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

Abstract

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-parameter Mittag-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.

Keywords

• Atangana-Baleanu derivative,advection-diffusion equation,Laplace integral transformation,Mittag-Leffler function,fundamental solution

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

Abstract

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-Leffler fonksiyonlarının lineer kombinasyonları olarak elde edilmiştir. Sonuç olarak, kesirli parametrenin ve sürüklenme hızı parametresinin çözümler üzerindeki etkileri bazı açıklayıcı grafikler yardımıyla yorumlanmıştır. 

Keywords

• Atangana-Baleanu türevi,adveksiyon-difüzyon denklemi,Laplace integral dönüşümü,Mittag-Leffler fonksiyonu,temel çözüm

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