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

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

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. $$u(x,0)=0,\quad u(x,b)=f(x),\quad 0<x<a,\\ u(0,y)=0,\quad u(a,y)=0,\quad 0<y<b \\ a=3,\quad b=2,\quad f(x)=x(3-x)$$

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

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

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

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

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

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

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

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

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

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

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

## 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

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

36. Show that the Neumann Problem

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

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 $$0<y<b$$.