Green's function for $-Delta$ on the plane is $-frac{1}{2pi}lnlVert x-y rVert$
$begingroup$
Green's function for $-Delta$ on the plane is $-frac{1}{2pi}lnlVert x-y rVert$
a) Using the method of reflection, compute the green's function for Laplace's equation $-Delta u=0$
b) Compute the green's function for the quadrant with the following other boundary condition Neuwann on the x-axis and dirichlet on the y-axis
My answers
a)$U(x)= Vlvert xrvert$ where, $x=(x,x^{2},x^{3},...x^{n})$
$-Delta u = -frac{partial^{2}u}{partial x^{2}} - frac{partial^{2}u}{partial y^{2}}=0$
taking derivative of $U$
$Ux_i = V'frac{x_i}{lvert xrvert}$
$Ux_ix_i = V''frac{x_i^{2}}{lvert xrvert^{2}}$ + $V'frac{lvert xrvert -frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}}$
therefore
$-Delta u= sum Ux_ix_i= sum V''frac{x_i^{2}}{lvert xrvert^{2}} + V'frac{lvert xrvert-frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}} $
I'm stuck here.
partial-derivative
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
$endgroup$
add a comment |
$begingroup$
Green's function for $-Delta$ on the plane is $-frac{1}{2pi}lnlVert x-y rVert$
a) Using the method of reflection, compute the green's function for Laplace's equation $-Delta u=0$
b) Compute the green's function for the quadrant with the following other boundary condition Neuwann on the x-axis and dirichlet on the y-axis
My answers
a)$U(x)= Vlvert xrvert$ where, $x=(x,x^{2},x^{3},...x^{n})$
$-Delta u = -frac{partial^{2}u}{partial x^{2}} - frac{partial^{2}u}{partial y^{2}}=0$
taking derivative of $U$
$Ux_i = V'frac{x_i}{lvert xrvert}$
$Ux_ix_i = V''frac{x_i^{2}}{lvert xrvert^{2}}$ + $V'frac{lvert xrvert -frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}}$
therefore
$-Delta u= sum Ux_ix_i= sum V''frac{x_i^{2}}{lvert xrvert^{2}} + V'frac{lvert xrvert-frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}} $
I'm stuck here.
partial-derivative
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
$endgroup$
$begingroup$
Note that $sum_i x_i^2 equiv |x|^2$ so you arrive at the ODE $V''(r) + frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension.
$endgroup$
– Winther
Dec 18 '18 at 15:41
$begingroup$
Thanks for the clarification here
$endgroup$
– Tariro Manyika
Feb 8 at 9:16
add a comment |
$begingroup$
Green's function for $-Delta$ on the plane is $-frac{1}{2pi}lnlVert x-y rVert$
a) Using the method of reflection, compute the green's function for Laplace's equation $-Delta u=0$
b) Compute the green's function for the quadrant with the following other boundary condition Neuwann on the x-axis and dirichlet on the y-axis
My answers
a)$U(x)= Vlvert xrvert$ where, $x=(x,x^{2},x^{3},...x^{n})$
$-Delta u = -frac{partial^{2}u}{partial x^{2}} - frac{partial^{2}u}{partial y^{2}}=0$
taking derivative of $U$
$Ux_i = V'frac{x_i}{lvert xrvert}$
$Ux_ix_i = V''frac{x_i^{2}}{lvert xrvert^{2}}$ + $V'frac{lvert xrvert -frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}}$
therefore
$-Delta u= sum Ux_ix_i= sum V''frac{x_i^{2}}{lvert xrvert^{2}} + V'frac{lvert xrvert-frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}} $
I'm stuck here.
partial-derivative
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
$endgroup$
Green's function for $-Delta$ on the plane is $-frac{1}{2pi}lnlVert x-y rVert$
a) Using the method of reflection, compute the green's function for Laplace's equation $-Delta u=0$
b) Compute the green's function for the quadrant with the following other boundary condition Neuwann on the x-axis and dirichlet on the y-axis
My answers
a)$U(x)= Vlvert xrvert$ where, $x=(x,x^{2},x^{3},...x^{n})$
$-Delta u = -frac{partial^{2}u}{partial x^{2}} - frac{partial^{2}u}{partial y^{2}}=0$
taking derivative of $U$
$Ux_i = V'frac{x_i}{lvert xrvert}$
$Ux_ix_i = V''frac{x_i^{2}}{lvert xrvert^{2}}$ + $V'frac{lvert xrvert -frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}}$
therefore
$-Delta u= sum Ux_ix_i= sum V''frac{x_i^{2}}{lvert xrvert^{2}} + V'frac{lvert xrvert-frac{x_i^{2}}{lvert xrvert}}{lvert xrvert^{2}} $
I'm stuck here.
partial-derivative
partial-derivative
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
edited Dec 18 '18 at 15:48
tomasz
23.7k23482
23.7k23482
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
asked Dec 18 '18 at 15:38
Tariro ManyikaTariro Manyika
638
638
$begingroup$
Note that $sum_i x_i^2 equiv |x|^2$ so you arrive at the ODE $V''(r) + frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension.
$endgroup$
– Winther
Dec 18 '18 at 15:41
$begingroup$
Thanks for the clarification here
$endgroup$
– Tariro Manyika
Feb 8 at 9:16
add a comment |
$begingroup$
Note that $sum_i x_i^2 equiv |x|^2$ so you arrive at the ODE $V''(r) + frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension.
$endgroup$
– Winther
Dec 18 '18 at 15:41
$begingroup$
Thanks for the clarification here
$endgroup$
– Tariro Manyika
Feb 8 at 9:16
$begingroup$
Note that $sum_i x_i^2 equiv |x|^2$ so you arrive at the ODE $V''(r) + frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension.
$endgroup$
– Winther
Dec 18 '18 at 15:41
$begingroup$
Note that $sum_i x_i^2 equiv |x|^2$ so you arrive at the ODE $V''(r) + frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension.
$endgroup$
– Winther
Dec 18 '18 at 15:41
$begingroup$
Thanks for the clarification here
$endgroup$
– Tariro Manyika
Feb 8 at 9:16
$begingroup$
Thanks for the clarification here
$endgroup$
– Tariro Manyika
Feb 8 at 9:16
add a comment |
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$begingroup$
Note that $sum_i x_i^2 equiv |x|^2$ so you arrive at the ODE $V''(r) + frac{V'(r)}{r}(n-1) = 0$ where $r = |x|$ and $n=2$ is the dimension.
$endgroup$
– Winther
Dec 18 '18 at 15:41
$begingroup$
Thanks for the clarification here
$endgroup$
– Tariro Manyika
Feb 8 at 9:16