• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.221-228
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• 2013

The method of upper and lower solutions and the generalized quasilinearization technique is developed for the existence and approximation of solutions to boundary value problems for higher order fractional differential equations of the type $^c\mathcal{D}^qu(t)+f(t,u(t))=0$, $t{\in}(0,1),q{\in}(n-1,n],n{\geq}2$ $u^{\prime}(0)=0,u^{\prime\prime}(0)=0,{\ldots},u^{n-1}(0)=0,u(1)={\xi}u({\eta})$, where ${\xi},{\eta}{\in}(0,1)$, the nonlinear function f is assumed to be continuous and $^c\mathcal{D}^q$ is the fractional derivative in the sense of Caputo. Existence of solution is established via the upper and lower solutions method and approximation of solutions uses the generalized quasilinearization technique.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.593-601
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• 2015

Let $f{\in}C^r$ ([-1,1]), $r{\geq}0$ and let $L^*$ be a linear left fractional differential operator such that $L^*$ $(f){\geq}0$ throughout [0, 1]. We can find a sequence of polynomials $Q_n$ of degree ${\leq}n$ such that $L^*$ $(Q_n){\geq}0$ over [0, 1], furthermore f is approximated left fractionally and simulta-neously by $Q_n$ on [-1, 1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for $f^{(r)}$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1869-1889
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• 2018

We are concerned with the following fractional p-Laplacian inclusion: $$(-{\Delta})^s_pu+V(x){\mid}u{\mid}^{p-2}u{\in}{\lambda}[{\underline{f}}(x,u(x)),\;{\bar{f}}(s,u(x))]$$ in ${\mathbb{R}}^N$, where $(-{\Delta})^s_p$ is the fractional p-Laplacian operator, 0 < s < 1 < p < $+{\infty}$, sp < N, and $f:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity f admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-{\Delta})^s_pu$ is replaced by any non-local integro-differential operator.

• Proceedings of the IEEK Conference
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• 2006.06a
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• pp.519-520
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• 2006

In this paper, we propose a voltage-controlled ring oscillator (VCO) for a 900 MHz low-noise fractional-N frequency synthesizer. The VCO delay cell is based on an nMOS source-coupled pair with load elements [1] and a combined tail current sources which consist of a large and a small current source to control the integer and fractional behaviors, respectively. The Spectre simulation results of the scheme in a 0.18um CMOS process show the accurate control of the KVCO better than the conventional one.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.215-228
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• 2007

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.811-819
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• 2023

In this research we have used the definition of standard fractional vector cross product to obtain fractional curl and fractional field of a standing wave, a travelling wave, a transverse wave, a vector field in xy plane, a complex vector field and an electric field. Fractional curl and fractional field for a complex order are also discussed. We have supported the study with calculation of impedance at γ = 0, 0 < γ < 1, γ = 1. The formula discussed in this paper are useful for study of polarization, reflection, impedance, boundary conditions where fractional solutions have applications.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1157-1163
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• 2009

Let G be a graph with vertex set V(G) and let f be a nonnegative integer-valued function defined on V(G). A spanning subgraph F of G is called a fractional f-factor if $d^h_G$(x)=f(x) for all x $\in$ for all x $\in$ V (G), where $d^h_G$ (x) = ${\Sigma}_{e{\in}E_x}$ h(e) is the fractional degree of x $\in$ V(F) with $E_x$ = {e : e = xy $\in$ E|G|}. In this paper it is proved that if ${\delta}(G){\geq}{\frac{b^2(k-1)}{a}},\;n>\frac{(a+b)(k(a+b)-2)}{a}$ and $|N_G(x_1){\cup}N_G(x_2){\cup}{\cdots}{\cup}N_G(x_k)|{\geq}\frac{bn}{a+b}$ for any independent subset ${x_1,x_2,...,x_k}$ of V(G), then G has a fractional f-factor. Where k $\geq$ 2 be a positive integer not larger than the independence number of G, a and b are integers such that 1 $\leq$ a $\leq$ f(x) $\leq$ b for every x $\in$ V(G). Furthermore, we show that the result is best possible in some sense.

• Proceedings of the KIEE Conference
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• 2008.10a
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• pp.161-162
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• 2008

본 논문에서는 820.11n 규격에 적합한 Fractional-N 주파수 합성기를 설계하였다. 본 논문에서 설계한 주파수 합성기의 특징은 PFD(Phase Frequency Detector) 뒷단에 잔여 펄스를 제거하는 Pulse Remover를 연결하여 이중 궤환 Charge Pump의 안정도를 향상시켰으며, Charge Pump에서 동시에 발생하는 Up/Down 전류로 인한 Spike성 전류를 없앰으로서 스퓨리어스를 최소화 시켰다. Pulse Removed RFD를 사용함으로서 발생하는 PFD Deadzon문제는 2N+2분주와 2N-2분주기를 3차의 ${\Delta}{\Sigma}$ Modulator가 선택해줌으로 해결하였다. 삼성 0.18u 공정을 이용하여 설계 하였으며 각 블록은 Cadence spectre를 이용하여 검증하였다.

• Proceedings of the KIEE Conference
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• 2008.07a
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• pp.1386-1388
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• 2008

본 논문에서는 820.11n 규격에 적합한 Fractional-N 주파수 합성기를 설계하였다. 본 논문에서 설계한 주파수 합성기의 특징은 PFD(Phase Frequency Detector) 뒷단에 잔여 펄스를 제거하는 Pulse Remover를 연결하여 이중 궤환 Charge Pump의 안정도를 향상시켰으며, Charge Pump에서 동시에 발생하는 Up/Down 전류로 인한 Spike성 전류를 없앰으로서 스퓨리어스를 최소화 시켰다. Pulse Removed PFD를 사용함으로서 발생하는 PFD Deadzon문제는 2N+2분주와 2N-2분주기를 3차의 ${\Delta}{\Sigma}$ Modulator가 선택해줌으로 해결하였다. 삼성 0.18u 공정을 이용하여 설계 하였으며 각 블락은 Cadence spectre 를 이용하여 검증하였다.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.31-48
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• 2007

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on bypassing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable function $f(x+h)=E_{\alpha}(h^{\alpha}{D_x^{\alpha})f(x)$.