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Why does laplace's equation only apply in limited regions, while Poisson's equation can apply in unbounded domains ?
HallsofIvy said:Where did you get the impression that Laplace's equation only applies in limited regions?
The function satisfying [itex]\nabla^2 f= 0[/itex], with f= 1 on the unit circle, in the region outside the circle is 1/r.
The Laplace equation is a second-order partial differential equation that describes the distribution of a scalar quantity in a given region. It applies in regions where there are no sources or sinks of the scalar quantity and the boundary conditions are well-defined.
No, Laplace equation only applies in limited regions that have well-defined boundaries. In an unbounded region, the boundary conditions cannot be specified and thus, the equation cannot be solved.
Laplace equation is commonly used in electrostatics, heat transfer, fluid mechanics, and potential theory. Some examples of limited regions where it can be applied are a parallel plate capacitor, a rectangular metal plate with fixed temperature boundaries, and a fluid flow between two parallel plates.
Yes, there are limitations to the applicability of Laplace equation. It cannot be used in regions where there are sources or sinks of the scalar quantity, such as in the presence of electric charges or heat sources. It also cannot be used in regions where the boundary conditions are not well-defined or there is a discontinuity in the boundary conditions.
Laplace equation can be solved using various mathematical methods such as separation of variables, Green's function, and numerical techniques. The specific method used depends on the boundary conditions and the geometry of the region. In some cases, analytical solutions may not be possible and approximations or numerical solutions are used.