1 Answer
The Laplace equation is a second-order partial differential equation given by:
\[
\nabla^2 \phi = 0
\]
where \(\nabla^2\) is the Laplacian operator, and \(\phi\) is a scalar field (such as the potential in electrostatics, temperature distribution, etc.).
The fundamental solution of the Laplace equation is a solution that represents the response of the equation to a point source, often used as a building block for solving more complex problems. This solution is also known as the Green's function for the Laplacian.
In 3 Dimensions (Euclidean space):
In 3D space, the fundamental solution of the Laplace equation, \( G(\mathbf{r}) \), is given by:
\[
G(\mathbf{r}) = \frac{1}{4\pi r}
\]
where \(r\) is the distance from the source point, and \(r = |\mathbf{r}|\).
This solution means that if you place a point source at a point (say, the origin), the potential at a distance \(r\) from the source decays inversely with \(r\).
In 2 Dimensions (Euclidean plane):
In 2D, the fundamental solution takes the form:
\[
G(\mathbf{r}) = -\frac{1}{2\pi} \ln r
\]
where \(r\) is again the distance from the source point.
Explanation:
In summary, the fundamental solution gives the potential generated by a unit point source in a domain satisfying the Laplace equation.
\[
\nabla^2 \phi = 0
\]
where \(\nabla^2\) is the Laplacian operator, and \(\phi\) is a scalar field (such as the potential in electrostatics, temperature distribution, etc.).
The fundamental solution of the Laplace equation is a solution that represents the response of the equation to a point source, often used as a building block for solving more complex problems. This solution is also known as the Green's function for the Laplacian.
In 3 Dimensions (Euclidean space):
In 3D space, the fundamental solution of the Laplace equation, \( G(\mathbf{r}) \), is given by:\[
G(\mathbf{r}) = \frac{1}{4\pi r}
\]
where \(r\) is the distance from the source point, and \(r = |\mathbf{r}|\).
This solution means that if you place a point source at a point (say, the origin), the potential at a distance \(r\) from the source decays inversely with \(r\).
In 2 Dimensions (Euclidean plane):
In 2D, the fundamental solution takes the form:\[
G(\mathbf{r}) = -\frac{1}{2\pi} \ln r
\]
where \(r\) is again the distance from the source point.
Explanation:
- The fundamental solution represents the effect of a point source in the domain where Laplace's equation is applied.
- In physical terms, if \(\phi\) is a potential (like electric potential or temperature), the fundamental solution shows how the potential behaves due to a "point" source in free space.
- This fundamental solution is key in solving problems involving the Laplace equation, as it can be used to construct solutions for more complex configurations (using superposition or Green's function methods).
In summary, the fundamental solution gives the potential generated by a unit point source in a domain satisfying the Laplace equation.