In summary, the Neumann Boundary Value Problem in a Half Plane is a mathematical problem that involves finding the solution to a partial differential equation in a half plane, subject to certain boundary conditions. It is named after Carl Neumann and has applications in physics and engineering, such as in heat transfer and fluid flow. The main difference between the Dirichlet and Neumann boundary conditions is that the former specifies the value of the unknown function at the boundary while the latter specifies its derivative. The problem is solved using various mathematical techniques, but it can present challenges in finding an appropriate method, handling non-unique solutions, and ensuring accuracy and satisfaction of boundary conditions.
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Find all bounded solutions to the PDE ##\Delta u(x,y) = 0## for ##x\in \mathbb{R}## and ##y > 0## with Neumann boundary condition ##u_y(x,0) = g(x)##.
Let ##\hat{u}(k,y) = \int_{-\infty}^\infty u(x,y)e^{ikx}\, dx##, the Fourier transform of ##u## with respect to the first variable. Applying this Fourier transform to the PDE with respect to the first variable yields ##\hat{u}_{yy} - k^2 \hat{u} = 0## with ##\hat{u}_y(k,0) = \hat{g}(0)##. We have general solution ##\hat{u}(k,y) = A(k)e^{ky} + B(k) e^{-ky}##. Assuming boundedness of ##u##, ##A(k) \equiv 0## if ##k > 0## and ##B(k) \equiv 0## if ##k < 0##. So we express ##\hat{u}(k,y) = C(k)e^{-|k|y}##. The condition ##\hat{u}_y(k,0) = \hat{g}(k)## forces ##-|k| C(k) = \hat{g}(k)##. Therefore ##\hat{u}_y(k,y) = -|k| C(k) e^{-|k|y} = \hat{g}(k) e^{-|k|y}##. By the convolution theorem we obtain $$u_y(x,y) = \frac{y}{\pi} \int_{-\infty}^\infty \frac{g(x-t)}{t^2 + y^2}\, dt$$ Integrating with respect to ##y## produces solution ##u(x,y) = \frac{1}{2\pi} \int_{-\infty}^\infty g(x-t) \log(t^2 + y^2)\, dt + c##.