PDE24

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\Delta u=u_{{xx}}+u_{{yy}}=0\,


u(x,0)=f(x)\,
\lim _{{x^{2}+y^{2}\rightarrow \infty }}u(x,y)=0\,

-\infty <x<\infty ,\,\,\,\,\,0<y<\infty \,



This is called the Dirichlet problem for the half-plane. Let u=XY\, so that

X''+\lambda ^{2}X=0,\,\,-\infty <x<\infty \,

Y''-\lambda ^{2}Y=0,\,\,0<y<\infty \,

X(x)={\begin{cases}{\frac  {1}{2}}A_{0}&\lambda =0\\c_{1}\cos \lambda x+c_{2}\sin \lambda x&\lambda >0\end{cases}}\,

Y(y)={\begin{cases}{\frac  {1}{2}}B_{0}&\lambda =0\\c_{3}e^{{\lambda y}}+c_{4}e^{{-\lambda y}}&\lambda >0\end{cases}}\,

The exponentials were chosen over the hyperbolic trig functions because one of the exponetials tends to zero as its argument tends to infinity, while both h-trig functions go to infinity.

To satisfy the limit condition, we must set c_{3}=A_{0}=B_{0}=0\,

Since we are dealing with \lambda ^{2}\, above, \lambda \, only needs to take on nonnegative values. This is reflected in the integrals in the solution below:

  • u(x,y)=\int _{0}^{\infty }\left[A_{\lambda }\cos \lambda x+B_{\lambda }\sin \lambda x\right]e^{{-\lambda y}}\,d\lambda \,

u(x,0)=\int _{0}^{\infty }A_{\lambda }\cos \lambda x+B_{\lambda }\sin \lambda x\,d\lambda =f(x)\,

The Fourier coefficients are:

A_{\lambda }={\frac  {1}{\pi }}\int _{{-\infty }}^{\infty }f(x)\cos \lambda x\,dx\,

B_{\lambda }={\frac  {1}{\pi }}\int _{{-\infty }}^{\infty }f(x)\sin \lambda x\,dx\,

Partial Differential Equations

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