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
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