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Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions

Year 2023, Volume: 11 Issue: 2, 148 - 154, 31.10.2023

Süleyman Çetinkaya Ali Demir

Abstract

In this research, we discuss the construction of analytic solution of homogenous initial boundary value problem including PDEs of fractional order. Since homogenous initial boundary value problem involves local fractional order derivative, it has classical initial and boundary conditions. By means of separation of variables method and the inner product defined on $L^2\left[0,l\right]$, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in local sense used in this study. Illustrative example presents the applicability and influence of separation of variables method on fractional mathematical problems.

Keywords

Local Fractional Derivative Dirichlet boundary conditions Spectral method Separation of variables

References

  • [1] D. Baleanu, A. Fernandez, A. Akg¨ul, On a Fractional Operator Combining Proportional and Classical Differintegrals, Mathematics Vol. 8, No. 360 (2020).
  • [2] J. Bisquert, Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Physical Review E Vol. 72, (2005), 011109.
  • [3] N. Sene, Solutions of Fractional Diffusion Equations and Cattaneo-Hristov Diffusion Model, International Journal of Analysis and Applications Vol. 17, No. 2 (2019), 191–207.
  • [4] J. F. G. Aguilar, M. M. Hernandez, Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative, Abstract and Applied Analysis, Vol. 2014, (2014), Article ID 283019.
  • [5] M. Naber, Distributed order fractional sub-diffusion, Fractals Vol. 12, No. 1 (2004), 23–32.
  • [6] E. Nadal, E. Abisset-Chavanne, E. Cueto, F. Chinesta, On the physical interpretation of fractional diffusion, Comptes Rendus Mecanique, Vol. 346 (2018), 581-589.
  • [7] W. Zhang and M. Yi, Sturm-Liouville problem and numerical method of fractional diffusion equation on fractals, Advances in Difference Equations Vol. 2016, No. 217 (2016).

Year 2023, Volume: 11 Issue: 2, 148 - 154, 31.10.2023

Süleyman Çetinkaya Ali Demir

Abstract

References

  • [1] D. Baleanu, A. Fernandez, A. Akg¨ul, On a Fractional Operator Combining Proportional and Classical Differintegrals, Mathematics Vol. 8, No. 360 (2020).
  • [2] J. Bisquert, Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Physical Review E Vol. 72, (2005), 011109.
  • [3] N. Sene, Solutions of Fractional Diffusion Equations and Cattaneo-Hristov Diffusion Model, International Journal of Analysis and Applications Vol. 17, No. 2 (2019), 191–207.
  • [4] J. F. G. Aguilar, M. M. Hernandez, Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative, Abstract and Applied Analysis, Vol. 2014, (2014), Article ID 283019.
  • [5] M. Naber, Distributed order fractional sub-diffusion, Fractals Vol. 12, No. 1 (2004), 23–32.
  • [6] E. Nadal, E. Abisset-Chavanne, E. Cueto, F. Chinesta, On the physical interpretation of fractional diffusion, Comptes Rendus Mecanique, Vol. 346 (2018), 581-589.
  • [7] W. Zhang and M. Yi, Sturm-Liouville problem and numerical method of fractional diffusion equation on fractals, Advances in Difference Equations Vol. 2016, No. 217 (2016).
There are 7 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Articles
Authors

Süleyman Çetinkaya

Ali Demir

Publication Date October 31, 2023
Submission Date October 21, 2020
Acceptance Date July 5, 2023
Published in Issue Year 2023 Volume: 11 Issue: 2

Cite

APA Çetinkaya, S., & Demir, A. (2023). Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions. Konuralp Journal of Mathematics, 11(2), 148-154.
AMA Çetinkaya S, Demir A. Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions. Konuralp J. Math. October 2023;11(2):148-154.
Chicago Çetinkaya, Süleyman, and Ali Demir. “Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions”. Konuralp Journal of Mathematics 11, no. 2 (October 2023): 148-54.
EndNote Çetinkaya S, Demir A (October 1, 2023) Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions. Konuralp Journal of Mathematics 11 2 148–154.
IEEE S. Çetinkaya and A. Demir, “Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions”, Konuralp J. Math., vol. 11, no. 2, pp. 148–154, 2023.
ISNAD Çetinkaya, Süleyman - Demir, Ali. “Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions”. Konuralp Journal of Mathematics 11/2 (October 2023), 148-154.
JAMA Çetinkaya S, Demir A. Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions. Konuralp J. Math. 2023;11:148–154.
MLA Çetinkaya, Süleyman and Ali Demir. “Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions”. Konuralp Journal of Mathematics, vol. 11, no. 2, 2023, pp. 148-54.
Vancouver Çetinkaya S, Demir A. Diffusion Equation Including Local Fractional Derivative and Dirichlet Boundary Conditions. Konuralp J. Math. 2023;11(2):148-54.
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