Fractional Calculus: Theory and Applications

We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ∞ ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined.

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“…Therefore, with the aid of the dominated convergence theorem one obtains from Equations (17) and (18) (λ…”

Section: Resultsmentioning

confidence: 99%

“…Very recent applications concerning Caputo fractional derivative operator are also discussed in [14] by the same authors using a completely different method than Sviridyuk's group (see [15,16]). Some related topics can be found in [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Fractional Cauchy Problems for Infinite Interval Case-II

et al. 2019

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“…From known experimental results, most of the processes associated with complex systems have non‐local dynamics involving long‐memory in time, and the fractional derivative operators do have some of those characteristics. The main advantage of considering fractional models is that these models describe the later state of the system upon all of its preceding states not only upon its current state, and therefore, these are more competent than integer order models 5,6 . Moreover, fractional order derivatives have the elegant property that, if the limit of the fractional parameter tends to integer, they coincide with the classic derivative of that order.…”

Section: Introductionmentioning

confidence: 99%

“…The main advantage of considering fractional models is that these models describe the later state of the system upon all of its preceding states not only upon its current state, and therefore, these are more competent than integer order models. 5,6 Moreover, fractional order derivatives have the elegant property that, if the limit of the fractional parameter tends to integer, they coincide with the classic derivative of that order. For the accurate modeling of physical and engineering processes, the fractional order derivative models and techniques are found to be the best and reliable to the experimental results.…”

Section: Introductionmentioning

confidence: 99%

Numerical investigation with stability analysis of time‐fractional Korteweg‐de Vries equations

Ullah

Butt

Buhader

2020

Math Methods in App Sciences

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In this paper, existing classical Korteweg‐de Vries (KdV) equations are converted into the corresponding time‐fractional KdV equations by using Caputo‐Fabrizio fractional derivative and then solved with appropriate initial conditions by implementing semi‐numerical technique, that is, Laplace transform together with an iterative scheme. The obtained solutions are novel, and previous literature lacks such derivations. The stability of implemented technique is analyzed by applying Banach contraction principle and S‐stable mapping. Efficiency of Caputo‐Fabrizio fractional derivative is exhibited through graphical illustrations, and fractional results are drafted in tabular form for specific values of fractional parameter to validate the numerical investigation.

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“…Fractional models have an important role in many fields of engineering and science, for instance, fluid flows, solute transport, electromagnetic theory, signal processing, biology, economics, physics, and geology, etc. [1][2][3][4][5][6]. Fractional theory has many applications in wireless networks [7,8].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

et al. 2019

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The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

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Fractional Calculus: Theory and Applications

Abstract: Fractional calculus is allowing integrals and derivatives of any positive order (the term fractional is kept only for historical reasons).[...]

Cited by 43 publications

References 18 publications

Fractional Cauchy Problems for Infinite Interval Case-II

Fractional Cauchy Problems for Infinite Interval Case-II

Numerical investigation with stability analysis of time‐fractional Korteweg‐de Vries equations

Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

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