$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 12.3E: Laplace's Equation in Rectangular Coordinates (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

In Exercises [exer:12.3.1}- [exer:12.3.16} apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem [thmtype:11.3.5} where it applies.) Where indicated by , graph the surface $$u=u(x,y)$$, $$0\le x\le a$$, $$0\le y\le b$$.

[exer:12.3.1] $$u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<1,$$
$$u(x,0)=x(1-x),\quad u(x,1)=0,\quad 0\le x\le 1,$$
$$u(0,y)=0,\quad u(1,y)=0,\quad 0\le y\le 1$$

[exer:12.3.2] $$u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<3,$$
$$u(x,0)=x^2(2-x),\quad u(x,3)=0,\quad 0\le x\le 2,$$
$$u(0,y)=0,\quad u(2,y)=0,\quad 0\le y\le 3$$

[exer:12.3.3] $$u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<2,$$
$$u(x,0)= \left\{\begin{array}{cl} x,& 0\le x\le 1,\\ 2-x,&1\le x\le2, \end{array}\right. \quad u(x,2)=0,\quad 0\le x\le 2,$$
$$u(0,y)=0,\quad u(2,y)=0,\quad 0\le y\le 2$$

[exer:12.3.4] $$u_{xx}+u_{yy}=0,\quad 0<x<\pi,\quad 0<y<1,$$
$$u(x,0)=x\sin x,\quad u(x,\pi)=0,\quad 0\le x\le \pi,$$
$$u(0,y)=0,\quad u(\pi,y)=0,\quad 0\le y\le 1$$

[exer:12.3.5] $$u_{xx}+u_{yy}=0,\quad 0<x<3,\quad 0<y<2,$$
$$u(x,0)=0,\quad u_y(x,2)=x^2,\quad 0\le x\le 3,$$
$$u_x(0,y)=0,\quad u_x(3,y)=0,\quad 0\le y\le 2$$

[exer:12.3.6] $$u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<2,$$
$$u(x,0)=0,\quad u_y(x,2)=1-x,\quad 0\le x\le 1,$$
$$u_x(0,y)=0,\quad u_x(1,y)=0,\quad 0\le y\le 2$$

[exer:12.3.7] $$u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<2,$$
$$u(x,0)=0,\quad u_y(x,2)=x^2-4,\quad 0\le x\le 2,$$
$$u_x(0,y)=0,\quad u_x(2,y)=0,\quad 0\le y\le 2$$

[exer:12.3.8] $$u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<1,$$
$$u(x,0)=0,\quad u_y(x,1)=(x-1)^2,\quad 0\le x\le 1,$$
$$u_x(0,y)=0,\quad u_x(1,y)=0,\quad 0\le y\le 1$$

[exer:12.3.9] $$u_{xx}+u_{yy}=0,\quad 0<x<3,\quad 0<y<2,$$
$$u(x,0)=0,\quad u_y(x,2)=0,\quad 0\le x\le 3,$$
$$u(0,y)=y(4-y),\quad u_x(3,y)=0,\quad 0\le y\le 2$$

[exer:12.3.10] $$u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<1,$$
$$u(x,0)=0,\quad u_y(x,1)=0,\quad 0\le x\le 2,$$
$$u(0,y)=y^2(3-2y),\quad u_x(2,y)=0,\quad 0\le y\le 1$$

[exer:12.3.11] $$u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<2,$$
$$u(x,0)=0,\quad u_y(x,2)=0,\quad 0\le x\le 2,$$
$$u(0,y)=(y-2)^3+8,\quad u_x(2,y)=0,\quad 0\le y\le 2$$

[exer:12.3.12] $$u_{xx}+u_{yy}=0,\quad 0<x<3,\quad 0<y<1,$$
$$u(x,0)=0,\quad u_y(x,1)=0,\quad 0\le x\le 3,$$
$$u(0,y)=y(2y^2-9y+12),\quad u_x(3,y)=0,\quad 0\le y\le 1$$

[exer:12.3.13] $$u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<\pi,$$
$$u_y(x,0)=0,\quad u(x,\pi)=0,\quad 0\le x\le 1,$$
$$u_x(0,y)=0,\quad u_x(1,y)=\sin y,\quad 0\le y\le \pi$$

[exer:12.3.14] $$u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<3,$$
$$u_y(x,0)=0,\quad u(x,3)=0,\quad 0\le x\le 2,$$
$$u_x(0,y)=0,\quad u_x(2,y)=y(3-y),\quad 0\le y\le 3$$

[exer:12.3.15] $$u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<\pi,$$
$$u_y(x,0)=0,\quad u(x,\pi)=0,\quad 0\le x\le 1,$$
$$u_x(0,y)=0,\quad u_x(1,y)=\pi^2-y^2,\quad 0\le y\le \pi$$

[exer:12.3.16] $$u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<1,$$
$$u_y(x,0)=0,\quad u(x,1)=0,\quad 0\le x\le 1,$$
$$u_x(0,y)=0,\quad u_x(1,y)=1-y^3,\quad 0\le y\le 1$$

[exer:12.3.17] $$u(x,0)=0,\quad u(x,b)=f(x),\quad 0<x<a,\\ u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b$$
$$a=3$$,$$b=2$$,$$f(x)=x(3-x)$$

[exer:12.3.18] $$u(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=0,\quad u_x(a,y)=0,\quad 0<y<b$$
$$a=2$$,$$b=1$$,$$f(x)=x^2(x-2)^2$$

[exer:12.3.19] $$u(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=0,\quad u(a,y)=0,\quad 0<y<b$$
$$a=1$$,$$b=2$$,$$f(x)=3x^3-4x^2+1$$

[exer:12.3.20] $$u(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\ u(0,y)=0,\quad u_x(a,y)=0,\quad 0<y<b$$
$$a=3$$,$$b=2$$,$$f(x)=x(6-x)$$

[exer:12.3.21] $$u(x,0)=f(x),\quad u_y(x,b)=0,\quad 0<x<a,\\ u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b$$
$$a=\pi$$,$$b=2$$,$$f(x)=x(\pi^2-x^2)$$

[exer:12.3.22] $$u_y(x,0)=0,\quad u(x,b)=f(x),\quad 0<x<a,\\ u_x(0,y)=0,\quad u_x(a,y)=0,\quad 0<y<b$$
$$a=\pi$$,$$b=1$$,$$f(x)=x^2(x-\pi)^2$$

[exer:12.3.23] $$u_y(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\ u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b$$
$$a=\pi$$,$$b=1$$, $$f(x)= \left\{\begin{array}{cl} x,&0\le x\le{\pi\over2},\\\pi-x,&{\pi\over2}\le x\le \pi \end{array}\right.$$

[exer:12.3.24] $$u(x,0)=0,\quad u(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=0,\quad u(a,y)=g(y),\quad 0<y<b$$
$$a=1$$,$$b=1$$,$$g(y)=y(y^3-2y^2+1)$$

[exer:12.3.25] $$u_y(x,0)=0,\quad u(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=0,\quad u(a,y)=g(y),\quad 0<y<b$$
$$a=2$$,$$b=2$$,$$g(y)=4-y^2$$

[exer:12.3.26] $$u(x,0)=0,\quad u(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=0,\quad u_x(a,y)=g(y),\quad 0<y<b$$
$$a=1$$, $$b=4$$, $$g(y)= \left\{\begin{array}{cl} y,&0\le y\le2,\\4-y,&2\le y\le 4 \end{array}\right.$$

[exer:12.3.27] $$u(x,0)=0,\quad u_y(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=g(y),\quad u_x(a,y)=0,\quad 0<y<b$$
$$a=1$$,$$b=\pi$$,$$g(y)=y^2(3\pi-2y)$$

[exer:12.3.28] $$u_y(x,0)=0,\quad u_y(x,b)=0,\quad 0<x<a,\\ u_x(0,y)=g(y),\quad u(a,y)=0,\quad 0<y<b$$
$$a=2$$,$$b=\pi$$,$$g(y)=y$$

[exer:12.3.29] $$u(x,0)=f(x),\quad 0<x<a$$,
$$u_x(0,y)=0,\quad u_x(a,y)=0,\quad y>0$$
$$a=\pi$$ $$f(x)=x^2(3\pi-2x)$$

[exer:12.3.30] $$u(x,0)=f(x),\quad 0<x<a$$,
$$u_x(0,y)=0,\quad u(a,y)=0,\quad y>0$$
$$a=3$$,$$f(x)=9-x^2$$

[exer:12.3.31] $$u(x,0)=f(x),\quad 0<x<a$$,
$$u(0,y)=0,\quad u_x(a,y)=0,\quad y>0$$
$$a=\pi$$,$$f(x)=x(2\pi-x)$$

[exer:12.3.32] $$u_y(x,0)=f(x),\quad 0<x<a$$,
$$u(0,y)=0,\quad u(a,y)=0,\quad y>0$$
$$a=\pi$$,$$f(x)=x^2(\pi-x)$$

[exer:12.3.33] $$u_y(x,0)=f(x),\quad 0<x<a$$,
$$u_x(0,y)=0,\quad u(a,y)=0,\quad y>0$$
$$a=7$$,$$f(x)=x(7-x)$$

[exer:12.3.34] $$u_y(x,0)=f(x),\quad 0<x<a$$,
$$u(0,y)=0,\quad u_x(a,y)=0,\quad y>0$$
$$a=5$$,$$f(x)=x(5-x)$$

[exer:12.3.35] Define the formal solution of the Dirichlet problem

$\begin{array}{c} u_{xx}+u_{yy}=0,\quad 0<x<a,\quad 0<y<b,\\ u(x,0)=f_0(x),\quad u(x,b)=f_1(x),\quad 0\le x\le a,\\ u(0,y)=g_0(y),\quad u(a,y)=g_1(y),\quad 0\le y\le b \end{array}$

[exer:12.3.36] Show that the Neumann Problem

$\begin{array}{c} u_{xx}+u_{yy}=0,\quad 0<x<a,\quad 0<y<b,\\ u_y(x,0)=f_0(x),\quad u_y(x,b)=f_1(x),\quad 0\le x\le a,\\ u_x(0,y)=g_0(y),\quad u_x(a,y)=g_1(y),\quad 0\le y\le b \end{array}$

has no solution unless

$\int_0^af_0(x)\,dx= \int_0^af_1(x)\,dx= \int_0^bg_0(y)\,dy= \int_0^bg_1(y)\,dy=0.$

In this case it has infinitely many formal solutions. Find them.

[exer:12.3.37] In this exercise take it as given that the infinite series $$\sum_{n=1}^\infty n^pe^{-qn}$$ converges for all $$p$$ if $$q>0$$, and, where appropriate, use the comparison test for absolute convergence of an infinite series.

Let

$u(x,y)=\sum_{n=1}^\infty \alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a},$

where

$\alpha_n={2\over a}\int_0^a f(x)\sin{n\pi x\over a}\,dx$

and $$f$$ is piecewise smooth on $$[0,a]$$.

Verify the approximations

${\sinh n\pi(b-y)/a\over\sinh n\pi b/a}\approx e^{-n\pi y/a},\quad y<b, \eqno{\rm(A)}$

and

${\cosh n\pi(b-y)/a\over\sinh n\pi b/a}\approx e^{-n\pi y/a},\quad y<b \eqno{\rm(B)}$

for large $$n$$.

Use (A) to show that $$u$$ is defined for $$(x,y)$$ such that $$0<y<b$$.

For fixed $$y$$ in $$(0,b)$$, use (A) and Theorem [thmtype:12.1.2} with $$z=x$$ to show that

$u_x(x,y)={\pi\over a}\sum_{n=1}^\infty n\alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \cos{n\pi x\over a},\quad -\infty<x< \infty.$

Starting from the result of

## b

, use (A) and Theorem [thmtype:12.1.2} with $$z=x$$ to show that, for a fixed $$y$$ in $$(0,b)$$,

$u_{xx}(x,y)=-{\pi^2\over a^2}\sum_{n=1}^\infty n^2\alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a},\quad -\infty<x< \infty.$

For fixed but arbitrary $$x$$, use (B) and Theorem [thmtype:12.1.2} with $$z=y$$ to show that

$u_y(x,y)=-{\pi\over a}\sum_{n=1}^\infty n\alpha_n {\cosh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a}$

if $$0<y_0<y<b$$, where $$y_0$$ is an arbitrary number in $$(0,b)$$. Then argue that since $$y_0$$ can be chosen arbitrarily small, the conclusion holds for all $$y$$ in $$(0,b)$$.

Starting from the result of

## e

, use (A) and Theorem [thmtype:12.1.2} to show that

$u_{yy}(x,y)={\pi^2\over a^2}\sum_{n=1}^\infty n^2\alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a},\quad 0<y<b.$

Conclude that $$u$$ satisfies Laplace’s equation for all $$(x,y)$$ such that $$0<y<b$$.

By repeatedly applying the arguments in

## f

, it can be shown that $$u$$ can be differentiated term by term any number of times with respect to $$x$$ and/or $$y$$ if $$0<y<b$$.