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1
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A remark on local fractional calculus and ordinary derivatives

100%
Almeida R., Guzowska M., Odzijewicz T.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1122-1124
EN
In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.
2
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Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition

100%
Povstenko Y.
Open Mathematics
|
2014
|
tom 12
|
nr 4
611-622
EN
The central symmetric time-fractional heat conduction equation with Caputo derivative of order 0 < α ≤ 2 is considered in a ball under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of values of temperature and values of its normal derivative at the boundary, and the physical condition with the prescribed linear combination of values of temperature and values of the heat flux at the boundary, which is a consequence of Newton’s law of convective heat exchange between a body and the environment. The integral transform technique is used. Numerical results are illustrated graphically.
3
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Fractional BVPs with strong time singularities and the limit properties of their solutions

76%
Staněk S.
Open Mathematics
|
2014
|
tom 12
|
nr 11
1638-1655
EN
In the first part, we investigate the singular BVP $$\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u$$, u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $$\mathcal{H}$$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems $$\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)$$, u(0) = A, u(1) = B, $$\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0$$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.
4
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Integral inequalities involving generalized Erdélyi-Kober fractional integral operators

76%
Baleanu D., Purohit S. D., Prajapati J. C.
Open Mathematics
|
2016
|
tom 14
|
nr 1
89-99
EN
Using the generalized Erdélyi-Kober fractional integrals, an attempt is made to establish certain new fractional integral inequalities, related to the weighted version of the Chebyshev functional. The results given earlier by Purohit and Raina (2013) and Dahmani et al. (2011) are special cases of results obtained in present paper.
5
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A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation

76%
Eghbali N., Kalvandi V., Rassias J. M.
Open Mathematics
|
2016
|
tom 14
|
nr 1
237-246
EN
In this paper, we have presented and studied two types of the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation. We prove that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms by two notions.
6
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lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

76%
Li P., Shang Y.
Open Mathematics
|
2017
|
tom 15
|
nr 1
578-586
EN
Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.
7
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Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

76%
Goufo E. F. D., Mugisha S.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
Classical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters’ fission process.We make use of the theory of strongly continuous solution operators for fractional models (analogues of C0-semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993) to analyze and show existence results for clusters’ splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He’s homotopy perturbation (He, 1999) and Kato’s type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive.
8
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Mild solution of fractional order differential equations with not instantaneous impulses

76%
Li P.-L., Xu C.-J.
Open Mathematics
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2015
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tom 13
|
nr 1
EN
In this paper, we investigate the boundary value problems of fractional order differential equations with not instantaneous impulse. By some fixed-point theorems, the existence results of mild solution are established. At last, one example is also given to illustrate the results.
9
Content available remote

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

76%
Klimek M., Błasik M.
Open Mathematics
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2012
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tom 10
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nr 6
1981-1994
EN
Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.
10
Content available remote

Fractional virus epidemic model on financial networks

64%
Balci M. A.
Open Mathematics
|
2016
|
tom 14
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nr 1
1074-1086
EN
In this study, we present an epidemic model that characterizes the behavior of a financial network of globally operating stock markets. Since the long time series have a global memory effect, we represent our model by using the fractional calculus. This model operates on a network, where vertices are the stock markets and edges are constructed by the correlation distances. Thereafter, we find an analytical solution to commensurate system and use the well-known differential transform method to obtain the solution of incommensurate system of fractional differential equations. Our findings are confirmed and complemented by the data set of the relevant stock markets between 2006 and 2016. Rather than the hypothetical values, we use the Hurst Exponent of each time series to approximate the fraction size and graph theoretical concepts to obtain the variables.
11
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Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain

64%
Jessada T., Ntouyas S. K., Asawasamrit S., Promsakon C.
Open Mathematics
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2017
|
tom 15
|
nr 1
645-666
EN
In this paper, we investigate the existence of positive solutions for Hadamard type fractional differential system with coupled nonlocal fractional integral boundary conditions on an infinite domain. Our analysis relies on Guo-Krasnoselskii’s and Leggett-Williams fixed point theorems. The obtained results are well illustrated with the aid of examples.
12
Content available remote

On the more general Hermite-Hadamard type inequalities for convex functions via conformable fractional integrals

64%
Set E., Mumcu I., Özdemir M. E.
Topological Algebra and its Applications
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2017
|
tom 5
|
nr 1
67-73
13
Content available remote

Hermite-Hadamard Type Inequalities for convex functions via generalized fractional integral operators

64%
Set E., Gözpinar A.
Topological Algebra and its Applications
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2017
|
tom 5
|
nr 1
55-62
EN
In this present work, the authors establish a new integral identity involving generalized fractional integral operators and by using this fractional-type integral identity, obtain some new Hermite-Hadamard type inequalities for functions whose first derivatives in absolute value are convex. Relevant connections of the results presented here with those earlier ones are also pointed out.
14
Content available remote

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

64%
Wang J., Zhou Y., Fečkan M.
Open Mathematics
|
2014
|
tom 12
|
nr 6
911-922
EN
In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.
15
Content available remote

Extended Riemann-Liouville type fractional derivative operator with applications

53%
Agarwal P., Nieto J. J., Luo M.-J.
Open Mathematics
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2017
|
tom 15
|
nr 1
1667-1681
EN
The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.
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