Unable to determine Sobolev space
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I have the following function:
$T ( textbf{r}) = frac{q}{4 pi k} int_{s_i=0}^{L_i} frac{1}{left| textbf{r}-textbf{r}'right|} ds_i = frac{q}{4 pi k} logleft( frac{tan(theta_2/2)}{tan(theta_1/2)} right)$
which is the exact temperature field produced by a heat source of magnitude $q$ applied over a line (see figure)
I would like to know what's the space this function lies on. To that end, I set the line source between $(-2,0)--(2,0)$ and I consider my domain as $[0,1]times[0,1]$ (since the solution is symmetric and thus I can multiply by $4$ the result. I decided to use Mathematica to see whether I could integrate the function squared, and indeed I get a finite value, for which I conclude the function is at least in $L^2$ ($H^0$). However, if I obtain the gradient of the function, it seems that the $y$-component does not give a finite value if I numerically integrate it with exponent $2$. If I integrate it numerically with exponent $1$ Mathematica gives me the following message:
NIntegrate: ""NIntegrate failed to converge to prescribed accuracy after 18
recursive bisections in y near {x,y} = {0.844124,2.*10^-323}.
NIntegrate obtained -118.498 and 0.010644969997020758` for the
integral and error estimates.""
-118.498
and any other exponent returns even complex numbers. So at this point I am clueless. How can I tell exactly what's the Sobolev space this function lies on?
sobolev-spaces fractional-sobolev-spaces
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
add a comment |
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0
down vote
favorite
I have the following function:
$T ( textbf{r}) = frac{q}{4 pi k} int_{s_i=0}^{L_i} frac{1}{left| textbf{r}-textbf{r}'right|} ds_i = frac{q}{4 pi k} logleft( frac{tan(theta_2/2)}{tan(theta_1/2)} right)$
which is the exact temperature field produced by a heat source of magnitude $q$ applied over a line (see figure)
I would like to know what's the space this function lies on. To that end, I set the line source between $(-2,0)--(2,0)$ and I consider my domain as $[0,1]times[0,1]$ (since the solution is symmetric and thus I can multiply by $4$ the result. I decided to use Mathematica to see whether I could integrate the function squared, and indeed I get a finite value, for which I conclude the function is at least in $L^2$ ($H^0$). However, if I obtain the gradient of the function, it seems that the $y$-component does not give a finite value if I numerically integrate it with exponent $2$. If I integrate it numerically with exponent $1$ Mathematica gives me the following message:
NIntegrate: ""NIntegrate failed to converge to prescribed accuracy after 18
recursive bisections in y near {x,y} = {0.844124,2.*10^-323}.
NIntegrate obtained -118.498 and 0.010644969997020758` for the
integral and error estimates.""
-118.498
and any other exponent returns even complex numbers. So at this point I am clueless. How can I tell exactly what's the Sobolev space this function lies on?
sobolev-spaces fractional-sobolev-spaces
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have the following function:
$T ( textbf{r}) = frac{q}{4 pi k} int_{s_i=0}^{L_i} frac{1}{left| textbf{r}-textbf{r}'right|} ds_i = frac{q}{4 pi k} logleft( frac{tan(theta_2/2)}{tan(theta_1/2)} right)$
which is the exact temperature field produced by a heat source of magnitude $q$ applied over a line (see figure)
I would like to know what's the space this function lies on. To that end, I set the line source between $(-2,0)--(2,0)$ and I consider my domain as $[0,1]times[0,1]$ (since the solution is symmetric and thus I can multiply by $4$ the result. I decided to use Mathematica to see whether I could integrate the function squared, and indeed I get a finite value, for which I conclude the function is at least in $L^2$ ($H^0$). However, if I obtain the gradient of the function, it seems that the $y$-component does not give a finite value if I numerically integrate it with exponent $2$. If I integrate it numerically with exponent $1$ Mathematica gives me the following message:
NIntegrate: ""NIntegrate failed to converge to prescribed accuracy after 18
recursive bisections in y near {x,y} = {0.844124,2.*10^-323}.
NIntegrate obtained -118.498 and 0.010644969997020758` for the
integral and error estimates.""
-118.498
and any other exponent returns even complex numbers. So at this point I am clueless. How can I tell exactly what's the Sobolev space this function lies on?
sobolev-spaces fractional-sobolev-spaces
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
I have the following function:
$T ( textbf{r}) = frac{q}{4 pi k} int_{s_i=0}^{L_i} frac{1}{left| textbf{r}-textbf{r}'right|} ds_i = frac{q}{4 pi k} logleft( frac{tan(theta_2/2)}{tan(theta_1/2)} right)$
which is the exact temperature field produced by a heat source of magnitude $q$ applied over a line (see figure)
I would like to know what's the space this function lies on. To that end, I set the line source between $(-2,0)--(2,0)$ and I consider my domain as $[0,1]times[0,1]$ (since the solution is symmetric and thus I can multiply by $4$ the result. I decided to use Mathematica to see whether I could integrate the function squared, and indeed I get a finite value, for which I conclude the function is at least in $L^2$ ($H^0$). However, if I obtain the gradient of the function, it seems that the $y$-component does not give a finite value if I numerically integrate it with exponent $2$. If I integrate it numerically with exponent $1$ Mathematica gives me the following message:
NIntegrate: ""NIntegrate failed to converge to prescribed accuracy after 18
recursive bisections in y near {x,y} = {0.844124,2.*10^-323}.
NIntegrate obtained -118.498 and 0.010644969997020758` for the
integral and error estimates.""
-118.498
and any other exponent returns even complex numbers. So at this point I am clueless. How can I tell exactly what's the Sobolev space this function lies on?
sobolev-spaces fractional-sobolev-spaces
sobolev-spaces fractional-sobolev-spaces
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
asked Nov 22 at 16:25
Alejandro Marcos Aragon
1084
1084
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