Why can't we use Poisson Integral Formula to solve the Laplace equation on the pie wedge?The form of the...
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Why can't we use Poisson Integral Formula to solve the Laplace equation on the pie wedge?
The form of the Poisson Integral Formula for a Temperature ProfileEquilibrium solution using polar coordinatesReducing the Laplace equation with inhomogeneous BC's to the Poisson equation with homogeneous BC'sHow to numerically solve the Poisson equation given Neumann boundary conditions?Representation formula for the Laplace equationSolve the Laplace equation on an annular regionUniqueness of the Homogeneous Neumann BC Laplace EquationUsing Poisson kernel to solve heat equationUse of the Poisson Kernel to solve Inhomogeneous Laplace EquationUse principle of minimum maximum for the equation of Laplace
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We know Poisson Integral Formula can solve the Dirichlet boundary value problem for the Laplace equation on the unit disk. But for such a problem as below:
Find the solution to the Dirichlet boundary value problem for the Laplace equation on the pie wedge $displaystyle W=left{0<theta<frac{pi}{4}, quad 0<r<1right}$, when the nonzero boundary data $u(1, theta)=h(theta)$ appears only on the curved portion of its boundary.
Why can't we use Poisson Integral Formula to solve the Laplace equation on this pie wedge? I mean, it seems that the solution shouldn't change when we use a sub-region of the original problem.
pde
edited 17 hours ago
Dylan
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asked yesterday
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$endgroup$
add a comment |
$begingroup$
We know Poisson Integral Formula can solve the Dirichlet boundary value problem for the Laplace equation on the unit disk. But for such a problem as below:
Find the solution to the Dirichlet boundary value problem for the Laplace equation on the pie wedge $displaystyle W=left{0<theta<frac{pi}{4}, quad 0<r<1right}$, when the nonzero boundary data $u(1, theta)=h(theta)$ appears only on the curved portion of its boundary.
Why can't we use Poisson Integral Formula to solve the Laplace equation on this pie wedge? I mean, it seems that the solution shouldn't change when we use a sub-region of the original problem.
pde
edited 17 hours ago
Dylan
13.6k31027
asked yesterday
ThinkpadThinkpad
61
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
You can if there is a Poisson kernal for this domain. Can you find it?
$endgroup$
– Dylan
17 hours ago
add a comment |
$begingroup$
We know Poisson Integral Formula can solve the Dirichlet boundary value problem for the Laplace equation on the unit disk. But for such a problem as below:
Find the solution to the Dirichlet boundary value problem for the Laplace equation on the pie wedge $displaystyle W=left{0<theta<frac{pi}{4}, quad 0<r<1right}$, when the nonzero boundary data $u(1, theta)=h(theta)$ appears only on the curved portion of its boundary.
Why can't we use Poisson Integral Formula to solve the Laplace equation on this pie wedge? I mean, it seems that the solution shouldn't change when we use a sub-region of the original problem.
pde
edited 17 hours ago
Dylan
13.6k31027
asked yesterday
ThinkpadThinkpad
61
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
We know Poisson Integral Formula can solve the Dirichlet boundary value problem for the Laplace equation on the unit disk. But for such a problem as below:
Find the solution to the Dirichlet boundary value problem for the Laplace equation on the pie wedge $displaystyle W=left{0<theta<frac{pi}{4}, quad 0<r<1right}$, when the nonzero boundary data $u(1, theta)=h(theta)$ appears only on the curved portion of its boundary.
Why can't we use Poisson Integral Formula to solve the Laplace equation on this pie wedge? I mean, it seems that the solution shouldn't change when we use a sub-region of the original problem.
pde
pde
edited 17 hours ago
Dylan
13.6k31027
asked yesterday
ThinkpadThinkpad
61
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 17 hours ago
Dylan
13.6k31027
asked yesterday
ThinkpadThinkpad
61
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 17 hours ago
Dylan
13.6k31027
edited 17 hours ago
Dylan
13.6k31027
edited 17 hours ago
Dylan
13.6k31027
13.6k31027
asked yesterday
ThinkpadThinkpad
61
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
ThinkpadThinkpad
61
asked yesterday
ThinkpadThinkpad
61
61
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thinkpad is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
You can if there is a Poisson kernal for this domain. Can you find it?
$endgroup$
– Dylan
17 hours ago
add a comment |
$begingroup$
You can if there is a Poisson kernal for this domain. Can you find it?
$endgroup$
– Dylan
17 hours ago
$begingroup$
You can if there is a Poisson kernal for this domain. Can you find it?
$endgroup$
– Dylan
17 hours ago
$begingroup$
You can if there is a Poisson kernal for this domain. Can you find it?
$endgroup$
– Dylan
17 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The Poisson integral formula is a particular case of Green's representation formula, which uses the Green function of a region (and/or its derivative) to represent the solution to Dirichlet/Neumann problems. Green's function is the solution of your problem with a delta-type source at some generic point in your domain and homogeneous Dirichlet data on the boundary. It depends critically on the shape of the boundary (think of the electrostatic potential created by a unit charge when you impose that certain curve (namely the boundary of your domain) is an equipotential line or surface). Thus, you get a different Poisson kernel for the upper half-plane and for the disk. In one case the $x$ axis is a zero-potential line, in the other one the unit circle is a zero-potential line, etc. In physical terms, you have different distributions of the induced charges on the boundary.
answered 23 hours ago
GReyesGReyes
1,77515
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The Poisson integral formula is a particular case of Green's representation formula, which uses the Green function of a region (and/or its derivative) to represent the solution to Dirichlet/Neumann problems. Green's function is the solution of your problem with a delta-type source at some generic point in your domain and homogeneous Dirichlet data on the boundary. It depends critically on the shape of the boundary (think of the electrostatic potential created by a unit charge when you impose that certain curve (namely the boundary of your domain) is an equipotential line or surface). Thus, you get a different Poisson kernel for the upper half-plane and for the disk. In one case the $x$ axis is a zero-potential line, in the other one the unit circle is a zero-potential line, etc. In physical terms, you have different distributions of the induced charges on the boundary.
answered 23 hours ago
GReyesGReyes
1,77515
$endgroup$
add a comment |
$begingroup$
The Poisson integral formula is a particular case of Green's representation formula, which uses the Green function of a region (and/or its derivative) to represent the solution to Dirichlet/Neumann problems. Green's function is the solution of your problem with a delta-type source at some generic point in your domain and homogeneous Dirichlet data on the boundary. It depends critically on the shape of the boundary (think of the electrostatic potential created by a unit charge when you impose that certain curve (namely the boundary of your domain) is an equipotential line or surface). Thus, you get a different Poisson kernel for the upper half-plane and for the disk. In one case the $x$ axis is a zero-potential line, in the other one the unit circle is a zero-potential line, etc. In physical terms, you have different distributions of the induced charges on the boundary.
answered 23 hours ago
GReyesGReyes
1,77515
$endgroup$
add a comment |
$begingroup$
The Poisson integral formula is a particular case of Green's representation formula, which uses the Green function of a region (and/or its derivative) to represent the solution to Dirichlet/Neumann problems. Green's function is the solution of your problem with a delta-type source at some generic point in your domain and homogeneous Dirichlet data on the boundary. It depends critically on the shape of the boundary (think of the electrostatic potential created by a unit charge when you impose that certain curve (namely the boundary of your domain) is an equipotential line or surface). Thus, you get a different Poisson kernel for the upper half-plane and for the disk. In one case the $x$ axis is a zero-potential line, in the other one the unit circle is a zero-potential line, etc. In physical terms, you have different distributions of the induced charges on the boundary.
answered 23 hours ago
GReyesGReyes
1,77515
$endgroup$
The Poisson integral formula is a particular case of Green's representation formula, which uses the Green function of a region (and/or its derivative) to represent the solution to Dirichlet/Neumann problems. Green's function is the solution of your problem with a delta-type source at some generic point in your domain and homogeneous Dirichlet data on the boundary. It depends critically on the shape of the boundary (think of the electrostatic potential created by a unit charge when you impose that certain curve (namely the boundary of your domain) is an equipotential line or surface). Thus, you get a different Poisson kernel for the upper half-plane and for the disk. In one case the $x$ axis is a zero-potential line, in the other one the unit circle is a zero-potential line, etc. In physical terms, you have different distributions of the induced charges on the boundary.
answered 23 hours ago
GReyesGReyes
1,77515
answered 23 hours ago
GReyesGReyes
1,77515
answered 23 hours ago
GReyesGReyes
1,77515
answered 23 hours ago
GReyesGReyes
1,77515
1,77515
add a comment |
add a comment |
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$begingroup$
You can if there is a Poisson kernal for this domain. Can you find it?
$endgroup$
– Dylan
17 hours ago