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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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A New Planck’s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram

A New Planck’s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram

This paper reports derivation of a new Planck’s formula of spectral density of black-body radiation, that was originated by modeling the interpolation formula of Planck’s law of obtaining the mean of energy of black-body cavity in 2nd order of Bernoulli equation. The new Planck’s formula is created by means AF(A) diagram of solving arctangent differential equation after transforming the Bernoulli equation into the arctangent differential equation The New Planck’s formula not only contains the terms of the photon energy and the energy difference between two states of the motion of harmonic oscillator (), but also contains both terms of the minimum energy of harmonics oscillator () and the phase differences (ωh2/ωh2/π) as representing the intermodes-orthogonality, hence it can answer why the explanation of black-body radiation has been associated with the harmonic oscillators

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Parameter terpenting untuk menentukan panjang kopling dan formula propagasi medan directional coupler adalah nilai tetapan propagasi efektif medan moda simetri dan moda asimetri. Pada makalah ini dilaporkan formulasi tetapan propagasi efektif dimaksud untuk cahaya modus TE (transverse electric) yang terpandu dalam directional coupler linier dalam bentuk analitis. Formulasi analitis diturunkan dengan menggunakan metode matrik karakteristik pandu gelombang berlapis jamak, dan berlaku untuk kedua moda simetri dan asimetri, baik untuk struktur directional coupler simetri maupun asimetri.

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pattern-of-normal-modes-for-s3cebcm-and-v2955-and-the-power-transfer-between-the-two-waveguides

pattern-of-normal-modes-for-s3cebcm-and-v2955-and-the-power-transfer-between-the-two-waveguides

This paper presents generalization of the dispersion relation for analysis of optical power transfer of transverse-electric (TE) mode in five-layer of linear symmetrical directional-coupler, which consists of two equal waveguides as guiding layers with a gap material sandwiched between a semi-infinite clad and a substrate. The dispersion equation formulated by normalizing the intensity of normal modes which involve phase-shift due to the internal reflection at each boundary of neighboring layers corresponds to the standing-wave form, i.e, contains the terms of overlapping the normal modes in gap region, the evanescent waves into the superstate regions, and effective lateral width of the directional-coupler. This generalized dispersion equation can overcome disability of coupled–mode approximation when the two waveguides get too close, evenly it can also be used for designing a zero-gap coupler. Simulation of beam propagation and analysis of optical power transfer in the linear directional-coupler are also presented.

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Makalah ini menyajikan prosedur dan hasil perhitungan tenaga gerak partikel yang terperangkap dalam sistem tenaga potensial sumur tanjak dengan pendekatan variasional menggunakan fungsi cobaan Gaussian-Hermite. Unjuk kerja pendekatan variasional ini diujicobakan pada sistem potensial sumur tanjak yang profil distribusi tenaga potensialnya mematuhi fungsi kuadrat secant hyperbolic (Sech2). Semua perhitungan dilakukan secara numerik dan diprogram menggunakan Matlab for Windows. Dari perhitungan normalisasi tenaga partikel untuk dua tingkat terendah (keadaan dasar n=0 dan tereksitasi pertama n=1), diketahui bahwa akurasi pendekatan variasional semakin meningkat untuk nilai normalisasi lebar sumur (L) yang semakin besar.

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A guided wave excited along the boundary between linear and nonlinear media known as initial inspiration for developing devices based on Kerr nonlinear optics, such as the nonlinear directional coupler, etc. Two important parameters for such structure are respectively the minimum amplitude of light required for the excitation, and the location of the peak of guided nonlinear boundary wave. Analytical procedure of derivation the two parameters commonly involved the Jacobi’s elliptic functions based on the numerical integration. In order to simplify the calculation procedure, in this paper we introduce optimization procedure based on applying the solitary wave solution for guided field inside the nonlinear media. The simulation of guided wave excitation at the interface between linear and nonlinear media is also presented.

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Pada makalah ini dilaporkan teknik sederhana dalam menormalisasi bentuk persamaan relasi dispersi pandu gelombang slab untuk cahaya yang dirambatkan pada modus transverse magnetic. Langkah normalisasi dilakukan dengan cara menormalisasi semua komponen transversal tetapan propagasi cahaya terhadap faktor reduksi amplitudo medan magnet cahaya yang ter-evanescent ke daerah substrat. Hasil perumusan sesuai dengan formulaai yang didapatkan dengan teknik lain menggunakan pendekatan sinar optik. Teknik normalisasi ini juga efektif digunakan untuk mendapatkan formulasi medan magnet cahaya terpandu dalam
bentuk ternormalisasi.

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