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  • SpringerOpen  (26)
  • 1
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    SpringerOpen
    Publication Date: 2014-11-01
    Description: In this manuscript we investigate the existence of the fractional finite difference equation (FFDE) Δμ−2μx(t)=g(t+μ−1,x(t+μ−1),Δx(t+μ−1)) via the boundary condition x(μ−2)=0 and the sum boundary condition x(μ+b+1)=∑k=μ−1αx(k) for order 1
    Print ISSN: 1687-1839
    Electronic ISSN: 1687-1847
    Topics: Mathematics
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  • 2
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    SpringerOpen
    Publication Date: 2015-07-17
    Description: In this work, we study the fractional order Lane-Emden differential equations by using the reproducing kernel method. The exact solution is shown in the form of a series in the reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of the present method. The results obtained from the method are compared with the exact solutions and another method. The obtained numerical results are better than the ones provided by the collocation method. Results of numerical examples show that the presented method is simple, effective, and easy to use.
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    Electronic ISSN: 1687-1847
    Topics: Mathematics
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  • 3
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    SpringerOpen
    Publication Date: 2015-08-07
    Description: In this paper, we discussed the existence of solutions for the fractional finite difference inclusion Δ ν x ( t ) ∈ F ( t , x ( t ) , Δ x ( t ) , Δ 2 x ( t ) ) via the boundary value conditions ξ x ( ν − 3 ) + β Δ x ( ν − 3 ) = 0 , x ( η ) = 0 , and γ x ( b + ν ) + δ Δ x ( b + ν ) = 0 , where η ∈ N ν − 2 b + ν − 1 , 2 〈 ν 〈 3 , and F : N ν − 3 b + ν + 1 × R × R × R → 2 R is a compact valued multifunction.
    Print ISSN: 1687-1839
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    Topics: Mathematics
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  • 4
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    SpringerOpen
    Publication Date: 2015-10-16
    Description: We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given byD ω ( x ( t ) H ( t , x ( t ) , z ( t ) ) ) = − K 1 ( t , x ( t ) , z ( t ) ) , ω ∈ ( 2 , 3 ] , D ϵ ( z ( t ) G ( t , x ( t ) , z ( t ) ) ) = − K 2 ( t , x ( t ) , z ( t ) ) , ϵ ∈ ( 2 , 3 ] , x ( t ) H ( t , x ( t ) , z ( t ) ) | t = 1 = 0 , D μ ( x ( t ) H ( t , x ( t ) , z ( t ) ) ) | t = δ 1 = 0 , x ( 2 ) ( 0 ) = 0 , z ( t ) G ( t , x ( t ) , z ( t ) ) | t = 1 = 0 , D ν ( z ( t ) G ( t , x ( t ) , z ( t ) ) ) | t = δ 2 = 0 , z ( 2 ) ( 0 ) = 0 ,where t ∈ [ 0 , 1 ] , δ 1 , δ 2 , μ , ν ∈ ( 0 , 1 ) , and D ω , D ϵ , D μ and D ν are Caputo’s fractional derivatives of order ω, ϵ, μ and ν, respectively, K 1 , K 2 ∈ C ( [ 0 , 1 ] × R × R , R ) and G , H ∈ C ( [ 0 , 1 ] × R × R , R − { 0 } ) . We use classical results due to Dhage and Banach’s contraction principle (BCP) for the existence and uniqueness of solutions. For applications of our results, we include examples.
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    Topics: Mathematics
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  • 5
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    SpringerOpen
    Publication Date: 2018-03-12
    Description: We establish analogues of the mean value theorem and Taylor’s theorem for fractional differential operators defined using a Mittag–Leffler kernel. We formulate a new model for the fractional Boussinesq equatio...
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  • 6
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    SpringerOpen
    Publication Date: 2016-09-23
    Description: We investigate in this manuscript an optimal control problem for a fractional wave equation involving the fractional Riemann-Liouville derivative and with missing initial condition. For this purpose, we use th...
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  • 7
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    SpringerOpen
    Publication Date: 2017-02-05
    Description: We apply an iterative reproducing kernel Hilbert space method to get the solutions of fractional Riccati differential equations. The analysis implemented in this work forms a crucial step in the process of dev...
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    Topics: Mathematics
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  • 8
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    SpringerOpen
    Publication Date: 2016-11-06
    Description: This paper presents alternative representations to traditional calculus of the Euler-Lagrangian equations, in the alternative representations these equations contain fractional operators. In this work, we cons...
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  • 9
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    SpringerOpen
    Publication Date: 2016-11-26
    Description: This paper investigates the existence of the unique solution for a Hadamard fractional integral boundary value problem of a Hadamard fractional integro-differential equation with the monotone iterative techniq...
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  • 10
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    SpringerOpen
    Publication Date: 2017-07-08
    Description: In this note we present some corrections to our previous paper (Nisar et al. in Adv. Differ. Equ. 2017:118, 2017).
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    Topics: Mathematics
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