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I have the following equation:

$$ \begin{cases} H_{xx} + H_{yy} = xy \\ H(x,0) = 0 \\ H(x,1) = x \\ H(0,y) = 0 \\ H(1,y) = 0 \end{cases} $$

We want to solve this, so from inspection of eigenvalues and eigenfunctions we see the solution takes the form of: $ H(x,y) = \sum\limits_{n=1}^{\infty} \! Y_{n}(y)\sin(n\pi x) $.

Using undertermined coefficients we see:

$$ \sum_{n=1}^{\infty}\left[ Y_{n}''-n^2\pi^2 Y_{n}(y) \right]\sin(n\pi x) = xy $$

$$ E_{n}(y) = 2\int_0^1 \! xy\sin(n\pi x)\,\mathrm{d}x = \frac{2y\Bigl(-n\pi\cos(n\pi )+\sin(n\pi)\Bigr)}{n^2 \pi ^2} = -\frac{2(-1)^n y}{n\pi} $$

Thus complementary solution being,

$$ Y_{n}(y) = a_{n}\cosh(n\pi y) + b_{n}\sinh(n\pi y) + c + dy $$

$c=0,\quad d = \dfrac{2}{\pi^3}\dfrac{\cos(n\pi)}{n^3}$

Applying conditions:

$ H(x,0) = 0 = \displaystyle\sum_{n=1}^{\infty} a_{n}\sin(n\pi x),\hspace{0.2cm} \rightarrow \hspace{0.1cm} a_{n}=0 $

$ H(x,1) = x = \displaystyle\sum_{n=1}^{\infty} \Bigl[ b_{n}\sinh(n\pi) + d \Bigr] \sin(n\pi x) $

How would we continue from here to determine the $b_{n}$ coefficient? Is there a more elegant way to solve the non-homogeneous Poisson problem in general besides method of undetermined coefficients?

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How do you square $H(1,y)=0$ with $H(x,1)=x$? What is $H(1,1)$? –  Thomas Andrews Aug 9 '12 at 18:53
    
@night owl not "to solve" the coefficient. Determine, calculate or so. –  vesszabo Aug 9 '12 at 20:43

2 Answers 2

Express $x$ is a Fourier sine series: if $x = \sum_{n=1}^\infty c_n \sin(n \pi x)$, then $b_n = (c_n - d)/\sinh(n \pi)$.

Alternatively, you could start by looking for a particular solution, which isn't hard to find (hint: try one of the form $h(x) y$). Then you need to solve Laplace's equation.

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Let $H=K+Ax^3y+Axy^3$ ,

Then $H_x=K_x+3Ax^2y+Ay^3$

$H_{xx}=K_{xx}+6Axy$

$H_y=K_y+Ax^3+3Axy^2$

$H_{yy}=K_{yy}+6Axy$

$\therefore K_{xx}+6Axy+K_{yy}+6Axy=xy$

$K_{xx}+K_{yy}=(1-12A)xy$

$\therefore1-12A=0$

$A=\dfrac{1}{12}$

For $K_{xx}+K_{yy}=0$ , although it has a nice form of the general solution $K(x,y)=C_1(x+iy)+C_2(x-iy)$ , i.e. $H_{xx}+H_{yy}=xy$ has a nice form of the general solution $H(x,y)=C_1(x+iy)+C_2(x-iy)+\dfrac{xy(x^2+y^2)}{12}$ , it is difficult to directly substitute $H(x,0)=0$ , $H(x,1)=x$ , $H(0,y)=0$ and $H(1,y)=0$ to find $C_1(u)$ and $C_2(v)$ (at least I have no idea to these).

So it seems that we are unavoidable to solve these by using separation of variables.

For $K_{xx}+K_{yy}=0$ with conditions of the types $K(x,0)$ , $K(x,1)$ , $K(0,y)$ and $K(1,y)$ , according to http://eqworld.ipmnet.ru/en/solutions/lpde/lpde301.pdf#page=2 , we have special consideration:

$K(x,y)=\sum\limits_{n=1}^\infty A(n)\sinh(n\pi(1-x))\sin n\pi y+\sum\limits_{n=1}^\infty B(n)\sinh n\pi x\sin n\pi y+\sum\limits_{n=1}^\infty C(n)\sin n\pi x\sinh(n\pi(1-y))+\sum\limits_{n=1}^\infty D(n)\sin n\pi x\sinh n\pi y$

$\therefore H(x,y)=\sum\limits_{n=1}^\infty A(n)\sinh(n\pi(1-x))\sin n\pi y+\sum\limits_{n=1}^\infty B(n)\sinh n\pi x\sin n\pi y+\sum\limits_{n=1}^\infty C(n)\sin n\pi x\sinh(n\pi(1-y))+\sum\limits_{n=1}^\infty D(n)\sin n\pi x\sinh n\pi y+\dfrac{xy(x^2+y^2)}{12}$

$H(x,0)=0$ :

$\sum\limits_{n=1}^\infty C(n)\sin n\pi x\sinh n\pi=0$

$C(n)=0$

$\therefore H(x,y)=\sum\limits_{n=1}^\infty A(n)\sinh(n\pi(1-x))\sin n\pi y+\sum\limits_{n=1}^\infty B(n)\sinh n\pi x\sin n\pi y+\sum\limits_{n=1}^\infty D(n)\sin n\pi x\sinh n\pi y+\dfrac{xy(x^2+y^2)}{12}$

$H(0,y)=0$ :

$\sum\limits_{n=1}^\infty A(n)\sinh n\pi\sin n\pi y=0$

$A(n)=0$

$\therefore H(x,y)=\sum\limits_{n=1}^\infty B(n)\sinh n\pi x\sin n\pi y+\sum\limits_{n=1}^\infty D(n)\sin n\pi x\sinh n\pi y+\dfrac{xy(x^2+y^2)}{12}$

$H(x,1)=x$ :

$\sum\limits_{n=1}^\infty D(n)\sin n\pi x\sinh n\pi+\dfrac{x(x^2+1)}{12}=x$

$\sum\limits_{n=1}^\infty D(n)\sinh n\pi\sin n\pi x=\dfrac{x(11-x^2)}{12}$

You are facing to find an unusual kernel inversion, so all the calculations should be start from first principle!

Luckily the method in fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series) still hold in this case.

$\sum\limits_{n=1}^\infty D(n)\sinh n\pi\sin n\pi x=\dfrac{x(11-x^2)}{12}$

$\sum\limits_{n=1}^\infty D(n)\sinh n\pi\sin n\pi x\sin m\pi x=\dfrac{x(11-x^2)\sin m\pi x}{12}$

$\int_k^{k+1}\sum\limits_{n=1}^\infty D(n)\sinh n\pi\sin n\pi x\sin m\pi x~dx=\int_k^{k+1}\dfrac{x(11-x^2)\sin m\pi x}{12}dx$ , $\forall k\in\mathbb{Z}$

$\sum\limits_{n=1}^\infty D(n)\sinh n\pi\int_k^{k+1}\sin n\pi x\sin m\pi x~dx=\int_k^{k+1}\dfrac{x(11-x^2)\sin m\pi x}{12}dx$ , $\forall k\in\mathbb{Z}$

$\because\int_k^{k+1}\sin n\pi x\sin m\pi x~dx$

$=\int_{k}^{k+1}\dfrac{\cos((n-m)\pi x)-\cos((n+m)\pi x)}{2}dx$

$=\begin{cases}\biggl[\dfrac{\sin((n-m)\pi x)}{2(n-m)\pi}-\dfrac{\sin((n+m)\pi x)}{2(n+m)\pi}\biggr]_{k}^{k+1}&\text{when}~n\neq m~\text{and}~n\neq-m\\\biggl[\dfrac{x}{2}-\dfrac{\sin((n+m)\pi x)}{2(n+m)\pi}\biggr]_{k}^{k+1}&\text{when}~n=m\\\biggl[\dfrac{\sin((n-m)\pi x)}{2(n-m)\pi}-\dfrac{x}{2}\biggr]_{k}^{k+1}&\text{when}~n=-m\\\left[0\right]_{k}^{k+1}&\text{when}~n=m~\text{and}~n=-m\end{cases}$

$=\begin{cases}0&\text{when}~n,m~\text{and}~k~\text{are integers and}~n\neq m~\text{and}~n\neq-m~\text{and}~(n=m~\text{and}~n=-m)\\\dfrac{1}{2}&\text{when}~n,m~\text{and}~k~\text{are integers and}~n=m\\-\dfrac{1}{2}&\text{when}~n,m~\text{and}~k~\text{are integers and}~n=-m\end{cases}$

$\therefore D(m)\sinh m\pi\left(\dfrac{1}{2}\right)=\int_k^{k+1}\dfrac{x(11-x^2)\sin m\pi x}{12}dx$ , $\forall k\in\mathbb{Z}$ , $x\in(k,k+1)$

$D(m)=\int_k^{k+1}\dfrac{x(11-x^2)\sin m\pi x}{6\sinh m\pi}dx$ , $\forall k\in\mathbb{Z}$ , $x\in(k,k+1)$

$D(n)=\int_k^{k+1}\dfrac{x(11-x^2)\sin n\pi x}{6\sinh n\pi}dx$ , $\forall k\in\mathbb{Z}$ , $x\in(k,k+1)$

According to http://integrals.wolfram.com/index.jsp?expr=%28x%2811-x%5E2%29sin%28n+pi+x%29%29%2F%286sinh%28n+pi%29%29&random=false,

$D(n)=\int_k^{k+1}\dfrac{x(11-x^2)\sin n\pi x}{6\sinh n\pi}dx$

$=\dfrac{(-1)^{n(k+1)}(k+1)(n^2\pi^2((k+1)^2-11)-6)-(-1)^{nk}k(n^2\pi^2(k^2-11)-6)}{6n^3\pi^3\sinh n\pi}$ , $\forall n,k\in\mathbb{Z}$ , $x\in(k,k+1)$

$H(1,y)=0$ :

$\sum\limits_{n=1}^\infty B(n)\sinh n\pi\sin n\pi y+\dfrac{y(y^2+1)}{12}=0$

$\sum\limits_{n=1}^\infty B(n)\sinh n\pi\sin n\pi y=-\dfrac{y(y^2+1)}{12}$

Note that you are facing to find an unusual kernel inversion again, so all the calculations should be again to start from first principle!

Luckily the method in fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series) still hold in this case.

$\sum\limits_{n=1}^\infty B(n)\sinh n\pi\sin n\pi y=-\dfrac{y(y^2+1)}{12}$

$\sum\limits_{n=1}^\infty B(n)\sinh n\pi\sin n\pi y\sin m\pi y=-\dfrac{y(y^2+1)\sin m\pi y}{12}$

$\int_k^{k+1}\sum\limits_{n=1}^\infty B(n)\sinh n\pi\sin n\pi y\sin m\pi y~dy=\int_k^{k+1}-\dfrac{y(y^2+1)\sin m\pi y}{12}dy$ , $\forall k\in\mathbb{Z}$

$\sum\limits_{n=1}^\infty B(n)\sinh n\pi\int_k^{k+1}\sin n\pi y\sin m\pi y~dy=\int_k^{k+1}-\dfrac{y(y^2+1)\sin m\pi y}{12}dy$ , $\forall k\in\mathbb{Z}$

$\because\int_k^{k+1}\sin n\pi y\sin m\pi y~dy$

$=\int_{k}^{k+1}\dfrac{\cos((n-m)\pi y)-\cos((n+m)\pi y)}{2}dy$

$=\begin{cases}\biggl[\dfrac{\sin((n-m)\pi y)}{2(n-m)\pi}-\dfrac{\sin((n+m)\pi y)}{2(n+m)\pi}\biggr]_{k}^{k+1}&\text{when}~n\neq m~\text{and}~n\neq-m\\\biggl[\dfrac{y}{2}-\dfrac{\sin((n+m)\pi y)}{2(n+m)\pi}\biggr]_{k}^{k+1}&\text{when}~n=m\\\biggl[\dfrac{\sin((n-m)\pi y)}{2(n-m)\pi}-\dfrac{y}{2}\biggr]_{k}^{k+1}&\text{when}~n=-m\\\left[0\right]_{k}^{k+1}&\text{when}~n=m~\text{and}~n=-m\end{cases}$

$=\begin{cases}0&\text{when}~n,m~\text{and}~k~\text{are integers and}~n\neq m~\text{and}~n\neq-m~\text{and}~(n=m~\text{and}~n=-m)\\\dfrac{1}{2}&\text{when}~n,m~\text{and}~k~\text{are integers and}~n=m\\-\dfrac{1}{2}&\text{when}~n,m~\text{and}~k~\text{are integers and}~n=-m\end{cases}$

$\therefore B(m)\sinh m\pi\left(\dfrac{1}{2}\right)=\int_k^{k+1}-\dfrac{y(y^2+1)\sin m\pi y}{12}dy$ , $\forall k\in\mathbb{Z}$ , $y\in(k,k+1)$

$B(m)=\int_k^{k+1}-\dfrac{y(y^2+1)\sin m\pi y}{6\sinh m\pi}dy$ , $\forall k\in\mathbb{Z}$ , $y\in(k,k+1)$

$B(n)=\int_k^{k+1}-\dfrac{y(y^2+1)\sin n\pi y}{6\sinh n\pi}dy$ , $\forall k\in\mathbb{Z}$ , $y\in(k,k+1)$

According to http://integrals.wolfram.com/index.jsp?expr=-%28x%28x%5E2%2B1%29sin%28n+pi+x%29%29%2F%286sinh%28n+pi%29%29&random=false,

$B(n)=\int_k^{k+1}-\dfrac{y(y^2+1)\sin n\pi y}{6\sinh n\pi}dy$

$=\dfrac{(-1)^{n(k+1)}(k+1)(n^2\pi^2((k+1)^2+1)-6)-(-1)^{nk}k(n^2\pi^2(k^2+1)-6)}{6n^3\pi^3\sinh n\pi}$ , $\forall n,k\in\mathbb{Z}$ , $y\in(k,k+1)$

$\therefore H(x,y)=\sum\limits_{k=-\infty}^\infty\sum\limits_{n=1}^\infty\dfrac{(-1)^{n(k+1)}(k+1)(n^2\pi^2((k+1)^2+1)-6)-(-1)^{nk}k(n^2\pi^2(k^2+1)-6)}{6n^3\pi^3\sinh n\pi}\prod_{k,k+1}(y)\sinh n\pi x\sin n\pi y$

$+\sum\limits_{k=-\infty}^\infty\sum\limits_{n=1}^\infty\dfrac{(-1)^{n(k+1)}(k+1)(n^2\pi^2((k+1)^2-11)-6)-(-1)^{nk}k(n^2\pi^2(k^2-11)-6)}{6n^3\pi^3\sinh n\pi}\prod_{k,k+1}(x)\sin n\pi x\sinh n\pi y+\dfrac{xy(x^2+y^2)}{12} , x,y\in\mathbb R$

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