12.3E: Laplace's Equation in Rectangular Coordinates (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
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].
- 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.
- 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 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
- 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
- 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).
- 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
- 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.