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Mathematics LibreTexts

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

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    18306
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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 \(0<y<b\).