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Posts Tagged ‘stable modulation technique’

Solution of the Stochastic Differential equation

Solution of the Stochastic Differential equation

The Bernoulli differential equation that has been used as primary modeling in many application branches commonly contains harmonic force function in the inhomogeneous term. The nhomogeneous Bernoulli differential equation (IBDE) is specified by nonlinearity of nth order. For instance, 3rd order IBDE is called a stochastic differential equation that has been commonly applied for describing the corrosion mechanism, the transport of fluxon, the generation of squeezed laser, etc. Due to the difficulty of applying a linearization procedure for IBDE, it has commonly been solved numerically. In this paper we introduced a so-called Stable Modulation Technique (SMT) which is able to solve a first order nonlinear differential equation that transformable into the homogeneous Bernoulli differential equation (BDE). SMT is employed by splitting BDE into linear and nonlinear parts. The solution of its nonlinear part has been found to be AF(A) where A is the initial value of nonlinear solution part and F is the modulation function whose phase is a function of A. The general solution of BDE obtained by substituting the linear solution part into initial value of the nonlinear solution part. Although IBDE can not be transformed into BDE completely, SMT gives nevertheless an approximation solution in AF(A) form, where the homogeneous solution part becomes its amplitude term. The AF(A) formula for stochastic differential equation with cosine function as inhomogeneous term can be decomposed as well into transient and steady state solutions. In addition, a special example of solving 2 ogeneous Bernoulli differential equation (BDE) has en used as a primary modeling scheme in many ation branches. A nonlinear BDE is specified by a nonlinearity of n. Although the stochastic differential equation is the first order differential equation, nevertheless the procedure of obtaining its analytical solution is very complicated as shown in utilizing of BWK (Brillouin, Wenzel, Kramer) and Reversion methods nd order BDE in creating a new Planck’s formula of black-body radiation is also presented.

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  1. New Science Future (http://rohedi.blogspot.com)
  2. ROHEDI Laboratory (http://rohedi.com)

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Solusi Persamaan diferensial Stochastic

Solusi Persamaan diferensial Stochastic

Albert Einstein is the greatest scientist in this century. Because of the greatness, he lifted by his follower as prophet of siences. So didn’t suprise if action of other scientist looked into justify his theory. Masya Allah, this is really abundant. Why? Ya, because in towards the prestisius throne of real sicence, Einstein is not only walks besides do small running („sai“ in term arabic), but he beforehand rides a delman Phytagoras and then takes a rides binomial bus. This statement not without a reason, because in mathematics context, the problem of special relativity only to be form of exploitation from Phytagoras theorem. Greatness of Einstein which is praiseworthy is his excess in exploiting the binomial approach of which is not thinking of other scientist. The fact, his special relativity only applied to movement of object which its speed (v) closing the velocity of light ().

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  1. New Science Future (http://rohedi.blogspot.com)
  2. ROHEDI Laboratory (http://rohedi.com)

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Analytical Solution Of The Ricatti Differential Equation For High Frequency Derived By Using The Stable Modulation Technique

Analytical Solution Of The Ricatti Differential Equation For High Frequency Derived By Using The Stable Modulation Technique

The Ricatti differential equation dy/dt = P(t)y2+Q(t)y+R(t) is a nonlinear differential equation which is of contemporary interest in various fields, including particle dynamics, optics, and petroleum exploration. The Ricatti differential equation is easily solved by numerical methods. But in order to obtain the exact solution in analytical form, the first order of nonlinear inhomogeneous differential equation is commonly convert into a second order linear differential equation by use of a change of the dependent variable. In this paper we introduce the stable modulation technique (SMT) to solve the Ricatti differential equation without of the use linearization procedure. The main principle of the SMT in solving a first order nonlinear differential equation is modulate the solution of the linear part into the initial value of the nonlinear part solution. Important to be stressed here that the solution of nonlinear part must be written in the modulation function, where the initial value acts as amplitude and also including in the total phase shift. For a special case, dy/dt = -by2+ay+Acos(2πft) where a and b of both are constants, while t is variable of the time (in s), frequency f in Herzt (Hz) we find that the analytical solution of the SMT is appropriate with numerical solution espescially for high frequency f≥10 Hz, amplitude value A≤1, and initial value of the y in the range 0.1≤y0≤1. The analytical solution above can be used as trial function when the Ricatti differential equation will be solved by using combination of the modulational instability technique and variational approximation.

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  1. New Science Future (http://rohedi.blogspot.com)
  2. ROHEDI Laboratory (http://rohedi.com)

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