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.
In this study, a linear advection–diffusion equation described by Atangana–Baleanu derivative with non-singular Mittag-Leffler kernel is considered. The Cauchy, Dirichlet and source problems are formulated on the half-line. The main motivation of this work is to find the fundamental solutions of prescribed problems. For this purpose, Laplace transform method with respect to time t and sine/cosine-Fourier transform methods with respect to spatial coordinate x are applied. It is remarkable that the obtained results are quite similar to the existing fundamental solutions of advection–diffusion equation with time-Caputo fractional derivative. Although the results are mathematically similar in both formulations, the AB derivative is a non-singular operator and provides a significant advantage in the computational processes. Therefore, it is preferable to replace the Caputo derivative in modelling such diffusive transports.
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