Solve Dirichlet Problem: Find u(x,y) Harmonic on Upper Half-Plane

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In summary, we are asked to find a function u(x,y) that is harmonic on the upper half-plane, with certain conditions at the boundaries. Using the Poisson integral formula for the upper half-plane, we can express u(x,y) as an integral over the real axis. After some calculations, we arrive at u(x,y) = (1/π)[2arctan((x-1)/y) - arctan(x/y) + (π/2)].
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skrat
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Homework Statement


Find function ##u(x,y)## that is harmonic on the upper half-plane ##0<Im(z)##. Note that
##u(x,0)=0##, ##x<0##
##u(x,0)=-1##, ##0<x<1## and
##u(x,0)=1##, ##x>1##.

Homework Equations



##u(z)=\frac{1}{2\pi i}\int _{I}\gamma '(s)u(\gamma (s))\frac{\alpha '(\gamma (s))}{\alpha (\gamma (s))}Re(\frac{\alpha (\gamma (s))+\alpha (z)}{\alpha (\gamma (s))-\alpha (z)})ds##

Where ##\alpha ## is holomorphic function that maps ##Im(z)>0## into open unit disk, and ##\gamma (s) ## parameterization of the edge.

The Attempt at a Solution



For this problem ##\alpha (z)=\frac{1-iz}{1+iz}## and it is also obvious that I will have to make at least 3 integrals.

Let's firstly take a look at ##u(x,0)=0##, ##x<0##:

This tells me that ##u(z)=\frac{1}{2\pi i}\int _{I}\gamma '(s)u(\gamma (s))\frac{\alpha '(\gamma (s))}{\alpha (\gamma (s))}Re(\frac{\alpha (\gamma (s))+\alpha (z)}{\alpha (\gamma (s))-\alpha (z)})ds## will be equal to ##0## for all ##x<0##.

But the question here is: What about ##y## ? None of the conditions tell anything about ##y##. What do I do here?

The same question is for ##u(x,0)=-1##, ##0<x<1##:

Here the integral for ##x## goes from ##0## to ##1##. But again, ##y## can be anything from ##0## to ##\infty ##. ?

I guess my question here is: What should I do with ##y## to get ##u(z)##?
 
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  • #2
Ok, I found out that my relevant equation in original post was a very stupid equation to begin with. I think it would wiser to use the Poisson integral formula for the upper half plane: ##u(x,y)=\frac{y}{\pi }\int_{-\infty }^{\infty }\frac{u(t,0)}{y^2+(x-t)^2}dt## where I am guessing ##t## stands for some kind of parameterization of real axis? http://www.diss.fu-berlin.de/diss/s...ionid=76E8A9CC9C7EE802A569873649AAF2F7?hosts= page 59.

Anyway, now with that in mind:

##u(x,y)=\int_{-\infty }^{0}...+\int_{0}^{1}...+\int_{1}^{\infty} ## where the first integral is obviously ##0##.

##u(x,y)=-\frac{y}{\pi }\int_{0}^{1}\frac{1}{y^2+(x-t)^2}dt+\frac{y}{\pi }\int_{1}^{\infty }\frac{1}{y^2+(x-t)^2}dt##

##u(x,y)=\frac{y}{\pi }[\frac{1}{y}arctan(\frac{u}{y})\mid _{u=x}^{u=x-1}-\frac{1}{y}arctan(\frac{u}{y})\mid _{u=x-1}^{u=-\infty }]##

and finally

##u(x,y)=\frac{1}{\pi }[2arctan(\frac{x-1}{y})-arctan(\frac{x}{y})+\frac{\pi }{2}]##

This should be a bit better than my first attempt.
 

1. What is the Dirichlet problem?

The Dirichlet problem is a mathematical concept in partial differential equations that involves finding a solution for a given function on a bounded domain, subject to certain boundary conditions.

2. What does it mean for u(x,y) to be harmonic?

A function u(x,y) is said to be harmonic if it satisfies Laplace's equation, which states that the sum of the second-order partial derivatives of u with respect to both x and y is equal to zero.

3. What is the upper half-plane in this context?

The upper half-plane refers to the region in the complex plane above the x-axis, where the imaginary part of a complex number is positive. In this context, it is the domain on which the Dirichlet problem is being solved.

4. How is the solution to the Dirichlet problem found?

The solution to the Dirichlet problem is found by using the method of separation of variables, where the given function u(x,y) is expressed as a product of two functions, one depending only on x and the other only on y. These functions are then solved separately and combined to form a solution for u(x,y).

5. What are the applications of solving the Dirichlet problem?

The Dirichlet problem has various applications in physics, engineering, and other fields, as it can be used to model and solve problems involving heat conduction, electrostatics, fluid flow, and more. It is also a fundamental concept in the theory of partial differential equations.

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