Undetermined coefficients for local fractional differential equations

Khalil, Roshdi; Horani, Mohammed Al; Anderson, Douglas R.

doi:10.22436/jmcs.016.02.02

Cited by 29 publications

(17 citation statements)

References 4 publications

(5 reference statements)

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“…The conformable time-scale fractional calculus of order 0 < < 1 is introduced in [14] and has been used to develop the fractional differentiation and fractional integration. After then, many authors got interested in this type of derivatives for their many nice behaviors [10,[15][16][17][18]. Motivated by the need of some new fractional derivatives with nice properties and that can be applied to more real world modeling, some authors introduced very recently new kinds of fractional derivatives whose kernel is nonsingular.…”

Section: Introductionmentioning

confidence: 99%

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Al-Refai

Abdeljawad

2017

Complexity

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We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

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Section: Introductionmentioning

confidence: 99%

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Al-Refai

Abdeljawad

2017

Complexity

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“…In this section, the dimensional system is first transformed to dimensionless form using non-similarity variables to reduce the number of variables and get rid of units. The dimensionless system is then artificially converted to time-fractional form or generalized form using the Caputo-Fabrizio fractional operator (see [2], p. [1][2][3][4][5][6][7][8][9][10][11][12][13]. It is worth to mention here that the fractional models are more general and convenient in the description of flow behavior and memory effect.…”

Section: Generalization Of Local Modelmentioning

confidence: 99%

“…Atangana and Baleanu developed fractional operators in the Caputo and Riemann-Liouville sense using the generalized Mittag-Leffler law E α (-φx α ) as a kernel (see [3], p. 763-769). All these fractional operators have some shortcomings and challenges but at the same time this area is growing fast, and researchers devoted their attention to this field (see [4][5][6][7][8][9][10] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms

2019

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This article deals with the generalization of natural convection flow of Cu -Al 2 O 3 -H 2 O hybrid nanofluid in two infinite vertical parallel plates. To demonstrate the flow phenomena in two parallel plates of hybrid nanofluids, the Brinkman type fluid model together with the energy equation is considered. The Caputo-Fabrizio fractional derivative and the Laplace transform technique are used to developed exact analytical solutions for velocity and temperature profiles. The general solutions for velocity and temperature profiles are brought into light through numerical computation and graphical representation. The obtained results show that the velocity and temperature profiles show dual behaviors for 0 < α < 1 and 0 < β < 1 where α and β are the fractional parameters. It is noticed that, for a shorter time, the velocity and temperature distributions decrease with increasing values of the fractional parameters, whereas the trend reverses for a longer time. Moreover, it is found that the velocity and temperature profiles oppositely behave for the volume fraction of hybrid nanofluids. MSC: 34A08; 76R10

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“…Moreover, the conformable partial derivative of the order α ∈ of the real value of several variables and conformable gradient vector are defined; and a conformable version of Clairaut's theorem for partial derivatives of conformable fractional orders is proved . In short time, many studies about theory and application of the fractional differential equations are based on this new fractional derivative definition.…”

Section: Introductionmentioning

confidence: 99%