On Dirichlet Problem of Time-Fractional Advection-Diffusion Equation



  • Changdev P. Jadhav
  • Tanisha Dale
  • Dr. Vaijanath chinchane Deogiri Institute of Engineering and Management Studies, Aurangabad


The significant motivation behind this research article is to utilize a technique depending upon a
certain variant of the integral transform (Fourier and Laplace) to investigate the basic solution for the
Dirichlet problem with constant boundary conditions. The time-fractional derivative one-dimensional,
the equation of advection-diffusion and the Liouville-Caputo fractional derivative in a line fragment are
introduced. We also illustrate the results using graphical representations


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

Povstenko, Y. & Klekot, J. (2016). The Dirichlet problem for the time-fractional advection-diffusion

equation in a line segment, Boundary Value Problems, Article ID 89, 2016.

Metzler, R. & Klafter, J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics

approach, Phys. Rep. 339, 1–77.

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

Springer, Berlin.

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

Povstenko, Y. (2015). Theory of diffusive stresses based on the fractional advection-diffusion equation,

In: Abi Zeid Daou, R.; Moreau, X. (eds.) Fractional Calculus: Applications, Nova Science Publishers,

New York, 227–241 .

Podlubny, I. (1999). Fractional Differential Equations, Academic Press, New York.

Srivastava, H. M. (2021). Some parametric and argument variations of the operators of fractional calculus

and related special functions and integral transformations, J. Nonlinear Convex Anal. 22, 1501–1520.

Srivastava, H. M. (2021). An introductory overview of fractional-calculus operators based upon the FoxWright and related higher transcendental functions, J. Adv. Engrg. Comput. 5, 135–166.

Bracewell, R. N. (1986). The Fourier Transform and Its Applications, McGraw-Hill, New York.

Gupta, V. G., Shrama, B. & Kilicman, A. (2010). A note on fractional Sumudu transform, J. Appl.

Math. 2010, Article ID 154189, 1–9. doi:10.1155/2010/154189.

Goswami, P.& Alqahtani R. T. (2016). Solutions of fractional differential equations by Sumudu transform

and variational iteration methods, J. Nonlinear Sci. Appl. 9, 1944–1951.

Srivastava. H. M. (2021). A survey of some recent developments on higher transcendental functions of

analytic number theory and applied mathematics, Symmetry 13, Article ID 2294, 1–22.

Povstenko, Y. (2005). Fractional heat conduction equation and associated thermal stress J. Therm.

Stresses 28, 83–102.

Povstenko, Y. (2009). Thermoelasticity which uses fractional heat conduction equation, J. Math. Sci.

, 296–305.

Povstenko, Y. (2014). Fundamental solutions to time-fractional advection diffusion equation in a case

of two space variables, Math. Probl. Engrg. 2014, 705364.

Gorenflo, R. & Mainardi, F. (1997). Fractional calculus: Integral and differential equations of fractional

order, In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics,

Springer, Wien, 223–276.

Srivastava, H. M. & Izadi, M.(2022). The Rothe-Newton approach to simulate the variable coefficient

convection-diffusion equations J. Mahani Math. Res. Cent. 11, 141–157.

Srivastava, H.M., Ahmad, H., Ahmad, I., P. Thounthong, P. & Khan, M. N. (2021) Numerical simulation

of 3-D fractional-order convection-diffusion PDE by a local meshless method, Thermal Sci. 25(1A), 347–

Tripathi, V. M., Srivastava, H. M., Singh, H. Swarup, C. & Aggarwal, S. (2021) Mathematical analysis

of non-isothermal reaction-diffusion models arising in spherical catalyst and spherical biocatalyst, Appl.

Sci. 11, Article ID 10423, 1–14.

Izadi, M. & Srivastava, H. M. (2022). An optimized second order numerical scheme applied to the nonlinear Fisher’s reaction-diffusion equation, J. Interdisciplinary Math. 25, 471–492.

Liu, F., Anh, V. V., Turner, I. W.& Zhuang, P. (2003). Time-fractional advection-dispersion equation

J. Appl. Math. Comput. 123, 233-245.



How to Cite

Jadhav, C. P. ., Dale, T., & chinchane, D. V. (2023). On Dirichlet Problem of Time-Fractional Advection-Diffusion Equation. Journal of Fractional Calculus and Nonlinear Systems, 4(2), 1–13. https://doi.org/10.48185/jfcns.v4i2.861