2}}+\lambda \phi-0\) with \(\phi(0)-0\) and \(\frac{d \phi}{d x}(1)+\phi(1)-0\) (See Exercise 5.8.10.)

A Sturm-Liouville eigenvalue problem is called self-adjoint if $$ \left.p\left(u \frac{d v}{d x}-v \frac{d u}{d x}\right)\right|_{a} ^{b}=0 $$ (since then \(\int_{a}^{b}[u L(v)-v L(u)] d x=0\) ) for any two functions \(u\) and \(v\) satisfying the boundary conditions. Show that the following yield self-adjoint problems. (a) \(\phi(0)=0\) and \(\phi(L)=0\) (b) \(\frac{d \phi}{d x}(0)=0\) and \(\phi(L)=0\) (c) \(\frac{d \phi}{d x}(0)-h \phi(0)=0 \quad\) and \(\quad \frac{d \phi}{d x}(L)=0\) (d) \(\phi(a)=\phi(b) \quad\) and \(\quad p(a) \frac{d \phi}{d x}(a)=p(b) \frac{d \phi}{d x}(b)\) (e) \(\phi(a)=\phi(b) \quad\) and \(\quad \frac{d \phi}{d x}(a)=\frac{d \phi}{d x}(b)\) [self-adjoint only if \(p(a)=p(b)\) ] [in the situation in which \(p(0)=0\) ] (f) \(\begin{aligned} \phi(L)-0 & \text { and } \end{aligned} \quad\left[\begin{array}{l}\text { [in the situation in which } p(0)=0] \\\ \phi(0) \text { bounded and } \lim _{x \rightarrow 0} p(x) \frac{d \phi}{d x}=0\end{array}\right.\) *(g) Under what conditions is the following self-adjoint (if \(p\) is constant)? $$ \begin{aligned} \phi(L)+\alpha \phi(0)+\beta \frac{d \phi}{d x}(0) &=0 \\ \frac{d \phi}{d x}(L)+\gamma \phi(0)+\delta \frac{d \phi}{d x}(0) &=0 \end{aligned} $$

Consider $$ c \rho \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(K_{0} \frac{\partial u}{\partial x}\right)+\alpha u $$ where \(c, \rho, K_{0}, \alpha\) are functions of \(x\), subject to $$ \begin{aligned} &u(0, t)=0 \\ &u(L, t)=0 \\ &u(x, 0)=f(x) \end{aligned} $$ Assume that the appropriate eigenfunctions are known. (a) Show that the eigenvalues are positive if \(\alpha<0\) (see Sec. 5.2.1). (b) Solve the initial value problem. (c) Briefly discuss \(\lim _{t \rightarrow \infty} u(x, t)\).

Obtain a formula for an infinite series using Parseval's equality applied to the: (a) Fouricr sine scrics of \(f(x)-1\) on the interval \(0

Consider $$ \frac{d^{2} \phi}{d x^{2}}+\lambda(1+x) \phi=0 $$ subject to \(\phi(0)-0\) and \(\phi(1)-0\). Roughly sketch the eigenfunctions for \(\lambda\) laige. Take into account amplitude and period variations.

Consider the cigenvalue problem $$ \frac{d^{\prime} \phi}{d x^{2}}+\left(\lambda-x^{2}\right) \phi=0 $$ subject to \(\frac{d \phi}{d x}(0)=0\) and \(\frac{d \phi}{d x}(1)=0 .\) Show that \(\lambda>0\) (be sure to show that \(\lambda \neq 0\) ).

Prove that the eigenfunctions corresponding to different eigenvalues (of the following eigenvalue problem) are orthogonal: $$ \frac{d}{d x}\left[p(x) \frac{d \phi}{d x}\right]+q(x) \phi+\lambda \sigma(x) \phi=0 $$ with the boundary conditions $$ \begin{array}{r} \phi(1)=0 \\ -2 \frac{d \phi}{d x}(2)=0 \end{array} $$ What is the weighting function?

Consider $$ c \rho \frac{\partial u}{\partial t}-\frac{\partial}{\partial x}\left(K_{0} \frac{\partial u}{\partial x}\right) $$ where \(c, \rho, K_{0}\) are functions of \(x\), subject to $$ \begin{aligned} \frac{\partial u}{\partial x}(0, t) &=0 \\ \frac{\partial u}{\partial x}(L, t) &=0 \\ u(x, 0) &=f(x) \end{aligned} $$ Assume that the appropriate eigenfunctions are known. Solve the initial value problem, briefly discussing \(\lim _{t \rightarrow \infty} u(x, t)\).

consider $$ \rho \frac{\partial^{2} u}{\partial t^{2}}=T_{0} \frac{\partial^{2} u}{\partial x^{2}}+\alpha u+\beta \frac{\partial u}{\partial t} . $$ (a) Give a brief physical interpretation. What signs do \(\alpha\) and \(\beta\) have to be physical? (b) Allow \(\rho, \alpha, \beta\) to be functions of \(x\). Show that separation of variables works only if \(\beta=c \rho\), where \(c\) is a constant. (c) If \(\beta=c \rho\), show that the spatial equation is a Sturm-Liouville differential equation. Solve the time equation.

Consider any function \(f(x)\) defined for \(a \leqslant x \leqslant b\). Approximate this function by a constant. Show that the best such constant (in the mean-square sense, i.e., minimizing the mean-square deviation) is the constant equal to the average of \(f(x)\) over the interval \(a \leqslant x \leqslant b\).

Consider the eigenvalue problem $$ \frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0 $$ subject to \(\frac{d \phi}{d x}(0)=0\) and \(\frac{d \phi}{d x}(L)+h \phi(L)=0\) with \(h>0\). (a) Prove that \(\lambda>0\) (without solving the differential equation). *(b) Determine all eigenvalues graphically. Obtain upper and lower bounds. Estimate the large eigenvalues. (c) Show that the \(n\)th eigenfunction has \(n-1\) zeros in the interior.

Prove that \((5.6 .10)\) is valid in the following way. Assume \(L(u) / \sigma\) is piecewise smooth so that $$ \frac{L(u)}{\sigma}=\sum_{n=1}^{\infty} b_{n} \phi_{n}(x) . $$ Determine \(b_{n}\). [Hint: Using Green's formula \((5.5 .5)\), show that \(b_{n}=-a_{n} \lambda_{n}\) if \(u\) and \(d u / d x\) are continuous and if \(u\) satisfies the same homogeneous boundary conditions as the eigenfunctions \(\left.\phi_{n}(x) .\right]\)

Consider the eigenvalue problem \(L(\phi)=-\lambda \sigma(x) \phi\), subject to a given set of homogeneous boundary conditions. Suppose that $$ \int_{a}^{b}[u L(v)-v L(u)] d x=0 $$ for all functions \(u\) and \(v\) satisfying the same set of boundary conditions. Prove that eigenfunctions corresponding to different eigenvalues are orthogonal (with what weight?).

Solve $$ \frac{\partial u}{\partial t}=k \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right) $$ with \(u(r, 0)=f(r), u(0, t)\) bounded, and \(u(a, t)=0\). You may assume that the corresponding eigenfunctions, denoted \(\phi_{n}(r)\), are known and are complete. (Hint: See Sec. 5.2.2.)

Consider the non-Sturm-Liouville differential equation $$ \frac{d^{2} \phi}{d x^{2}}+\alpha(x) \frac{d \phi}{d x}+[\lambda \beta(x)+\gamma(x)] \phi=0 . $$ Multiply this equation by \(H(x)\). Determine \(H(x)\) such that the equation may be reduced to the standard Sturm-Liouville form: $$ \frac{d}{d x}\left[p(x) \frac{d \phi}{d x}\right]+[\lambda \sigma(x)+q(x)] \phi=0 . $$ Given \(\alpha(x), \beta(x)\), and \(\gamma(x)\), what are \(p(x), \sigma(x)\), and \(q(x)\) ?

(a) Using Parseval's equality, express the error in terms of the tail of a series. (b) Redo part (a) for a Fourier sine series on the interval \(0 \leqslant x \leqslant L\). (c) If \(f(x)\) is piecewise smooth, estimate the tail in part (b). (Hint: Use integration by parts.)

Give an example of an eigenvalue problem with more than one eigenfunction corresponding to an cigenvalue.

Consider the following boundary value problem: $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} \quad \text { with } \quad \frac{\partial u}{\partial x}(0, t)=0 \quad \text { and } \quad u(L, t)=0 \text {. } $$ Solve such that \(u(x, 0)=\sin \pi x / L\) (initial condition). (Hint: If necessary, use a table of integrals.)

Consider heat flow with convection (see Exercise 1.5.2): $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}-V_{0} \frac{\partial u}{\partial x} $$ (a) Show that the spatial ordinary differential equation obtained by separation of variables is not in Sturm-Liouville form. *(b) Solve the initial boundary value problem $$ \begin{aligned} &u(0, t)=0 \\ &u(L, t)=0 \\ &u(x, 0)=f(x) \end{aligned} $$ (c) Solve the initial boundary value problem $$ \begin{aligned} \frac{\partial u}{\partial x}(0, t) &=0 \\ \frac{\partial u}{\partial x}(L, t) &=0 \\ u(x, 0) &=f(x) \end{aligned} $$

Show that if $$ L(f)-\frac{d}{d x}\left(p \frac{d f}{d x}\right)+4 f $$ then $$ -\int_{a}^{b} f L(f) d x=-\left.p f \frac{d f}{d x}\right|_{a} ^{b}+\int_{a}^{b}\left[p\left(\frac{d f}{d x}\right)^{2}-q f^{2}\right] d x $$ if \(f\) and \(d f / d x\) are continuous.

Consider $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} $$ with \(\frac{\partial u}{\partial x}(0, t)=0, \frac{\partial u}{\partial x}(L, t)=-h u(L, t)\), and \(u(x, 0)=f(x)\). (a) Solve if \(h>0\). (b) Solve if \(h<0\).

Consider $$ L=\frac{d^{2}}{d x^{2}}+6 \frac{d}{d x}+9 $$ (a) Show that \(L\left(e^{r x}\right)=(r+3)^{2} e^{r \pi}\). (b) Use part (a) to obtain solutions of \(L(y)-0\) (a second order constant coefficient differential equation). (c) If \(z\) depends on \(x\) and a parameter \(r\), show that $$ \frac{\partial}{\partial r} L(z)=L\left(\frac{\partial z}{\partial r}\right) \text {. } $$ (d) Using part (c), evaluate \(L(\partial z / \partial r)\) if \(z-e^{r x}\). (e) Obtain a second solution of \(L(y)=0\), using part (d).

Consider $$ \rho \frac{\partial^{2} u}{\partial t^{2}}=T_{0} \frac{\partial^{2} u}{\partial x^{2}}+\alpha u, $$ where \(\rho(x)>0, \alpha(x)<0\), and \(T_{0}\) is constant, subject to $$ \begin{array}{rlrl} u(0, t) & =0 & u(x, 0) & =f(x) \\ u(L, t) & =0 & \frac{\partial u}{\partial t}(x, 0) & =g(x) \end{array} $$ Assume that the appropriate eigenfunctions are known. Solve the initial value problem.

For the Sturm-Liouville eigenvalue problem, \(\frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0 \quad\) with \(\quad \frac{d \phi}{d x}(0)=0 \quad\) and \(\quad \frac{d \phi}{d x}(L)=0\) verify the following general properties: (a) There are an infinite number of eigenvalues with a smallest but no largest. (b) The \(n\)th eigenfunction has \(n-1\) zeros. (c) The eigenfunctions are complete and orthogonal. (d) What does the Rayleigh quotient say concerning negative and zero eigenvalues?

Assuming that the operations of summation and integration can be interchanged, show that if $$ f=\sum \alpha_{n} \phi_{n} \quad \text { and } \quad g=\sum \beta_{n} \phi_{n} \text {, } $$ then for normalized eigenfunctions $$ \int_{a}^{b} f g \sigma d x=\sum_{n=1}^{\infty} \alpha_{n} \beta_{n} . $$ a generalization of Parseval's equality.

Consider (with \(h>0\) ) $$ \begin{array}{rlr} \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}} & \\ \frac{\partial u}{\partial x}(0, t)-h u(0, t)=0 & u(x, 0)=f(x) \\ \frac{\partial u}{\partial x}(L, t)=0 & \frac{\partial u}{\partial t}(x, 0)=g(x) . \end{array} $$ (a) Show that there are an infinite number of different frequencies of oscillation. (b) Estimate the large frequencies of oscillation. (c) Solve the initial value problem.

Prove that if \(x\) is a root of a sixth-order polynomial with real coefficients, then \(x\) is also a root.

Consider the vibrations of a nonuniform string of mass density \(\rho_{0}(x)\). Suppose that the left end at \(x=0\) is fixed and the right end obeys the elastic boundary condition: \(\partial u / \partial x=-\left(k / T_{0}\right) u\) at \(x=L\). Suppose that the string is initially at rest with a known initial position \(f(x)\). Solve this initial value problem. (Hints: Assume that the appropriate eigenvalues and corresponding eigenfunctions are known. What differential equations with what boundary conditions do they satisfy? The cigenfunctions arc orthogonal with what weighting function?)

Consider the eigenvalue problem $$ \frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0, \text { subject to } \phi(0)=0 \text { and } \phi(\pi)-2 \frac{d \phi}{d x}(0)=0 \text {. } $$ (a) Show that usually $$ \int_{0}^{\pi}\left(u \frac{d^{2} v}{d x^{2}}-n \frac{d^{2} u}{d x^{2}}\right) d x \neq 0 $$ for any two functions \(u\) and \(v\) satisfying these homogeneous boundary conditions. (b) Determine all positive eigenvalues. (c) Determine all negative eigenvalues. (d) Is \(\lambda=0\) an eigenvalue? (e) Is it possible that there are other eigenvalues besides those determined in parts (b) through (d)? Briefty explain.

For $$ L-\frac{d}{d x}\left(p \frac{d}{d x}\right)+q $$ with \(p\) and \(q\) real, carefully show that $$ \overline{L(\phi)}=L(\bar{\phi}) $$

Consider the boundary value problem $$ \frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0 \quad \text { with } $$ $$ \phi(1)+\frac{d \phi}{d x}(1)=0 . $$ (a) Using the Rayleigh quotient, show that \(\lambda \approx 0 .\) Why is \(\lambda>0\) ? (b) Prove that eigenfunctions corresponding to different eigenvalues arc (c) Show that $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}} $$ $$ u(0, t)-\frac{\partial u}{\partial x}(0, t)-0 $$ (b) Prove that eigenfunctions corresponding to different eigenvalues are orthogonal. *(c) Show that Determine the eigenvalues graphically. Estimate the large eigenvalues. (d) Solve with $$ \begin{aligned} u(0, t)-\frac{\partial u}{\partial x}(0, t) &=0 \\ u(1, t)+\frac{\partial u}{\partial x}(1, t) &=0 \\ u(x, 0) &=f(x) \end{aligned} $$ You may call the relevant eigenfunctions \(\phi_{n}(x)\) and assume that they are known.

Consider a fourth-order linear differential operator, $$ L=\frac{d^{4}}{d x^{4}} . $$ (a) Show that \(u L(v)-v L(u)\) is an exact differential. (b) Evaluate \(\int_{0}^{1}[u L(v)-v L(u)] d x\) in terms of the boundary data for any functions \(u\) and \(v\). (c) Show that \(\int_{0}^{1}[u I .(v)-v I .(u)] d x=0\) if \(u\) and \(v\) are any two functions satisfying the boundary conditions $$ \begin{aligned} \phi(0) &=0 & \phi(1) &=0 \\ \frac{d \phi}{d x}(0) &=0 & \frac{d^{2} \phi}{d x^{2}}(1) &=0 \end{aligned} $$ (d) Give another example of boundary conditions such that $$ \int_{0}^{1}[u L(v)-v L(u)] d x=0 \text {. } $$ (e) For the eigenvalue problem [using the boundary conditions in part (c)] $$ \frac{d^{4} \phi}{d x^{4}}+\lambda e^{\top} \phi=0 $$ show that the eigenfunctions corresponding to different eigenvalues are orthogonal. What is the weighting function?

Show that \(\lambda \geqslant 0\) for the eigenvalue problem \(\frac{d^{2} \phi}{d x^{2}}+\left(\lambda-x^{2}\right) \phi=0 \quad\) with \(\quad \frac{d \phi}{d x}(0)=0, \quad \frac{d \phi}{d x}(1)=0 .\) Is \(\lambda=0\) an eigenvalue?

For the eigenvalue problem $$ \frac{d^{4} \phi}{d x^{4}}+\lambda e^{x} \phi=0 $$ subject to the boundary conditions $$ \begin{aligned} \phi(0) &=0 & \phi(1) &=0 \\ \frac{d \phi}{d x}(0) &=0 & \frac{d^{2} \phi}{d x^{2}}(1) &=0 \end{aligned} $$ show that the eigenvalues are less than or equal to zero \((\lambda \leqslant 0)\). (Don't worry; in a physical context that is exactly what is expected.) Is \(\lambda=0\) an eigenvalue?

Consider the eigenvalue problem \(x^{2} \frac{d^{2} \phi}{d x^{2}}+x \frac{d \phi}{d x}+\lambda \phi=0\) with \(\phi(1)=0 \quad\) and \(\quad \phi(b)=0\) (a) Show that multiplying by \(1 / x\) puts this in the Sturm-Liouville form. (This multiplicative factor is derived in Exercise 5.3.3.) (b) Show that \(\lambda \geqslant 0\). *(c) Since (5.3.9) is an equidimensional equation, determine all positive eigenvalues. Is \(\lambda-0\) an eigenvalue? Show that there are an infinite number of eigenvalues with a smallest, but no largest. (d) The eigenfunctions are orthogonal with what weight according to SturmLiouville theory? Verify the orthogonality using properties of integrals. (e) Show that the \(n\)th eigenfunction has \(n-1\) zeros.

Consider the special case of the eigenvalue problem of Sec. 5.8: $$ \frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0 \text { with } \phi(0)=0 \text { and } \frac{d \phi}{d x}(1)+\phi(1)=0 \text {. } $$ *(a) Determine the lowest eigenvalue to at least two or three significant figures using tables or a calculator. "(b) Determine the lowest eigenvalue using a root finding algorithm (e.g., Newton's method) on a computer. (c) Compare either part (a) or (b) to the bound obtained using the Rayleigh quotient [see Exercise 5.6.1(c)].

Determine all negative eigenvalues for $$ \frac{d^{2} \phi}{d x^{2}}+5 \phi=-\lambda \phi \text { with } \phi(0)=0 \text { and } \phi(\pi)=0 \text {. } $$

Consider \(\partial^{2} u / \partial t^{2}=c^{2} \partial^{2} u / \partial x^{2}\) with the boundary conditions $$ \begin{aligned} u &=0 & \text { at } x &=0 \\ n \frac{\partial^{2} u}{\partial t^{2}} &=-T_{0} \frac{\partial u}{\partial x}-k u & \text { at } x &=L . \end{aligned} $$ (a) Give a brief physical interpretation of the boundary conditions. (b) Show how to determine the frequencies of oscillation. Estimate the large frequencies of oscillation. (c) Without attempting to use the Rayleigh quotient, explicitly determine if there are any separated solutions that do not oscillate in time. (Hint: There are none.) (d) Show that the boundary condition is not self-adjoint; that is, show $$ \int_{0}^{L}\left(u_{n} \frac{d^{2} u_{m}}{d x^{2}}-u_{m} \frac{d^{2} u_{n}}{d x^{2}}\right) d x \neq 0 $$ even when \(u_{n}\) and \(u_{m}\) are eigenfunctions corresponding to different eigenvalues.

Simplify \(\int_{0}^{L} \sin ^{2} \sqrt{\lambda} x d x\) when \(\lambda\) is given hy (5.8.12).