91Ó°ÊÓ

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

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) \(\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}\)

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.

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

Solve Laplace's equation inside a circular annulus \((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 $$ \begin{aligned} u(0, y) &=g(y) & & u(x, 0)=0 \\ u(L, y) &=0 & & u(x, H)=0 \end{aligned} $$

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

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