separation of variables

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separation of variables

[‚sep·ə′rā·shən əv ′ver·ē·ə·bəlz]
(mathematics)
A technique where certain differential equations are rewritten in the form ƒ(x) dx = g (y) dy which is then solvable by integrating both sides of the equation.
A method of solving partial differential equations in which the solution is written in the form of a product of functions, each of which depends on only one of the independent variables; the equation is then arranged so that each of the terms involves only one of the variables and its corresponding function, and each of these terms is then set equal to a constant, resulting in ordinary differential equations. Also known as product-solution method.
References in periodicals archive ?
The solution of the homogeneous equation (8) can be easily found by a method of separation of variables, whereby the above equation can be written as:
[1] successfully determined the first ten natural frequencies of a clamped-free beam with a finite mass at the free end using the standard method of separation of variables. Rama Bhat and Wagner [2] considered the frequencies of a uniform cantilever with an end mass using a power series expansion.
In this paper, we reinforced the method of separation of variables from [1] and developed the forced vibration method from [3].
The Reinforced Method of Separation of Variables. In this section, the natural frequencies and mode shapes of the cantilever model (as shown in Figure 3) will be derived by the reinforced method of separation of variables.
The solutions of (8) subjected to four boundary conditions and two initial conditions can be obtained conveniently by the traditional method of separation of variables. The method regards the response as a superposition of the system eigen-functions multiplied by corresponding time dependent generalized coordinates.
It is obvious that the characteristic equation derived by the reinforced method of separation of variables is equal to the equation obtained by the developed method of forced vibration.
1) by using the method of separation of variables and Fourier series analysis.
However, only Laplace Equation (1) is solved in all sub-domains by using the method of separation of variables.
The study is done as above with solving Equations (1) and (2) by using the method of separation of variables.
In the next section, we illustrate the sinc-convolution algorithm using the method of separation of variables.
Here, we enlist the method of separation of variables for solving the two-dimensional convolution-type integrals defined in (3.8) below, used in the radiosity integral equations of (2.1), and rewritten in the form

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