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

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Contributed by William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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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

c

–

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 $0<y<b$.