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Separation of variables may be possible in some coordinate systems but not others, [2] and which coordinate systems allow for separation depends on the symmetry properties of the equation. [3] Below is an outline of an argument demonstrating the applicability of the method to certain linear equations, although the precise method may differ in ...
Laplace's equation on is an example of a partial differential equation that admits solutions through -separation of variables; in the three-dimensional case this uses 6-sphere coordinates. (This should not be confused with the case of a separable ODE, which refers to a somewhat different class of problems that can be broken into a pair of ...
Download as PDF; Printable version; In other projects ... which can be solved by separation of variables. The result is the Michell solution. 2-dimensional space ...
Here is the time variable expressed in units of length using some characteristic velocity (e.g., speed of light or sound), is a constant originated from the separation of variables, and (,) represents a part of the source term in the initial wave equation that remains after application of the variable-separation procedures (a series coefficient ...
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve. This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals).
Separable differential equation, in which separation of variables is achieved by various means; Separable extension, in field theory, an algebraic field extension; Separable filter, a product of two or more simple filters in image processing; Separable ordinary differential equation, a class of equations that can be separated into a pair of ...
When =, =, we can easily see the separatrix and the two basins of attraction by solving for the trajectories backwards in time.. In the FitzHugh–Nagumo model, when the linear nullcline pierces the cubic nullcline at the left, middle, and right branch once each, the system has a separatrix.
The ML "model" includes a specification of a pdf, which in this case is the pdf of the unknown source signals . Using ML ICA , the objective is to find an unmixing matrix that yields extracted signals y = W x {\displaystyle y=\mathbf {W} x} with a joint pdf as similar as possible to the joint pdf p s {\displaystyle p_{s}} of the unknown source ...