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

Derya Avcı, Aylin Yetim


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.

Tam Metin:



Carslaw, H.S., Jaeger, J.C., Conduction of Heat in Solids, Oxford University Press, 1959.

Crank, J., The Mathematics of Diffusion, Oxford Science Publications, (1980).

Ozisik, M.N., Heat Conduction, 2nd edn., Wiley, (1993).

Kaviany, M., Principles of Heat Transfer in Porous Media, 2nd edn., Springer, (1995).

Podlubny, I., Fractional Differential Equations. Academic Press, Inc., San Diego, CA, (1999).

Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)

Uchaikin, V.V., Fractional Derivatives for Physicists and Engineers, Background and Theory, Springer, Berlin, (2013).

Povstenko, Y., Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Heidelberg, Birkhuser, (2015).

Povstenko, Y., Fractional Thermoelasticity, Springer, New York, (2015).

Bulut, H., Baskonus, H.M. and Pandir, Y., The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstract and Applied Analysis, ArticleID 636802, 8 pages, (2013).

Yavuz, M., Özdemir, N., Numerical Inverse Laplace Homotopy Technique for Fractional Heat Equations, Thermal Science, 22(1), 185-194, (2018).

Yavuz, M. and Özdemir, N., A quantitative approach to fractional option pricing problems with decomposition series. Konuralp Journal of Mathematics, 6(1), 102-109, (2018).

Gürbüz, B., and Sezer, M., Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method, Biomath Communications, 6(1), 1-5, (2017).

Gürbüz, B., and Sezer, M., Modified Laguerre collocation method for solving 1-dimensional parabolic convection-diffusion problems, Mathematical Methods in the Applied Sciences, 1-7, (2017).

Sarp, U., Evirgen, F. and Ikikardes, S., Applications of differential transformation method to solve systems of ordinary and partial differential equations, Journal of Balıkesir University Institute of Science and Technology, 20(2), 135-156, (2018).

Povstenko, Y., Theory of diffusive stresses based on the fractional advection-diffusion equation. In: Abi Zeid Daou, R, Moreau, X (eds.) Fractional Calculus: Applications, pp. 227-241. Nova Science Publishers, New York (2015)

Liu, F., Anh, V., Turner, I. and Zhuang, P., Time fractional advection dispersion equation, Journal of Applied Mathematics and Computing, 13(1-2), 233245, (2003).

Huang, F. and Liu, F., The time fractional diffusion equation and the advection-dispersion equation, ANZIAM Journal, 46(3), 317-330, (2005).

Povstenko, Y. and Klekot, J., Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation, Journal of Applied Mathematics and Comutational Mechanics, 13, 1, 95-102, (2014).

Povstenko, Y., Generalized boundary conditions for the time-fractional advection diffusion equation, Entropy, 17, 4028-4039, (2015).

Povstenko, Y., and Klekot, J., The Dirichlet problem for the time-fractional advection-diffusion equation in a line-segment, Boundary Value Problems, 2016, 89 (2016).

Povstenko, Y., and Klekot, J., The Cauchy problem for the time-fractional advection diffusion equation in a layer, Technical Sciences, 19, 3, 231-244, (2016).

Caputo, M. and Fabrizio, M., A New Definition of Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Applications, 1(2), 73-85 (2015).

Losada, J. and Nieto, J.J., Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), 87-92, (2015).

Caputo, M. and Fabrizio, M., Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, Progress in Fractional Differentiation and Applications, 2(2), 1-11, (2016).

Hristov, J., Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, 1, 270-342, (2017).

Baleanu, D. Agheli, B. and Al Qurashi, M.M., Fractional advection differential equation within Caputo and Caputo-Fabrizio derivatives, Advances in Mechanical Engineering, 8, 1-8, (2016).

Rubbab, Q., Mirza, I. A. and Qureshi, M.Z.A., Analytical solutions to the fractional advection diffusion equation with time-dependent pulses on the boundary, AIP Advances, 6, 075318, (2016).

Hristov, J., Transient heat diffusion with a non-singular fading memory: from the cattaneo constitutive equation with Jeffrey’s kernel to the Caputo Fabrizio time-fractional derivative, Thermal Sciences, 20(2), 757-762, (2016).

Singh, J., Kumar, D., Hammouch, Z. and Atangana, A., A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316, 504-515, (2018).

Yavuz, M. and Evirgen, F., An Alternative Approach for Nonlinear Optimization Problem with Caputo-Fabrizio Derivative, ITM Web of Conferences, 22, 01009, (2018).

Atangana, A. and Baleanu, D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sciences, 20(2), 763-769, (2016).

Atangana, A. and Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons and Fractals, 89, 447-454, (2016).

Yavuz, M., Ozdemir, N. and Baskonus, H.M., Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133(6), 215, (2018).

Alqahtani, R.T., Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, Journal of Nonlinear Science and Applications., 9, 3647-3654, (2016).

Hristov, J., On the Atangana-Baleanu derivative and its relation to fading memory concept: The diffusion equation formulation, Trends in theory and applications of fractional derivatives with Mittag-Leffler kernel (edited by José Francisco Gómez, Lizeth Torres and Ricardo Escobar), Springer, (2019).

Mirza, I.A. and Vieru, D., Fundamental solutions to advection-diffusion equation with time-fractional Caputo-Fabrizio derivative, Computers and Mathematics with Applications, 73, 1-10, (2017).


  • Şu halde refbacks yoktur.

Telif Hakkı (c) 2018 DERYA AVCI

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.