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  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  and developed the forced vibration method from .
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