Do you have some hints as to how to solve the following PDE?

Question

I have the following Boundary-value problem for $u = u(x,y)$: $$ u_{xx} + u_{yy} = 0 , $$ and $ \frac{\partial u(x,0) }{ \partial y } = h(x) $. Also, $y>0$ and $ - \infty < x < \infty $.

I thought that the general solution of this PDE is: $$ u(x,y) = c_1 (y + ix) + c_2 (y - ix) , $$ from which it follows that $$ \frac{ \partial u }{ \partial y } (x,0) = c_1 + c_2 = h(x). $$ Therefore, we have $$u(x,y) = (c_1 + c_2) (y + ix) - 2 c_2 ix = h(x)(y+ ix) - 2 c_2 i x .$$ I have the feeling that this isn't the correct solution, though. Or at least that it isn't complete. Is this correct? If so, how can I improve my solution?

Answer 1

Note that since this problem has only one condition, this is in fact an under-determining problem.

You can let one more dummy condition so that this becomes a just-determining problem.

For example, letting the dummy condition be $u(x,0)=g(x)$ , the general solution is more conventient to consider as $u(x,y)=c_1(x+iy)+c_2(x-iy)$ rather than in $u(x,y)=c_1(y+ix)+c_2(y-ix)$ , and the solution with such conditions can be expressed by using D’Alembert’s formula:

$u(x,y)=\dfrac{g(x+iy)+g(x-iy)}{2}-\dfrac{i}{2}\int_{x-iy}^{x+iy}h(t)~dt$

Note that this solution suitable for $x,y\in\mathbb{C}$ , not only suitable for $-\infty<x<\infty$ and $y>0$ .

Note that the ranges stated in the questions are only provide the minimum requirements of the domain of the solutions required, you are always welcomed if you smart enough to find the solutions which the domain larger than the ranges stated in the questions.

Answer 2

Your general solution is not correct. The differential equation you are dealing with is the Laplace equation.

For the general theory of Laplace equation, please have a look at the introductory exposition in

http://en.wikipedia.org/wiki/Laplace's_equation

Once the setting becomes familiar, you should get used to the method called "separation of variables" and the Fourier series expansions of periodic functions.

Some good references (to start with):

http://en.wikipedia.org/wiki/Separation_of_variables (paragraph: PDEs)

http://en.wikipedia.org/wiki/Fourier_series

Edit I would add some explicit fomulae. Let us assume a separable solution $u(x,y)=X(x)Y(y)$: the Laplace equation leads to $X''(x)=k X(x)$ and $Y''(y)=-k Y(y)$ for some real constant $k$. Depending whether $k<0, k=0$ or $k>0$ the solutions to the ordinary diff. eqs for $X$ and $Y$ are different. If we had some extra boundary conditions (periodic for example) we could select a choice for $k$ at this stage. In our case, however, we should consider all the combinations of $u(x,y)=X(x)Y(y)$ for $k<0, k=0$ or $k>0$. Any of these choices must satisfy the boundary condition $\frac{\partial u(x,0)}{\partial y}=h(x)$, for all $x\in\mathbb R$. Depending on the form of $h(x)$ we could, in general, determine the "right" $k$. As you can see I am pretty vague, as Iwe do not know anything about $h(x)$ and we cannot choose $k$ using other boundary conditions.

Let us move to a more explicit example. We select the case $k:=\omega^2>0$, obtaining through superposition the separable solution $u(x,y)=\sum_{i}(A_ie^{\omega_i x}+B_ie^{-\omega_i x})(C_ie^{i\omega_i y}+D_ie^{-i\omega_i y})$, for real coefficients $A_i,B_i,C_i,D_i$. We have to impose the boundary condition obtaining

$\sum_i(A_ie^{\omega_i x}+B_ie^{-\omega_i x})(C_i-D_i)i\omega_i=h(x)$,

for all $x\in \mathbb R$. At this stage a Fourier analysis of $h(x)$ is needed. Note that the Fourier analysis is done using

$\sum_i(\tilde{A}_ie^{\omega_i x}+\tilde{B}_ie^{-\omega_i x})=h(x)$,

where $\tilde{A}_i=i\omega_iA_i(C_i-D_i)$ and $\tilde{B}_i=i\omega_iB_i(C_i-D_i)$.