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Note on the convergence of fractional conformable diffusion equation with linear source term

Year 2022, Volume: 5 Issue: 3, 387 - 392, 30.09.2022
https://doi.org/10.53006/rna.1144709

Abstract

In this paper, we study the diffusion equation with conformable derivative. The main goal is to prove the convergence of the mild solution to our problem when the order of fractional Laplacian tends to $1^-$. The principal techniques of our paper is based on some useful evaluations for exponential kernels.

Supporting Institution

FPT University HCM

References

  • [1] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
  • [2] A.A. Abdelhakim, J.A. T. Machado, A critical analysis of the conformable derivative, Nonlinear Dynamics, Volume 95, Issue 4, (2019), 3063–3073.
  • [3] W.S. Chung, Fractional Newton mechanics with conformable fractional derivative, Journal of Computational and Applied Mathematics, Volume 290 (2015), Pages 150–158.
  • [4] A. Jaiswal, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach Spaces, Differ. Equ. Dyn. Syst. 27 , no. 1-3, (2019), 313–325.
  • [5] V.F. Morales-Delgado, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, M.A. Taneco-Hernández, Fractional conformable derivatives of Liouville-Caputo type with low-fractionality, Physica A: Statistical Mechanics and its Applications, Volume 503 (2018), 424–438.
  • [6] S. He, K. Sun, X. Mei, B. Yan, S. Xu, Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative, Eur. Phys. J. Plus, (2017) 132: 36. https://doi.org/10.1140/epjp/i2017-11306-3.
  • [7] F.M. Alharbi, D. Baleanu, A. Ebaid, Physical properties of the projectile motion using the conformable derivative, Chinese Journal of Physics, Volume 58, (2019), Pages 18–28.
  • [8] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66.
  • [9] A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889–898.
  • [10] Y. Çenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers’ type equations with conformable derivative, Waves Random Complex Media, 27 (2017), no. 1, 103–116.
  • [11] N.H. Tuan, T.B. Ngoc, D. Baleanu, D. O’Regan, On well-posedness of the sub-diffusion equation with conformable derivative model, Communications in Nonlinear Science and Numerical Simulation Volume 89, October 2020, 105332.
  • [12] Y. Çakmak, Inverse nodal problem for a conformable fractional diffusion operator, Inverse Probl. Sci. Eng. 29 (2021), no. 9, 1308–1322.
  • [13] A.M. Bayrak, A. Demir, E. Ozbilge, On the numerical solution of conformable fractional diffusion problem with small delay, Numer. Methods Partial Differential Equations 38 (2022), no. 2, 177–189.
  • [14] A. Jaiswal, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach Spaces, Differ. Equ. Dyn. Syst. 27 (2019), no. 1-3, 313–325.
  • [15] E. Karapinar, A.Fulga,M. Rashid, L.Shahid, H. Aydi, Large Contractions on Quasi-Metrics Spaces with a Application to Nonlinear Fractional Differential-Equations, Mathematics 2019, 7, 444.
  • [16] E.Karapinar, Ho Duy Binh, Nguyen Hoang Luc, and Nguyen Huu Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Difference Equations (2021) 2021:70.
  • [17] J. E. Lazreg, S. Abbas, M. Benchohra, and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces , Open Mathematics 2021; 19: 363-372.
  • [18] N. D. Phuong, Note on a Allen-Cahn equation with Caputo-Fabrizio derivative, Results in Nonlinear Analysis 4 (2021), 179–185.
  • [19] N. D. Phuong, N. H. Luc and L. D. Long, Modified Quasi Boundary Value method for inverse source problem of the bi-parabolic equation, Advances in the Theory of Nonlinear Analysis and its Applications 4 (2020), 132–142.
Year 2022, Volume: 5 Issue: 3, 387 - 392, 30.09.2022
https://doi.org/10.53006/rna.1144709

Abstract

References

  • [1] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
  • [2] A.A. Abdelhakim, J.A. T. Machado, A critical analysis of the conformable derivative, Nonlinear Dynamics, Volume 95, Issue 4, (2019), 3063–3073.
  • [3] W.S. Chung, Fractional Newton mechanics with conformable fractional derivative, Journal of Computational and Applied Mathematics, Volume 290 (2015), Pages 150–158.
  • [4] A. Jaiswal, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach Spaces, Differ. Equ. Dyn. Syst. 27 , no. 1-3, (2019), 313–325.
  • [5] V.F. Morales-Delgado, J.F. Gómez-Aguilar, R.F. Escobar-Jiménez, M.A. Taneco-Hernández, Fractional conformable derivatives of Liouville-Caputo type with low-fractionality, Physica A: Statistical Mechanics and its Applications, Volume 503 (2018), 424–438.
  • [6] S. He, K. Sun, X. Mei, B. Yan, S. Xu, Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative, Eur. Phys. J. Plus, (2017) 132: 36. https://doi.org/10.1140/epjp/i2017-11306-3.
  • [7] F.M. Alharbi, D. Baleanu, A. Ebaid, Physical properties of the projectile motion using the conformable derivative, Chinese Journal of Physics, Volume 58, (2019), Pages 18–28.
  • [8] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66.
  • [9] A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889–898.
  • [10] Y. Çenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers’ type equations with conformable derivative, Waves Random Complex Media, 27 (2017), no. 1, 103–116.
  • [11] N.H. Tuan, T.B. Ngoc, D. Baleanu, D. O’Regan, On well-posedness of the sub-diffusion equation with conformable derivative model, Communications in Nonlinear Science and Numerical Simulation Volume 89, October 2020, 105332.
  • [12] Y. Çakmak, Inverse nodal problem for a conformable fractional diffusion operator, Inverse Probl. Sci. Eng. 29 (2021), no. 9, 1308–1322.
  • [13] A.M. Bayrak, A. Demir, E. Ozbilge, On the numerical solution of conformable fractional diffusion problem with small delay, Numer. Methods Partial Differential Equations 38 (2022), no. 2, 177–189.
  • [14] A. Jaiswal, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach Spaces, Differ. Equ. Dyn. Syst. 27 (2019), no. 1-3, 313–325.
  • [15] E. Karapinar, A.Fulga,M. Rashid, L.Shahid, H. Aydi, Large Contractions on Quasi-Metrics Spaces with a Application to Nonlinear Fractional Differential-Equations, Mathematics 2019, 7, 444.
  • [16] E.Karapinar, Ho Duy Binh, Nguyen Hoang Luc, and Nguyen Huu Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Difference Equations (2021) 2021:70.
  • [17] J. E. Lazreg, S. Abbas, M. Benchohra, and E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces , Open Mathematics 2021; 19: 363-372.
  • [18] N. D. Phuong, Note on a Allen-Cahn equation with Caputo-Fabrizio derivative, Results in Nonlinear Analysis 4 (2021), 179–185.
  • [19] N. D. Phuong, N. H. Luc and L. D. Long, Modified Quasi Boundary Value method for inverse source problem of the bi-parabolic equation, Advances in the Theory of Nonlinear Analysis and its Applications 4 (2020), 132–142.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Tien Nguyen 0000-0002-0975-9131

Publication Date September 30, 2022
Published in Issue Year 2022 Volume: 5 Issue: 3

Cite

APA Nguyen, T. (2022). Note on the convergence of fractional conformable diffusion equation with linear source term. Results in Nonlinear Analysis, 5(3), 387-392. https://doi.org/10.53006/rna.1144709
AMA Nguyen T. Note on the convergence of fractional conformable diffusion equation with linear source term. RNA. September 2022;5(3):387-392. doi:10.53006/rna.1144709
Chicago Nguyen, Tien. “Note on the Convergence of Fractional Conformable Diffusion Equation With Linear Source Term”. Results in Nonlinear Analysis 5, no. 3 (September 2022): 387-92. https://doi.org/10.53006/rna.1144709.
EndNote Nguyen T (September 1, 2022) Note on the convergence of fractional conformable diffusion equation with linear source term. Results in Nonlinear Analysis 5 3 387–392.
IEEE T. Nguyen, “Note on the convergence of fractional conformable diffusion equation with linear source term”, RNA, vol. 5, no. 3, pp. 387–392, 2022, doi: 10.53006/rna.1144709.
ISNAD Nguyen, Tien. “Note on the Convergence of Fractional Conformable Diffusion Equation With Linear Source Term”. Results in Nonlinear Analysis 5/3 (September 2022), 387-392. https://doi.org/10.53006/rna.1144709.
JAMA Nguyen T. Note on the convergence of fractional conformable diffusion equation with linear source term. RNA. 2022;5:387–392.
MLA Nguyen, Tien. “Note on the Convergence of Fractional Conformable Diffusion Equation With Linear Source Term”. Results in Nonlinear Analysis, vol. 5, no. 3, 2022, pp. 387-92, doi:10.53006/rna.1144709.
Vancouver Nguyen T. Note on the convergence of fractional conformable diffusion equation with linear source term. RNA. 2022;5(3):387-92.