Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation

Povstenko, Yuriy; Klekot, Joanna

doi:10.17512/jamcm.2014.1.10

Cited by 8 publications

(4 citation statements)

References 14 publications

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“…The first term ( ) t δ is a natural result also encountered in the fundamental solutions arising from the ADE model with CF derivative [37]. Moreover, it is a remarkable difference between the fundamental solutions of ADE models with Caputo fractional derivative [19][20][21][22], and the CF models [28,37] with the current AB model for ADE. By using the definition ( ) 0 for 0 t t δ = > and taking the limit for α γ = → ∞ 1, i.e.…”

Section: Casementioning

confidence: 97%

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Sonlu bir bölge üzerinde Atangana-Baleanu türevli adveksiyon-difüzyon denklemine analitik çözümler

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: Casementioning

confidence: 97%

“…The analytical solutions of onedimensional fractional advection-diffusion equation have been obtained in terms of Hfunction [17,18]. The fundamental solutions to time-fractional advection-diffusion equation have been analyzed in different domains by using integral transform techniques [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Sonlu bir bölge üzerinde Atangana-Baleanu türevli adveksiyon-difüzyon denklemine analitik çözümler

Avcı¹,

Yetim²

2018

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…The comprehensive survey of literature on the fractional advection--diffusion equation can be found in [11]. In the previous paper [12] the fundamental solution to the Cauchy problem for time-fractional advection-diffusion equation with one spatial variable was obtained in the domain x −∞ < < ∞. In the present paper we study the Dirichlet problem for this equation in a half-line 0 x < < ∞.…”

Section: Introductionmentioning

confidence: 94%

The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space

Povstenko¹,

Klekot²

2015

J. Appl. Math. Comput. Mech.

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“…References [2][3][4] provide information about fractional calculus, fractional differential equations, approximations of fractional derivatives and numerical methods. There are also a lot of articles about fractional calculus-for example, [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation

et al. 2017

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Abstract:The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction inverse problem. Additional information for the inverse problem was the temperature measurements obtained from porous aluminum. In this paper, the authors used a finite difference method to solve direct problems and the Real Ant Colony Optimization algorithm to find a minimum of certain functional (solve the inverse problem). Finally, the authors present the temperature values computed from the model and compare them with the measured data from real objects.

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Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation

Abstract: Abstract. The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered. The fundamental solution to the Cauchy problem is obtained using the integral transform technique. The numerical results are illustrated graphically.

Cited by 8 publications

References 14 publications

Sonlu bir bölge üzerinde Atangana-Baleanu türevli adveksiyon-difüzyon denklemine analitik çözümler

Sonlu bir bölge üzerinde Atangana-Baleanu türevli adveksiyon-difüzyon denklemine analitik çözümler

The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space

Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation

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