Tuesday, August 26, 2008

Application of Stable Modulation Scheme for Solving Bernoulli Differential Equation

Solving of Bernoulli differential equation traditionally always is done applies linearization procedure by using Bernoulli transformation function. This paper introduces a new technique of solving the Bernoulli differential equation without using linearization by application of stable modulation scheme.

Application of the method named Stable Modulation Technique (called as SMT) is started by splitting the Bernoulli differential equation to parts of linear and nonlinear, then writes down the solution of nonlinear part in the form of modulation function which its initial value besides played the part of as amplitude A and also is modulated into a phase function F(A). The exact solution of Bernoulli differential equation given in AF(A) formula obtained after replacing the linear solution part into initial value of its nonlinear part solution. In this paper presented the usage of SMT for solving the storage model of magnetic energy into inductor.

I. Introduction
The homogeneous Bernoulli differential equation that commonly called Bernoulli differential equation (BDE) to become as primary model in so many application branches. The BDE is distinguished to the degree of its nonlinearity (n). For instance, the BDE having degree of two commonly applied to model growth of logistic in Biology[1] and the behavior of chaos[2], while for the degree of three (n=3)
the BDE forms Gizbun or quartic equation commonly used to analyze corrosion process[3]. The BDEalso is nonlinear part of Klein Gordon partial differential equation which is the usage widely, among these are in studying the dynamics of elementary particles and stochastic resonances4], the transportation of fluxon[5], the excitation of squeezed laser[6], etc.


As commonly explained in mathematical handbook[7],[8], solving of BDE always is done through linearization procedure as in recommending by Jacob Bernoulli. The transformation from the form of nonlinear to the linear differential equation is performed by using Bernoulli transformation function, and hereinafter solved by using the common method of solving a linear differential equation. Recently, Rohedi[9] has reported verification of the Bernoulli transfomation function, and justify the general solution of Bernoulli differential equation which written in mathematical handbook. At the paper was introduced stable modulation technique (called as SMT) focused to solve BDE of constant coefficients, especially which its solution is started from ordinary point. Rohedi[10] has also reported application of SMT for solving a Ricatti differential equation of constant coefficients which its inhomogeneous term in form of sinusoidal function that also was started from ordinary point. In this paper, applying the SMT is developed to solve BDE for arbitrary value of its linear and nonlinear coefficients, either and also constant valuable and varying as function of its dependent variable. In mathematics, this differential equation is known as the general homogeneous Bernoulli differential
equation.

For Detail Visit http://rohedi.com

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