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On the concept of general solution for impulsive differential equations of fractional-orderq∈ (2,3)

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Zhang X., Shu T., Liu Z., Ding W., Peng H., He J.
Open Mathematics
|
2016
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tom 14
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nr 1
452-473
EN
In this paper, we find the formula of general solution for a generalized impulsive differential equations of fractional-order q ∈ (2, 3).
2
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The general solution of impulsive systems with Riemann-Liouville fractional derivatives

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Zhang X., Ding W., Peng H., Liu Z., Shu T.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1125-1137
EN
In this paper, we study a kind of fractional differential system with impulsive effect and find the formula of general solution for the impulsive fractional-order system by analysis of the limit case (as impulse tends to zero). The obtained result shows that the deviation caused by impulses for fractional-order system is undetermined. An example is also provided to illustrate the result.
3
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Oscillation of impulsive conformable fractional differential equations

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Tariboon J., Ntouyas S. K.
Open Mathematics
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2016
|
tom 14
|
nr 1
497-508
EN
In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…. $$\left\{ \begin{array}{l} {t_k}{D^\alpha }\left( {p\left( t \right)\left[ {{t_k}{D^\alpha }x\left( t \right) + r\left( t \right)x\left( t \right)} \right]} \right) + q\left( t \right)x\left( t \right) = 0,\quad t \ge {t_0},\;t \ne {t_k},\\ x\left( {t_k^ + } \right) = {a_k}x(t_k^ - ),\quad {t_k}{D^\alpha }x\left( {t_k^ + } \right) = {b_{k\;{t_{k - 1}}}}{D^\alpha }x(t_k^ - ),\quad \;k = 1,2, \ldots. \end{array} \right.$$ Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.
4
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The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)

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Zhang X., Agarwal P., Liu Z., Peng H.
Open Mathematics
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2015
|
tom 13
|
nr 1
EN
In this paper we consider the generalized impulsive system with Riemann-Liouville fractional-order q ∈ (1,2) and obtained the error of the approximate solution for this impulsive system by analyzing of the limit case (as impulses approach zero), as well as find the formula for a general solution. Furthermore, an example is given to illustrate the importance of our results.
5
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A detailed analysis for the fundamental solution of fractional vibration equation

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Liu L.-L., Duan J.-S.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.
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