Fourier transform-pde: ...then x^2 + y^2 goes to infinity.

Grow112

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Solve this using fourier transform of sine function

\(\displaystyle v_{xx}\, +\, v_{yy}\, =\, 0\, \mbox{ for }\, x\, >\, 0,\, y\, >\, 0\)

\(\displaystyle v_x (0,\, y)\, -\, hv(0,\, y)\, =\, 0\, \mbox{ for }\, y\, >\, 0\)

\(\displaystyle v(x,\, 0)\, =\, T_0,\, \mbox{ for }\, x\, >\, 0\)

\(\displaystyle v(x,\, y)\, \rightarrow\, 0,\, v_x(x,\, y)\, \rightarrow\, 0\, v_y(x,\, y)\, \rightarrow\, 0\)

\(\displaystyle \mbox{Then }\, x^2\, +\, y^2\, \rightarrow\, \infty\)
 
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Solve this using fourier transform of sine function

\(\displaystyle v_{xx}\, +\, v_{yy}\, =\, 0\, \mbox{ for }\, x\, >\, 0,\, y\, >\, 0\)

\(\displaystyle v_x (0,\, y)\, -\, hv(0,\, y)\, =\, 0\, \mbox{ for }\, y\, >\, 0\)

\(\displaystyle v(x,\, 0)\, =\, T_0,\, \mbox{ for }\, x\, >\, 0\)

\(\displaystyle v(x,\, y)\, \rightarrow\, 0,\, v_x(x,\, y)\, \rightarrow\, 0\, v_y(x,\, y)\, \rightarrow\, 0\)

\(\displaystyle \mbox{Then }\, x^2\, +\, y^2\, \rightarrow\, \infty\)

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting


If you are haveing difficulty starting, you might review
http://www.math.ubc.ca/~feldman/m267/pdeft.pdf
which solves a different problem but should give you the idea of how to do your problem
 
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problem with substitute \(\displaystyle v\left ( 0,y \right )\) in \(\displaystyle F_{s}\left \{v_{xx\left ( x,y \right )} \right \}=-\lambda ^{2}v_{s}\left ( \lambda ,y \right )+\sqrt{2/\pi }\lambda v\left ( 0,y \right )\)
 
Solve this using fourier transform of sine function
Which transform do you understand this to be?

\(\displaystyle v_x (0,\, y)\, -\, hv(0,\, y)\, =\, 0\, \mbox{ for }\, y\, >\, 0\)

\(\displaystyle v(x,\, 0)\, =\, T_0,\, \mbox{ for }\, x\, >\, 0\)
What is your understanding of what are the functions "T0" and "hv"? How have you applied this information?

problem with substitute \(\displaystyle v\left ( 0,y \right )\) in \(\displaystyle F_{s}\left \{v_{xx\left ( x,y \right )} \right \}=-\lambda ^{2}v_{s}\left ( \lambda ,y \right )+\sqrt{2/\pi }\lambda v\left ( 0,y \right )\)
What is the "problem" that you're having? How did you get to this point?

Please reply showing all of your work and reasoning so far, clearly specifying where you are getting into difficulties. Thank you! ;)
 
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