Cauchy and source problems for an advection-diffusion equation with Atangana–Baleanu derivative on the real line. (January 2019)

• Record Type: Journal Article
• Authors: Avcı, Derya; Yetim, Aylin
• Is Part Of: Chaos, solitons and fractals. Volume 118(2019)
• Page Start: 361
• Page End: 365
• Publication Date: 2019-01
• Subjects: Atangana–Baleanu derivative -- Advection-diffusion -- Fundamental solutions -- Cauchy problem -- Mittag–Leffler kernel; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2018.11.035
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: What is the effect of Atangana–Baleanu derivative on advection-diffusion process? The fundamental solutions to Cauchy and source problems for an advection-diffusion equation equipped with Atangana–Baleanu (AB) derivative are researched. Integral transform techniques can be successfully applied. AB derivative is a remarkable alternative to the Caputo fractional derivative to model an advection-diffusion phenomena Abstract: In this paper, a linear advection-diffusion equation involving Atangana–Baleanu derivative described with the Mittag–Leffler kernel is considered on the real line. Different kinds of diffusive transports in the nature obey the exponential/generalized exponential and Mittag–Leffler functions rather than the power law. By this reality, the current study is devoted to investigate the fundamental solutions of the Cauchy and source problems. For this purpose, Laplace and exponential Fourier transforms are applied. The results are achieved in terms of one and two-parameter Mittag–Leffler functions. The results show that the Atangana–Baleanu derivative is an effective alternative to Caputo derivative to model the diffusion with advection processes because the continuous structure of Mittag–Leffler kernel removes the computational complexities. Thus, it is rather practical to achieve analytical solutions.