Home Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
Article
Licensed
Unlicensed Requires Authentication

Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations

  • Mokhtar Kirane EMAIL logo and Berikbol T. Torebek
Published/Copyright: May 11, 2019

Abstract

In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.

Acknowledgements

The second author was financially supported by a Grant No. AP05131756 from the Ministry of Science and Education of the Republic of Kazakhstan. No new data was collected or generated during the course of research.

References

[1] S. Abbas, M. Benchohra, N. Hamidi, J. Henderson, Caputo-Hadamard fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 21, No 4 (2018), 1027–1045; /10.1515/fca-2018-0056; https://www.degruyter.com/view/j/fca.2018.21.issue-4/fca-2018-0056/fca-2018-0056.xml.Search in Google Scholar

[2] A.A. Alikhanov, A time-fractional diffusion equation with generalized memory kernel in differential and difference settings with smooth solutions. Comp Meth. App. Math. 17, No 4 (2017), 647–660.10.1515/cmam-2017-0035Search in Google Scholar

[3] M. Al-Refai, On the fractional derivatives at extreme points. Electr. J. Qual. Theory Diff. Eq. 2012, No 55 (2012), 1–5.Search in Google Scholar

[4] M. Al-Refai, Y. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications. Fract. Calc. Appl. Anal. 17, No 2 (2014), 483–498; 10.2478/s13540-014-0181-5; https://www.degruyter.com/view/j/fca.2014.17.issue-2/s13540-014-0181-5/s13540-014-0181-5.xml.Search in Google Scholar

[5] M. Al-Refai, Y. Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives. Appl. Math. Comput. 257 (2015), 40–51.10.1016/j.amc.2014.12.127Search in Google Scholar

[6] M. Al-Refai, Comparison principles for differential equations involving Caputo fractional derivative with Mittag-Leffler non-singular kernel. Electr. J. Diff. Eq. 2018 (2018), 1–10.Search in Google Scholar

[7] A. Alsaedi, B. Ahmad and M. Kirane, Maximum principle for certain generalized time and space fractional diffusion equations. Quart. Appl. Math. 73, No 1 (2015), 163–175.10.1090/S0033-569X-2015-01386-2Search in Google Scholar

[8] M. Borikhanov, M. Kirane, B.T. Torebek, Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions. Appl. Math. Lett. 81 (2018), 14–20.10.1016/j.aml.2018.01.012Search in Google Scholar

[9] X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principle and Hamiltonian estimates. Ann. Inst. Henri Poincaré, Anal. Non Linéaire31 (2014), 23–53.10.1016/j.anihpc.2013.02.001Search in Google Scholar

[10] A. Capella, J. Dávila, L. Dupaigne, Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Part. Diff. Equat. 36 (2011), 1353–138410.1080/03605302.2011.562954Search in Google Scholar

[11] C.Y. Chan, H.T. Liu, A maximum principle for fractional diffusion equations. Quart. Appl. Math. 74, No 3 (2016), 421–427.10.1090/qam/1433Search in Google Scholar

[12] T. Cheng, G. Huang, C. Li, The maximum principles for fractional Laplacian equations and their applications. Comm. Contemp. Math. 19, No 6 (2017), 1750018-1–1750018-12.10.1142/S0219199717500183Search in Google Scholar

[13] L.M. Del Pezzo, A. A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian. J. Diff. Equat. 263, No 1 (2017), 765–778.10.1016/j.jde.2017.02.051Search in Google Scholar

[14] J. Jia, K. Li, Maximum principles for a time-space fractional diffusion equation. Appl. Math. Lett. 62 (2016), 23–28.10.1016/j.aml.2016.06.010Search in Google Scholar

[15] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies, 2006.Search in Google Scholar

[16] Z. Liu, S. Zeng, Y. Bai, Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications. Fract. Calc. Appl. Anal. 19, No 1 (2016), 188–211; 10.1515/fca-2016-0011; https://www.degruyter.com/view/j/fca.2016.19.issue-1/fca-2016-0011/fca-2016-0011.xml.Search in Google Scholar

[17] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218–223.10.1016/j.jmaa.2008.10.018Search in Google Scholar

[18] Y. Luchko, Some uniqueness and existence results for the initial boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59 (2010), 1766–1772.10.1016/j.camwa.2009.08.015Search in Google Scholar

[19] Y. Luchko, Initial-boundary-value problems for the generalized multiterm time-fractional diffusion equation. J. Math. Anal. Appl. 374 (2011), 538-548.10.1016/j.jmaa.2010.08.048Search in Google Scholar

[20] Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations. Fract. Calc. Appl. Anal. 14, No 1 (2011), 110–124; 10.2478/s13540-011-0008-6; https://www.degruyter.com/view/j/fca.2011.14.issue-1/s13540-011-0008-6/s13540-011-0008-6.xml.Search in Google Scholar

[21] Y. Luchko, M. Yamamoto, On the maximum principle for a time-fractional diffusion equation. Fract. Calc. Appl. Anal. 20, No 5 (2017), 1131–1145; 10.1515/fca-2017-0060; https://www.degruyter.com/view/j/fca.2017.20.issue-5/fca-2017-0060/fca-2017-0060.xml.Search in Google Scholar

[22] J. Mu, B. Ahmad, S. Huang, Existence and regularity of solutions to time-fractional diffusion equations. Comput. Math. Appl. 73, No 6 (2017), 985–996.10.1016/j.camwa.2016.04.039Search in Google Scholar

[23] J.J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23 (2010), 1248–1251.10.1016/j.aml.2010.06.007Search in Google Scholar

[24] H. Ye, F. Liu, V. Anh, I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations. Appl. Math. Comput. 227 (2014), 531–540.10.1016/j.amc.2013.11.015Search in Google Scholar

[25] L. Zhang, B. Ahmad, G. Wang, Analysis and application of diffusion equations involving a new fractional derivative without singular kernel. Electr. J. Differential Equations2017 (2017), 1–6.10.1186/s13662-017-1356-2Search in Google Scholar

Received: 2018-06-05
Published Online: 2019-05-11
Published in Print: 2019-04-24

© 2019 Diogenes Co., Sofia

Downloaded on 16.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2019-0022/html
Scroll to top button