91Ó°ÊÓ

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).

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 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.

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