91Ó°ÊÓ

Solve the heat equation with time-independent sources and boundary conditions $$ \begin{aligned} \frac{\partial u}{\partial t} &=k \frac{\partial^{2} u}{\partial x^{2}}+Q(x) \\\ u(x, 0) &=f(x) \end{aligned} $$ if an equilibrium solution exists. Analyze the limits as \(t \rightarrow \infty\). If no equilibrium exists, explain why and reduce the problem to one with homogeneous boundary conditions (but do not solve). Assume *(a) \(Q(x)=0, \quad u(0, t)=A, \quad \frac{\partial u}{\partial x}(L, t)=B\) (b) \(Q(x)=0, \quad \frac{\partial u}{\partial x}(0, t)=0, \quad \frac{\partial u}{\partial x}(L, t)=B \neq 0\) (c) \(Q(x)=0, \quad \frac{\partial u}{\partial x}(0, t)=A \neq 0, \quad \frac{\partial u}{\partial x}(L, t)=A\) k(d) \(Q(x)=k, \quad u(0, t)=A, \quad u(L, t)=B\) Other choices for \(r(x, t)\) yield equivalent solutions to the original nonhomogeneous problem. (e) \(Q(x)=k\) \(\frac{\partial u}{\partial x}(0, t)=0, \quad \frac{\partial u}{\partial x}(L, t)=0\) (f) \(Q(x)=\sin \frac{2 \pi x}{L}, \quad \frac{\partial u}{\partial x}(0, t)=0\), \(\frac{\partial u}{\partial x}(L, t)=0\)

Consider the heat equation with a steady source $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}+Q(x) $$ subject to the initial and boundary conditions described in this section: $$ u(0, t)=0, u(L, t)-0, \text { and } u(x, 0)=f(x) . $$ Obtain the solution by the method of eigenfunction expansion. Show that the solution approaches a steady-state solution.

Consider a vibrating string with time-dependent forcing: $$ \begin{array}{rr} \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}+Q(x, t) \\ u(0, t)=0 & u(x, 0)=f(x) \\ u(L, t)=0 & \frac{\partial u}{\partial t}(x, 0)=0 . \end{array} $$ (a) Solve the initial value problem. (b) Solve the initial value problem if \(Q(x, t)=g(x) \cos \omega t\). For what values of \(\omega\) does resonance occur?

Use the method of eigenfunction expansions to solve, without reducing to homogeneous boundary conditions: $$ \left.\begin{array}{rl} \frac{\partial u}{\partial t} & =k \frac{\partial^{2} u}{\partial x^{2}} \\ u(0, t) & =A \\ u(x, 0)=f(x) & u(L, t)=B \end{array}\right\\} \text { constants. } $$

Solve the two-dimensional heat equation with circularly symmetric time- independent sources, boundary conditions, and initial conditions (inside a circle): $$ \frac{\partial u}{\partial t}=\frac{k}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+Q(r) $$ with $$ u(r, 0)=f(r) \quad \text { and } \quad u(a, t)=T $$

Solve the following example of Poisson's equation: $$ \nabla^{2} u=e^{2 y} \sin x $$ subject to the following boundary conditions: $$ \begin{array}{ll} u(0, y)=0 & u(x, 0)=0 \\ u(\pi, y)=0 & u(x, L)=f(x) \end{array} $$