A maximum principle for a fractional boundary value problem with convection term and applications
Abstract
We consider a fractional boundary value problem with Caputo-Fabrizio fractional derivative of order 1 < α < 2 We prove a maximum principle for a general linear fractional boundary value problem. The proof is based on an estimate of the fractional derivative at extreme points and under certain assumption on the boundary conditions. A prior norm estimate of solutions of the linear fractional boundary value problem and a uniqueness result of the nonlinear problem have been established. Several comparison principles are derived for the linear and nonlinear fractional problems.
First Published Online: 21 Nov 2018
Keyword : fractional differential equations, Caputo-Fabrizio fractional derivative, maximum principle
How to Cite
Al-Refai, M., & Pal, K. (2019). A maximum principle for a fractional boundary value problem with convection term and applications. Mathematical Modelling and Analysis, 24(1), 62-71. https://doi.org/10.3846/mma.2019.005
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[17] E.F.D Goufo. Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation. Mathematical Modeling and Analysis, 21(2):188–198, 2016. https://doi.org/10.3846/13926292.2016.1145607.
[18] J. Hristov. Transient heal diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s kernel to the CaputoFabrizio time-fractional derivative. Thermal Science, 20(2):757–762, 2016. https://doi.org/10.2298/TSCI160112019H.
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[20] X. Meng and M. Stynes. The Green’s function and maximum principle for a Caputo two-point boundary value problem with a convection term. J. Math. Anal. Appl., 461(1):198–218, 2018. https://doi.org/10.1016/j.jmaa.2018.01.004.
[21] E. Karimov N. Al-Salti and K. Sadarangani. On a differential equation with Caputo-Fabrizio fractional derivative of order 1 < β ≤ 2 and application to mass-spring-damper system. Progr. Fract. Differ. Appl., 2(4):257–263, 2016.
[22] A. Pedas and E. Tamme. Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math., 263(13):3349–3359, 2012. https://doi.org/10.1016/j.cam.2012.03.002.
[23] M. Stynes and J. L. Gracia. A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA Journal of Numerical Analysis, 35(2):698–721, 2015. https://doi.org/10.1093/imanum/dru011.
[24] V. Tarasov. No nonlocality, no fractional derivative. Communications in Nonlinear Science and Numerical Simulation, 62:157–163, 2018. https://doi.org/10.1016/j.cnsns.2018.02.019.
[2] Thabet Abdeljawad. Fractional operators with exponential kernels and a Lyapunov type inequality. Advances in Difference Equations, 2017(1):313, Oct 2017. https://doi.org/10.1186/s13662-017-1285-0.
[3] M. Al-Refai. Basic results on nonlinear eigenvalue problems of fractional order. Electronic Journal of Differential Equations, 2012(191):1–12, 2012.
[4] M. Al-Refai. On the fractional derivatives at extreme points. Electronic Journal of Qualitative Theory of Differential Equations, 2012(55):1–5, 2012. https://doi.org/10.14232/ejqtde.2012.1.55.
[5] M. Al-Refai. Reduction of order formula and fundamental set of solutions for linear fractional differential equations. Applied Mathematics Letters, 2018(82):8– 13, 2018. https://doi.org/10.1016/j.aml.2018.02.014.
[6] M. Al-Refai and T. Abdeljawad. Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel. Advances in Difference Equations, 2017(315), 2017. https://doi.org/10.1186/s13662-017-1356-2.
[7] M. Al-Refai and Y. Luchko. Maximum principles for the fractional diffusion equations with the Riemann-Liouville fractional derivative and their applications. Frac. Cal. Appl. Anal., 17(2):483–498, 2014. https://doi.org/10.2478/s13540- 014-0181-5.
[8] M. Al-Refai and Yu. Luchko. Analysis of fractional diffusion equations of distributed order: Maximum principles and its applications. Analysis, 2015. https://doi.org/10.1515/anly-2015-5011.
[9] B.S.T. Alkahtani and A. Atangana. Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order. Chaos, Solitons and Fractals, 89:539–546, 2016. https://doi.org/10.1016/j.chaos.2016.03.012.
[10] A. Atangana. On the new fractional derivative and application to nonlinear fisher’s reaction-diffusion equation. Appl. Math. Comput., 273(15):948–956, 2016. https://doi.org/10.1016/j.amc.2015.10.021.
[11] D. Baleanu, K. Diethelm, E. Scalas and J. Trujillo. Fractional calculus: Models and numerical methods, nonlinearity and chaos. Series on Complexity, Boston, 2016. World Scientific.
[12] M. Caputo and M. Fabrizio. A new definition of fractional derivative without singlular kernel. Prog. Fract. Differ. Appl., 1(2):73–85, 2015. https://doi.org/10.12785/pfda/010201.
[13] M. Caputo and M. Fabrizio. Applications of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl., 2(1):1–11, 2016. https://doi.org/10.18576/pfda/020101.
[14] J.F. G´omez-Aguilar, M.G. L´opez-L´opez, V.M. Alvarado-Mart´ınez, J. ReyesReyes and M. Adam-Medina. Modeling diffusive transport with a fractional derivative without singular kernel. Physica A, 447(1):467–481, 2016. https://doi.org/10.1016/j.physa.2015.12.066.
[15] J.F. G´omez-Aguilar, L. Torres, H. Y´epez-Mart´ınez, D. Baleanu, J.M. Reyes and I.O. Sosa. Fractional Li´enard type model of a pipeline within the fractional derivative without singular kernel. Advances in Difference Equations, 2016(1):173, Jul 2016. https://doi.org/10.1186/s13662-016-0908-1.
[16] J.F. G´omez-Aguilar, L. Torres, H. Y´epez-Mart’inez, C. Calder´on-Ram´on, I. Cruz-Orduna, R.F. Escobar-Jim´enez and V.H. Olivares-Peregrino. Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy, 17:6289–6303, 2015. https://doi.org/10.3390/e17096289.
[17] E.F.D Goufo. Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation. Mathematical Modeling and Analysis, 21(2):188–198, 2016. https://doi.org/10.3846/13926292.2016.1145607.
[18] J. Hristov. Transient heal diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s kernel to the CaputoFabrizio time-fractional derivative. Thermal Science, 20(2):757–762, 2016. https://doi.org/10.2298/TSCI160112019H.
[19] Y. Luchko. Maximum principle for the generalized time-fractional diffusion equations. J. Math. Anal. Appl., 351:18–223, 2009. https://doi.org/10.1016/j.jmaa.2008.10.018.
[20] X. Meng and M. Stynes. The Green’s function and maximum principle for a Caputo two-point boundary value problem with a convection term. J. Math. Anal. Appl., 461(1):198–218, 2018. https://doi.org/10.1016/j.jmaa.2018.01.004.
[21] E. Karimov N. Al-Salti and K. Sadarangani. On a differential equation with Caputo-Fabrizio fractional derivative of order 1 < β ≤ 2 and application to mass-spring-damper system. Progr. Fract. Differ. Appl., 2(4):257–263, 2016.
[22] A. Pedas and E. Tamme. Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math., 263(13):3349–3359, 2012. https://doi.org/10.1016/j.cam.2012.03.002.
[23] M. Stynes and J. L. Gracia. A finite difference method for a two-point boundary value problem with a Caputo fractional derivative. IMA Journal of Numerical Analysis, 35(2):698–721, 2015. https://doi.org/10.1093/imanum/dru011.
[24] V. Tarasov. No nonlocality, no fractional derivative. Communications in Nonlinear Science and Numerical Simulation, 62:157–163, 2018. https://doi.org/10.1016/j.cnsns.2018.02.019.