A class of fractional evolution equations and optimal controls

Wang, JinRong; Zhou, Yong

doi:10.1016/j.nonrwa.2010.06.013

Cited by 349 publications

(151 citation statements)

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“…Fractional calculus has been widely applied to describing various engineering and physical problems, such as the periodic problem [2], random and disordered media [3,23,26], anomalous diffusion [8,12,31], medicine [11], mechanics [21], control [27,28], and so on. The fractional functional differential equations are an important class of fractional order differential equations.…”

Section: Introductionmentioning

confidence: 99%

Waveform relaxation methods for fractional functional differential equations

Ding

Jiang

2013

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In this paper, we use waveform relaxation method to solve fractional functional differential equations. Under suitable conditions imposed on the so-called splitting functions the convergence results of the waveform relaxation method are given. Delay dependent error estimates for the method are derived. Error bounds for some special cases are considered. Numerical examples illustrate the feasibility and efficiency of the method. It is the first time for applying the method in the fractional functional differential equations.MSC 2010 : Primary 26A33; Secondary 33E12

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Section: Introductionmentioning

confidence: 99%

Waveform relaxation methods for fractional functional differential equations

Ding

Jiang

2013

fcaa

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“…Particular, Wang et al [30,31,32,33,34,35] and Zhou et al [36,37] also introduced two characteristic solutions operators and give a suitable concept on a mild solution by applying Laplace transform and probability density functions and discussed the optimal control problems for some classes of fractional controlled systems.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we continue to combine these two areas and our works [16,17,33,36] to extend the study to the Cauchy problems for impulsive fractional evolution equations:…”

Section: Introductionmentioning

confidence: 99%

On the new concept of solutions and existence results for impulsive fractional evolution equations

Fečkan

Wang

Zhou

et al. 2011

Dynamics of Partial Differential Equations

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Abstract. In this paper we discuss the existence of P C-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative. By utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a P C-mild solution for our problem is introduced. Our main techniques based on fractional calculus and fixed point theorems. Some concrete applications to partial differential equations are considered.

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“…For more details, one can see the monographs of Kilbas et al [6], Lakshmikantham et al [7], Miller and Ross [8], Podlubny [12]. Very recently, fractional differential equations and optimal controls in Banach spaces are studied by Balachandran et al [3,4], N'Guérékata [9,10], Mophou and N'Guérékata [11], Wang et al [13][14][15][16][17][18][19][20], Zhou et al [22][23][24] and etc.…”

Section: Introductionmentioning

confidence: 99%