Chapter 8

Consider $$ L(u)=f(x) \quad \text { with } \quad L=\frac{d}{d x}\left(p \frac{d}{d x}\right)+q $$ subject to two homogeneous boundary conditions. All homogeneous solutions \(\phi_{h}\) (if they exist) satisfy \(L\left(\phi_{h}\right)=0\) and the same two homogeneous boundary conditions. Apply Green's formula to prove that there are no solutions \(u\) if \(f(x)\) is not orthogonal (weight 1) to all \(\phi_{\mu}(x)\).

Consider $$ \begin{aligned} \frac{\partial u}{\partial t} &=k \frac{\partial^{2} u}{\partial x^{2}}+Q(x, t) \\ u(x, 0) &=g(x) . \end{aligned} $$ In all cases obtain formulas similar to \((8.2 .12)\) by introducing a Green's function. (a) Use Green's formula instead of term-by-term spatial differentiation if (b) Modify part (a) if \(u(0, t)=0\) and \(u(L, t)=0\). $$ u(0, t)=A(t) \quad \text { and } \quad u(L, t)=B(t) $$ Do not reduce to a problem with homogeneous boundary conditions. (c) Solve using any method if $$ \frac{\partial u}{\partial x}(0, t)=0 \quad \text { and } \quad \frac{\partial u}{\partial x}(L, t)=0 \text {. } $$ *(d) Use Green's formula instead of term-by-term differentiation if \(\frac{\partial u}{\partial x}(0, t)=A(t) \quad\) and \(\quad \frac{\partial u}{\partial x}(L, t)=B(t)\)

(a) Solve $$ \nabla^{2} u=f(x, y) $$ on a rectangle \((0

Solve by the method of eigenfunction expansion $$ c \rho \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(K_{0} \frac{\partial u}{\partial x}\right)+Q(x, t) $$ subject to \(u(0, t)=0, u(L, t)=0\), and \(u(x, 0)=g(x)\), if \(c \rho\) and \(K_{0}\) are functions of \(x\). Assume that the eigenfunctions are known. Obtain a formula similar to (8.2.12) by introducing a Green's function.

Using the method of (multidimensional) eigenfunction expansion, determine \(G\left(\mathbf{x}, \mathbf{x}_{0}\right)\) if $$ \nabla^{2} G=\delta\left(\mathbf{x}-\mathbf{x}_{0}\right) $$ and (a) on the rectangle \((0

Without determining \(u(x)\) how many solutions are there of $$ \frac{d^{2} u}{d x^{2}}+\gamma u=\sin x $$ (a) If \(\gamma=1\) and \(u(0)=u(\pi)=0\) ? "(b) If \(\gamma=1 \quad\) and \(\quad \frac{d u}{d x}(0)=\frac{d u}{d x}(\pi)=0\) ? (c) If \(\gamma=-1 \quad\) and \(\quad u(0)=u(\pi)=0\) ? (d) If \(\gamma=2 \quad\) and \(\quad u(0)=u(\pi)=0\) ?

Consider $$ \begin{aligned} \frac{\partial u}{\partial t} &=k \frac{\partial^{2} u}{\partial x^{2}}+Q(x, t) \\ t, t) &=0, \text { and } u(x, 0)=g(x) \end{aligned} $$ subject to \(u(0, t)=0, \frac{\partial u}{\partial x}(L, t)=0\), and \(u(x, 0)=g(x) .\) (a) Solve by the method of eigenfunction expansion. (b) Determine the Green's function for this time-dependent problem. (c) If \(Q(x, t)=Q(x)\), take the limit as \(t \rightarrow \infty\) of part (b) in order to determine the Green's function for $$ \frac{d^{2} u}{d x^{2}}=f(x) \quad \text { with } \quad u(0)=0 \quad \text { and } \quad \frac{d u}{d x}(L)=0 \text {. } $$

Consider in some three-dimensional region $$ \nabla^{2} u=f $$ with \(u=h(\mathbf{x})\) on the boundary. Represent \(u(\mathbf{x})\) in terms of the Green's function (assumed to be known).

Consider the nonlinearly perturbed eigenvalue problem: $$ \frac{d^{2} \phi}{d x^{2}}+\lambda \phi=\varepsilon \phi^{3} $$ with \(\phi(0)=0\) and \(\phi(L)=0\). Determine the perturbation of the eigenvalue \(\lambda_{1}\). Since the problem is nonlinear, the amplitude is important. Assume \(\int_{0}^{L} \phi^{2} d x=\) \(a^{2}\). Sketch \(a\) as a function of \(\lambda\).

Consider inside a circle of radius \(a\) $$ \nabla^{2} u=f $$ with $$ \begin{aligned} u(a, \theta) &=h_{1}(\theta) & & \text { for } 0<\theta<\pi \\ \frac{\partial u}{\partial r}(a, \theta) &=h_{2}(\theta) & & \text { for }-\pi<\theta<0 \end{aligned} $$ Represent \(u(r, \theta)\) in terms of the Green's function (assumed to be known).

Are there any values of \(\beta\) for which there are solutions of $$ \begin{gathered} \frac{d^{2} u}{d x^{2}}+u=\beta+x \\ u(-\pi)=u(\pi) \quad \text { and } \quad \frac{d u}{d x}(-\pi)=\frac{d u}{d x}(\pi) ? \end{gathered} $$

Consider a vibrating string with approximately uniform tension \(T\) and mass density \(\rho_{0}+\varepsilon \rho_{1}(x)\) subject to fixed boundary conditions. Determine the changes in the natural frequencies induced by the mass variation.

Consider \(\nabla^{2} u=f(x)\) in two dimensions, satisfying homngenenus boundary conditions. Suppose that \(\phi_{h}\) is a homogeneous solution, $$ \nabla^{2} \phi_{h}=0 $$ satisfying the same homogeneous boundary conditions. There may be more than one function \(\phi_{\hat{a}}\). (a) Show that there are no solutions \(u(\mathbf{x})\) if \(\iint f(\mathbf{x}) \phi_{h}(\mathbf{x}) d A \neq 0\) for any \(\phi_{h}(\mathbf{x})\). (b) Show that there are an infinite number of solutions if \(\iint f(\mathbf{x}) \phi_{h}(\mathbf{x}) d A=0\).

Consider $$ \frac{d^{2} u}{d x^{2}}+u=1 $$ (a) Find the general solution of this differential equation. Determine all solutions with \(u(0)=u(\pi)=0\). Is the Fredholm alternative consistent with your result? (b) Redo part (a) if \(\frac{d u}{d x}(0)=\frac{d u}{d x}(\pi)=0\). (c) Redo part (a) if \(\frac{d u}{d x}(-\pi)=\frac{d u}{d x}(\pi)\) and \(u(-\pi)=u(\pi)\).

Consider a uniform membrane of fixed shape with known frequencies and known natural modes of vibration. Suppose the mass density is perturbed. Determine how the frequencies are perturbed. You may assume there is only one mode of vibration for each frequency.

Consider $$ \begin{aligned} \frac{d^{2} u}{d x^{2}}+4 u &=\cos x \\ u(0) &=u(\pi)=0 \end{aligned} $$ (a) Determine all solutions using the hint that a particular solution of the differential equation is in the form, \(u_{p}=A \cos x\). (h) Determine all solutions using the eigenfunction expansion method. (c) Apply the Fredholm alternative. Is it consistent with parts (a) and (b)?

Consider $$ \frac{d^{2} u}{d x^{2}}+u=\cos x $$ which has a particular solution of the form, \(u_{p}-A x \sin x\). *(a) Suppose that \(u(0)=u(\pi)=0\). Explicitly attempt to obtain all solutions. Is your result consistent with the Fredholm alternative? (b) Answer the same questions as in part (a) if \(u(-\pi)=u(\pi)\) and \(\frac{d u}{d x}(-\pi)=\) \(\frac{d u}{d x}(\pi)\).

Using the method of (one-dimensional) eigenfunction expansion, determine \(G\left(\mathrm{x}, \mathrm{x}_{0}\right)\) if $$ \nabla^{2} G=\delta\left(\mathbf{x}-\mathbf{x}_{0}\right) $$ and (a) On the rectangle \((0

a) Since (8.4.14) (with homogeneous boundary conditions) is solvable, there are an infinite number of solutions. Suppose that \(g_{m}\left(x, x_{0}\right)\) is one such solution that is not orthogonal to \(\phi_{k}(x)\). Show that there is a unique modified Green's function \(G_{m}\left(x, x_{0}\right)\) which is orthogonal to \(\phi_{k}(x)\). (b) Assume that \(G_{m}\left(x, x_{0}\right)\) is the modifiel \(\mathrm{G}_{1}\) een's function which is orthogonal to \(\phi_{k}(x)\). Prove that \(G_{m}\left(x, x_{0}\right)\) is symmetric. [Hint: Apply Green's formula with \(G_{m}\left(x, x_{1}\right)\) and \(G_{m}\left(x, x_{2}\right)\).] 0\. Determine the modified Green's function that is needed to solve $$ \begin{aligned} & \frac{d^{2} u}{d x^{2}}+u=f(x) \\ =& \alpha \quad \text { and } \quad u(\pi)=\beta . \end{aligned} $$ Assume that \(f(x)\) satisfies the solvability condition (see Exercise 8.4.2). Obtain a representation of the solution \(u(x)\) in terms of the modified Green's function.

Consider \(\frac{d^{2} u}{d x^{3}}+u=f(x) \quad\) with \(\quad u(0)=0 \quad\) and \(\quad u(L)=0\). Assume that \((n \pi / L)^{2} \neq 1\) (i.e., \(L \neq n \pi\) for any \(n\) ). (a) Solve by the method of variation of parameters. *(b) Determine the Green's function so that \(u(x)\) may be represented in terms of it [see (8.3.12)].

Consider the wave equation with a periodic source of frequency \(\omega>0\) $$ \frac{\partial^{2} \phi}{\partial t^{7}}=c^{2} \nabla^{2} \phi+g(\mathbf{x}) e^{-i \omega} $$ Show that a particular solution at the same frequency, \(\phi=u(\mathbf{x}) e^{-i \text { ior }}\), satisfies a nonhomogeneous Helmholtz equation \(\nabla^{2} u+k^{2} u=f(x) . \quad\) [What are \(k^{2}\) and \(f(x)\) ?] The Green's function satisfies $$ \nabla^{2} G+k^{2} G=\delta\left(\mathbf{x}-\mathbf{x}_{0}\right) $$ (a) What is Green's formula for the operator \(\nabla^{2}+k^{2} ?\) (b) In infinite threedimensional space, shuw that $$ G=\frac{c_{1} e^{i k \rho}+c_{2} e^{-i k \varphi}}{\rho} $$ Choose \(c_{1}\) and \(c_{2}\) so that the corresponding \(\phi(\mathbf{x}, t)\) is an outward-propagating wave. (c) In infinite two-dimensional space, show that the Green's function is a linear combination of Bessel functions. Determine the constants so that the corresponding \(\phi(\mathbf{x}, t)\) is an outward-propagating wave for \(r\) sufficiently large. [Hint; See \((6.7 .28)\) and \((6.8 .3)\).

Determine the modified Green's function that is needed to solve $$ \begin{aligned} & \frac{d^{2} u}{d x^{2}}+u=f(x) \\ =& \alpha \quad \text { and } \quad u(\pi)=\beta . \end{aligned} $$ Assume that \(f(x)\) satisfies the solvability condition (see Exercise 8.4.2). Obtain a representation of the solution \(u(x)\) in terms of the modified Green's function.

(a) Determine the Green's function for \(y>0\) (in two dimensions) for \(\nabla^{2} G=\) \(\delta\left(\mathbf{x}-\mathbf{x}_{n}\right)\) subject to \(\partial G / \partial y=0\) on \(y=0\). (Hint: Consider a positive source at \(\mathbf{x}_{0}\) and a positive image source at \(\mathbf{x}_{0}^{*}\).) (b) Use part (a) to solve \(\nabla^{2} u=f(\mathbf{x})\) with $$ \frac{\partial u}{\partial y}=h(x) \quad \text { at } y=0 \text {. } $$ Ignore the contribution at \(\infty\).

The alternate modified Green's function (Neumann function) satisfies $$ \begin{aligned} &\frac{d^{2} G_{a}}{d x^{2}}=\delta\left(x-x_{0}\right) \\ &\frac{d G_{a}}{d x}(0)=-c \\ &\frac{d G_{a}}{d x}(L)=c, \quad \text { where we have shown } c=\frac{1}{2} . \end{aligned} $$ (a) Determine all possible \(G_{d}\left(x, x_{0}\right)\). (b) Determine all symmetric \(G_{d}\left(x, x_{0}\right)\). (c) Determine all \(G_{a}\left(x, x_{a}\right)\) which are orthogonal to \(\phi_{k}(x)\). (d) What relationship exists between \(\beta\) and \(\gamma\) for there to be a solution to \(\frac{d^{2} u}{d x^{2}}=f(x) \quad\) with \(\quad \frac{d u}{d x}(0)=\beta \quad\) and \(\quad \frac{d u}{d x}(L)=\gamma\) ? In this case, derive the solution \(u(x)\) in terms of a Neumann function, defined above.

Consider the one-dimensional infinite space wave equation with a periodic source of frequency \(\omega\) : $$ \frac{\partial^{2} \psi}{\partial t^{2}}=c^{2} \frac{\partial^{2} \psi}{\partial x^{2}}+g(x) e^{-i \omega} $$ (a) Show that a particular solution \(\phi-u(x) c^{-\text {ine }}\) of \((8.3 .46)\) is obtained if \(u\) satisfies a nonhomogeneous Helmholtz equation $$ \frac{d^{2} u}{d x^{2}}+k^{2} u=f(x) $$ *(b) The Green's function \(G\left(x, x_{0}\right)\) satisfies $$ \frac{d^{2} G}{d x^{2}}+k^{2} G=\delta\left(x-x_{0}\right) $$ Determine this infinite space Green's function so that the corresponding \(\phi(x, t)\) is an outward propagating wave. (c) Determine a particular solution of (8.3.46) above.

Using the method of images, solve $$ \nabla^{2} G=\delta\left(x-x_{0}\right) $$ in the first quadrant \((x \geqslant 0\) and \(y \geqslant 0)\) with \(G=0\) on the boundaries.

Consider \(L(u)=f(x)\) with \(L=\frac{d}{d x}\left(p \frac{d}{d x}\right)+q\). Assume that the appropriate Green's function exists. Determine the representation of \(u(x)\) in terms of the Green's function if the boundary conditions are nonhomogeneous: (a) \(u(0)=\alpha \quad\) and \(\quad u(L)=\beta\) (b) \(\frac{d u}{d x}(0)=\alpha \quad\) and \(\quad \frac{d u}{d x}(L)=\beta\) (c) \(u(0)=\alpha \quad\) and \(\quad \frac{d u}{d x}(L)=\beta\) *(d) \(u(0)=\alpha \quad\) and \(\quad \frac{d u}{d x}(L)+h u(L)=\beta\)

Consider \(L(G)=\delta\left(x-x_{0}\right)\) with \(L=\frac{d}{d x}\left(p \frac{d}{d x}\right)+q\) subject to the boundary conditions \(G\left(0, x_{0}\right)=0\) and \(G\left(I, x_{0}\right)=0\). Introduce for all \(r\) two homogeneous solutions, \(y_{1}\) and \(y_{2}\), such that each solves one of the homogeneous boundary conditions: $$ \begin{array}{rlrl} L\left(y_{1}\right) & =0 & L\left(y_{2}\right) & =0 \\ y_{1}(0) & =0 & y_{2}(L) & =0 \\ \frac{d y_{1}}{d x}(0) & =1 & \frac{d y_{2}}{d x}(L) & =1 \end{array} $$ Even if \(y_{1}\) and \(y_{2}\) cannot he explicitly ohtained, they can be easily calculated numerically on a computer as two initial value problems. Any homogeneous solution must be a linear combination of the two. *(a) Solve for \(G\left(x, x_{0}\right)\) in terms of \(y_{1}(x)\) and \(y_{2}(x)\). You may assume that \(y_{1}(x)\) \(\neq c y_{2}(x)\). (b) What goes wrong if \(y_{1}(x)=c y_{2}(x)\) for all \(x\) and why?

(a) Using the method of images, solve $$ \nabla^{2} G=\delta\left(\mathbf{x}-\mathbf{x}_{0}\right) $$ in the \(60^{\circ}\) wedge-shaped region \((0<\theta<\pi / 3,0

A modified Green's function \(G_{\mathrm{m}}\left(\mathbf{x}, \mathbf{x}_{0}\right)\) satisfies $$ \nabla^{2} G_{m}=\delta\left(x-x_{0}\right)+c $$ with $$ \nabla G_{m} \cdot \hat{\mathbf{f}}=0 $$ on the boundary of the rectangle \((0 \sim x \sim L, 0 \sim y \sim H)\). (a) Show that the method of eigenfunction expansion (two-dimensional) only works for \(c=-1 / L H\). For this \(c\), determine \(G_{m}\left(\mathbf{x}, \mathbf{x}_{0}\right)\). If possible, make \(G_{m}\left(x, x_{0}\right)\) symutetric. (b) Show that the method of eigenfunction expansion (one-dimensional) works only for \(c=-1 / L H\). For this \(c\), determine \(G_{m}\left(\mathbf{x}, \mathbf{x}_{0}\right)\). If possible, make \(G_{m}\left(\mathbf{x}, \mathbf{x}_{0}\right)\) symmetric.

Determine the Green's function \(G\left(\mathbf{x}, \mathbf{x}_{u}\right)\) inside the semicircle \((0 \sim r

(a) If a concentrated source is placed at a node of some mode (eigenfunction), show that the amplitude of the response of that mode is zero. [Hint: Use the result of the method of cigenfunction cxpansion and recall that a nodc \(x^{*}\) of an eigenfunction means anyplace where \(\phi_{n}\left(x^{*}\right)=0\).] (b) If the eigenfunctions are \(\sin n \pi x / L\) and the source is located in the middle, \(x_{0}=L / 2\), show that the response will have no even harmonics.

Determine the Green's function \(G\left(\mathbf{x}, \mathbf{x}_{0}\right)\) inside the sphere of radius \(a\) $$ \nabla^{2} G=\delta\left(\mathrm{x}-\mathrm{x}_{0}\right) $$ with \(G=0\) on the boundary.

Solve $$ \frac{d G}{d x}=\delta\left(x-x_{0}\right) \quad \text { with } \quad G\left(0, x_{0}\right)=0 $$ Show that \(G\left(x, x_{0}\right)\) is not symmetric even though \(\delta\left(x-x_{0}\right)\) is.

Use the method of multiple images to obtain the Green's function \(G\left(\mathbf{x}, \mathbf{x}_{0}\right)\) \(\nabla^{2} G=\delta\left(\mathbf{x}-\mathbf{x}_{0}\right)\) (a) on the rectangle \((0

Solve $$ \frac{d G}{d x}+G=\delta\left(x-x_{0}\right) \quad \text { with } \quad G\left(0, x_{0}\right)=0 . $$ Show that \(G\left(x, x_{0}\right)\) is not symmetric even though \(\delta\left(x-x_{0}\right)\) is.

Determine a particular solution of $$ \nabla^{2} u=f(\mathbf{x}) $$ in infinite two-dimensional space if \(f(\mathbf{x})=g(r)\), where \(r=|\mathbf{x}|\) : (a) Use the infinite space Green's function (8.5.24a). (b) Use a Green's function for the ordinary differential equation $$ \frac{1}{r} \frac{d}{d r}\left(r \frac{d u}{d r}\right)=g(r) . $$ (c) Compare parts (a) and (b).

Solve $$ \begin{gathered} \frac{d^{4} G}{d x^{4}}=\delta\left(x-x_{0}\right) \\ G\left(0, x_{0}\right)=0 \quad G\left(L, x_{0}\right)=0 \\ \frac{d G}{d x}\left(0, x_{0}\right)=0 \quad \frac{d^{2} G}{d x^{2}}\left(L, x_{0}\right)=0 . \end{gathered} $$