Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives

Băleanu, Dumitru; Agheli, Bahram; Qurashi, Maysaa Mohamed Al

doi:10.1177/1687814016683305

Cited by 45 publications

(11 citation statements)

References 18 publications

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“…In order to compare the effects of Caputo and Caputo -Fabrizio derivatives on an advection partial differential equation, Baleanu et al [27] gave a detailed analysis by using some iterative techniques. Rubbab et al [28] studied the analytical solutions of the Dirichlet problem for an ADE with CF derivative.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions to the advection-diffusion equation with the Atangana-Baleanu derivative over a finite domain

Avcı¹,

Yetim²

2018

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this paper, an advection-diffusion equation with Atangana-Baleanu derivative is considered. Cauchy and Dirichlet problems have been described on a finite interval. The main aim is to scrutinize the fundamental solutions for the prescribed problems. The Laplace and the finite sin-Fourier integral transformation techniques are applied to determine the concentration profiles corresponding to the fundamental solutions. Results have been obtained as linear combinations of one or bi-parameterMittag-Leffler functions. Consequently, the effects of the fractional parameter and drift velocity parameter on the fundamental solutions are interpreted by the help of some illustrative graphics. Sonlu bir bölge üzerinde Atangana-Baleanu türevli adveksiyondifüzyon denklemine analitik çözümler Özet Bu çalışmada Atangana-Baleanu türevli bir adveksiyon-difüzyon denklemi ele alınmıştır. Cauchy ve Dirichlet problemleri sonlu bir aralıkta tanımlanmıştır. Asıl amaç, belirlenen problemler için temel çözümleri irdelemektir. Temel çözümlere karşılık gelen konsantrasyon profillerini belirlemek için Laplace ve sonlu sin-Fourier integral dönüşüm teknikleri uygulanmıştır. Sonuçlar, bir veya iki parametreli Mittag-* Derya AVCI, dkaradeniz@balikesir.edu.tr, https://orcid.org/0000-0003-3662-0474 Aylin YETİM, ayetim72@gmail.com, https://orcid.org/0000-0002-6961-9114 Leffler fonksiyonlarının lineer kombinasyonları olarak elde edilmiştir. Sonuç olarak, kesirli parametrenin ve sürüklenme hızı parametresinin çözümler üzerindeki etkileri bazı açıklayıcı grafikler yardımıyla yorumlanmıştır. Anahtar Kelimeler: Atangana-Baleanu türevi, adveksiyon-difüzyon denklemi, Laplace integral dönüşümü, Mittag-Leffler fonksiyonu, temel çözüm.

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Section: Introductionmentioning

confidence: 99%

Analytical solutions to the advection-diffusion equation with the Atangana-Baleanu derivative over a finite domain

Avcı¹,

Yetim²

2018

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…The aim of this chapter is to prove the uniqueness of solutions to the system (10). So the uniqueness of the solution is presented as below,…”

Section: Uniqueness Of Solutionmentioning

confidence: 99%

Analysis of Keller-Segel model with Atangana-Baleanu fractional derivative

Dokuyucu¹,

Băleanu²,

Çelık³

2018

Filomat

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The new definition of the fractional derivative was defined by Atangana and Baleanu in 2016. They used the generalized Mittag-Leffler function with the non-singular and non-local kernel. Further, their version provides all properties of fractional derivatives. Our aim is to analyse the Keller-Segel model with Caputo and Atangana-Baleanu fractional derivative in Caputo sense. Using fixed point theory, we first show the existence of coupled solutions. We then examine the uniqueness of these solutions. Finally, we compare our results numerically by modifying our model according to both definitions, and we demonstrate these results on the graphs in detail. All computations were done using Mathematica.

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“…The reason of introducing this new type of derivative was to search for fractional derivative with nonsingular kernel and without the Gamma function. Since then many researchers discussed this and applied this new fractional derivative to several real world phenomena and excellent results were reported [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. On the other side the discrete version of fractional derivative is one of the interesting topics nowadays [19,[36][37][38][39][40].…”

Section: )mentioning

confidence: 99%