Search results
Results from the WOW.Com Content Network
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 ...
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).
Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts [18] (,) = (), where () is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and () is a function of time only.
For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry. However as a computational tool, the partial differential equations are notoriously complicated to solve except when is it possible to separate the independent variables; in this ...
A simple example is Newton's second law of motion—the relationship between the displacement and the time of an object under ... Separation of variables ...
For example, in three dimensional ... which can be solved by separation of variables. The result is the ... of a biharmonic function in 2 variables can also be ...
The method of separation of variables suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t). For such a function W the partial differential equation becomes X″ / X + Y″ / Y = 1 / c 2 T″ / T . Since the three terms of this equation are functions of x, y, t separately, they must be ...