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

Q12.3.1

In Exercises 12.3.1-12.3.16 apply the definition developed in Example 12.3.1 to solve the boundary value problem. (Use Theorem 11.3.5 where it applies.) Graph the surface $u=u(x,y)$, $0\le x\le a$, $0\le y\le b$ for Exercises 12.3.3, 12.3.9, and 12.3.13.

1. $u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<1,$
$u(x,0)=x(1-x),\quad u(x,1)=0,\quad 0\le x\le 1,$
$u(0,y)=0,\quad u(1,y)=0,\quad 0\le y\le 1$

2. $u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<3,$
$u(x,0)=x^2(2-x),\quad u(x,3)=0,\quad 0\le x\le 2,$
$u(0,y)=0,\quad u(2,y)=0,\quad 0\le y\le 3$

3. $u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<2,$
$u(x,0)= \left\{\begin{array}{cl} x,& 0\le x\le 1,\\[4pt] 2-x,&1\le x\le2, \end{array}\right. \quad u(x,2)=0,\quad 0\le x\le 2,$
$u(0,y)=0,\quad u(2,y)=0,\quad 0\le y\le 2$

4. $u_{xx}+u_{yy}=0,\quad 0<x<\pi,\quad 0<y<1,$
$u(x,0)=x\sin x,\quad u(x,\pi)=0,\quad 0\le x\le \pi,$
$u(0,y)=0,\quad u(\pi,y)=0,\quad 0\le y\le 1$

5. $u_{xx}+u_{yy}=0,\quad 0<x<3,\quad 0<y<2,$
$u(x,0)=0,\quad u_y(x,2)=x^2,\quad 0\le x\le 3,$
$u_x(0,y)=0,\quad u_x(3,y)=0,\quad 0\le y\le 2$

6. $u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<2,$
$u(x,0)=0,\quad u_y(x,2)=1-x,\quad 0\le x\le 1,$
$u_x(0,y)=0,\quad u_x(1,y)=0,\quad 0\le y\le 2$

7. $u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<2,$
$u(x,0)=0,\quad u_y(x,2)=x^2-4,\quad 0\le x\le 2,$
$u_x(0,y)=0,\quad u_x(2,y)=0,\quad 0\le y\le 2$

8. $u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<1,$
$u(x,0)=0,\quad u_y(x,1)=(x-1)^2,\quad 0\le x\le 1,$
$u_x(0,y)=0,\quad u_x(1,y)=0,\quad 0\le y\le 1$

9. $u_{xx}+u_{yy}=0,\quad 0<x<3,\quad 0<y<2,$
$u(x,0)=0,\quad u_y(x,2)=0,\quad 0\le x\le 3,$
$u(0,y)=y(4-y),\quad u_x(3,y)=0,\quad 0\le y\le 2$

10. $u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<1,$
$u(x,0)=0,\quad u_y(x,1)=0,\quad 0\le x\le 2,$
$u(0,y)=y^2(3-2y),\quad u_x(2,y)=0,\quad 0\le y\le 1$

11. $u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<2,$
$u(x,0)=0,\quad u_y(x,2)=0,\quad 0\le x\le 2,$
$u(0,y)=(y-2)^3+8,\quad u_x(2,y)=0,\quad 0\le y\le 2$

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

13. $u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<\pi,$
$u_y(x,0)=0,\quad u(x,\pi)=0,\quad 0\le x\le 1,$
$u_x(0,y)=0,\quad u_x(1,y)=\sin y,\quad 0\le y\le \pi$

14. $u_{xx}+u_{yy}=0,\quad 0<x<2,\quad 0<y<3,$
$u_y(x,0)=0,\quad u(x,3)=0,\quad 0\le x\le 2,$
$u_x(0,y)=0,\quad u_x(2,y)=y(3-y),\quad 0\le y\le 3$

15. $u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<\pi,$
$u_y(x,0)=0,\quad u(x,\pi)=0,\quad 0\le x\le 1,$
$u_x(0,y)=0,\quad u_x(1,y)=\pi^2-y^2,\quad 0\le y\le \pi$

16. $u_{xx}+u_{yy}=0,\quad 0<x<1,\quad 0<y<1,$
$u_y(x,0)=0,\quad u(x,1)=0,\quad 0\le x\le 1,$
$u_x(0,y)=0,\quad u_x(1,y)=1-y^3,\quad 0\le y\le 1$

Q12.3.2

In Exercises 12.3.17-12.3.28 define the formal solution of $$u_{xx}+u_{yy}=0,\quad 0<x<a,\quad 0<y<b\nonumber$$ that satisfies the given boundary conditions for general $a, b$, and $f$ or $g$. Then solve the boundary value problem for the specified $a, b$, and $f$ or $g$. (Use Theorem 11.3.5 where it applies.) Graph the surface $u = u(x, y),\: 0 ≤ x ≤ a,\: 0 ≤ y ≤ b$ for Exercises 12.3.17, 12.3.23, and 12.3.25.

17. $u(x,0)=0,\quad u(x,b)=f(x),\quad 0<x<a,\\[4pt] u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b \\[4pt] a=3,\quad b=2,\quad f(x)=x(3-x)$

18. $u(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\[4pt] u_x(0,y)=0,\quad u_x(a,y)=0,\quad 0<y<b\\[4pt] a=2,\quad b=1,\quad f(x)=x^2(x-2)^2$

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

20. $u(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\[4pt] u(0,y)=0,\quad u_x(a,y)=0,\quad 0<y<b\\[4pt] a=3,\quad b=2,\quad f(x)=x(6-x)$

21. $u(x,0)=f(x),\quad u_y(x,b)=0,\quad 0<x<a,\\[4pt] u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b\\[4pt] a=\pi ,\quad b=2,\quad f(x)=x(\pi^2-x^2)$

22. $u_y(x,0)=0,\quad u(x,b)=f(x),\quad 0<x<a,\\[4pt] u_x(0,y)=0,\quad u_x(a,y)=0,\quad 0<y<b\\[4pt] a=\pi ,\quad b=1,\quad f(x)=x^2(x-\pi)^2$

23. $u_y(x,0)=f(x),\quad u(x,b)=0,\quad 0<x<a,\\[4pt] u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b\\[4pt] a=\pi,\quad b=1,\quad (f(x)= \left\{\begin{array}{cl} x,&0\le x\le{\pi\over2},\\[4pt]\pi-x,&{\pi\over2}\le x\le \pi \end{array}\right.$

24. $u(x,0)=0,\quad u(x,b)=0,\quad 0<x<a,\\[4pt] u_x(0,y)=0,\quad u(a,y)=g(y),\quad 0<y<b\\[4pt] a=1,\quad b=1,\quad g(y)=y(y^3-2y^2+1)$

25. $u_y(x,0)=0,\quad u(x,b)=0,\quad 0<x<a,\\[4pt] u_x(0,y)=0,\quad u(a,y)=g(y),\quad 0<y<b\\[4pt] a=2,\quad b=2,\quad g(y)=4-y^2$

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

27. $u(x,0)=0,\quad u_y(x,b)=0,\quad 0<x<a,\\[4pt] u_x(0,y)=g(y),\quad u_x(a,y)=0,\quad 0<y<b\\[4pt] a=1,\quad b=\pi ,\quad g(y)=y^2(3\pi-2y)$

28. $u_y(x,0)=0,\quad u_y(x,b)=0,\quad 0<x<a,\\[4pt] u_x(0,y)=g(y),\quad u(a,y)=0,\quad 0<y<b\\[4pt] a=2,\quad b=\pi ,\quad g(y)=y$

Q12.3.3

In Exercises 12.3.29-12.3.34 define the bounded formal solution of $$u_{xx}+u_{yy}=0,\quad 0<x<a,\quad y>0\nonumber$$ that satisfies the given boundary conditions for general $a$ and $f$. Then solve the boundary value problem for the specified $a$ and $f$.

29. $u(x,0)=f(x),\quad 0<x<a$,
$u_x(0,y)=0,\quad u_x(a,y)=0,\quad y>0$
$a=\pi$ $f(x)=x^2(3\pi-2x)$

30. $u(x,0)=f(x),\quad 0<x<a$,
$u_x(0,y)=0,\quad u(a,y)=0,\quad y>0$
$a=3$,$f(x)=9-x^2$

31. $u(x,0)=f(x),\quad 0<x<a$,
$u(0,y)=0,\quad u_x(a,y)=0,\quad y>0$
$a=\pi$,$f(x)=x(2\pi-x)$

32. $u_y(x,0)=f(x),\quad 0<x<a$,
$u(0,y)=0,\quad u(a,y)=0,\quad y>0$
$a=\pi$,$f(x)=x^2(\pi-x)$

33. $u_y(x,0)=f(x),\quad 0<x<a$,
$u_x(0,y)=0,\quad u(a,y)=0,\quad y>0$
$a=7$,$f(x)=x(7-x)$

34. $u_y(x,0)=f(x),\quad 0<x<a$,
$u(0,y)=0,\quad u_x(a,y)=0,\quad y>0$
$a=5$,$f(x)=x(5-x)$

Q12.3.4

35. Define the formal solution of the Dirichlet problem

$$\begin{array}{c} u_{xx}+u_{yy}=0,\quad 0<x<a,\quad 0<y<b,\\[4pt] u(x,0)=f_0(x),\quad u(x,b)=f_1(x),\quad 0\le x\le a,\\[4pt] u(0,y)=g_0(y),\quad u(a,y)=g_1(y),\quad 0\le y\le b \end{array}\nonumber$$

36. Show that the Neumann Problem

$$\begin{array}{c} u_{xx}+u_{yy}=0,\quad 0<x<a,\quad 0<y<b,\\[4pt] u_y(x,0)=f_0(x),\quad u_y(x,b)=f_1(x),\quad 0\le x\le a,\\[4pt] u_x(0,y)=g_0(y),\quad u_x(a,y)=g_1(y),\quad 0\le y\le b \end{array}\nonumber$$

has no solution unless

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

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

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

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

where

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

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

1. Verify the approximations $${\sinh n\pi(b-y)/a\over\sinh n\pi b/a}\approx e^{-n\pi y/a},\quad y<b, \tag{A}$$ and $${\cosh n\pi(b-y)/a\over\sinh n\pi b/a}\approx e^{-n\pi y/a},\quad y<b \tag{B}$$ for large $n$.
2. Use (A) to show that $u$ is defined for $(x,y)$ such that $0<y<b$.
3. For fixed $y$ in $(0,b)$, use (A) and Theorem 12.1.2 with $z=x$ to show that $$u_x(x,y)={\pi\over a}\sum_{n=1}^\infty n\alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \cos{n\pi x\over a},\quad -\infty<x< \infty.\nonumber$$
4. Starting from the result of (b), use (A) and Theorem 12.1.2 with $z=x$ to show that, for a fixed $y$ in $(0,b)$, $$u_{xx}(x,y)=-{\pi^2\over a^2}\sum_{n=1}^\infty n^2\alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a},\quad -\infty<x< \infty.\nonumber$$
5. For fixed but arbitrary $x$, use (B) and Theorem 12.1.2 with $z=y$ to show that $$u_y(x,y)=-{\pi\over a}\sum_{n=1}^\infty n\alpha_n {\cosh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a}\nonumber$$ 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 $(0,b)$.
6. Starting from the result of (e), use (A) and Theorem 12.1.2 to show that $$u_{yy}(x,y)={\pi^2\over a^2}\sum_{n=1}^\infty n^2\alpha_n {\sinh n\pi(b-y)/a\over\sinh n\pi b/a} \sin{n\pi x\over a},\quad 0<y<b.\nonumber$$
7. 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 $0<y<b$.