Positive properties of the Green function for two-term fractional differential equations and its application

Volume 10, Issue 4, pp 2094--2102 http://dx.doi.org/10.22436/jnsa.010.04.63

Publication Date: April 20, 2017 Submission Date: December 01, 2016

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Authors

Yongqing Wang - School of Statistics, Qufu Normal University, Qufu 273165, Shandong, P. R. China. - School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China. Lishan Liu - School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China. - Department of Mathematics and Statistics, Curtin University, Perth, WA6845, Australia.

Abstract

In this paper, we study the positive properties of the Green function for the following two-term fractional differential equation \[ \begin{cases} -D^\alpha_{0^+}u(t)+bu(t)=f(t,u(t)),\,\,\,\,\, 0<t<1,\\ u(0)=0,\,\,\,\,\, u(1)=0, \end{cases} \] where \(1 < \alpha < 2, b > 0, D^\alpha_{0^+}\) is the standard Riemann-Liouville derivative. As an application, the existence and uniqueness of positive solution are obtained under the singular conditions. Moreover, an iterative scheme is established to approximate the unique positive solution.

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Keywords

• Multi-term fractional differential equation
• Green function
• iterative solution
• boundary value problems.

MSC

•  34A08
•  34B27

References

• [1] R. L. Bagley, P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294–298.
  • View Article
  • Google Scholar
  • MATH


• [2] Z.-B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916–924.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] Z.-B. Bai, On solutions of some fractional m-point boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 15 pages.
  • MathSciNet
  • Google Scholar
  • MATH


• [4] Z.-B. Bai, Solvability for a class of fractional m-point boundary value problem at resonance, Comput. Math. Appl., 62 (2011), 1292–1302.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] Z.-B. Bai, Y.-H. Zhang, Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 218 (2011), 1719–1725.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] J. Čermák, T. Kisela, Stability properties of two-term fractional differential equations, Nonlinear Dynam., 80 (2015), 1673–1684.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] Y.-J. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51 (2016), 48–54.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] V. Daftardar-Gejji, S. Bhalekar, Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl., 345 (2008), 754–765.
  • View Article
  • Google Scholar


• [9] E. F. Elshehawey, E. M. A. Elbarbary, N. A. S. Afifi, M. El-Shahed, On the solution of the endolymph equation using fractional calculus, Appl. Math. Comput., 124 (2001), 337–341.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] N. J. Ford, J. A. Connolly, Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations, J. Comput. Appl. Math., 229 (2009), 382–391.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] J.-Q. Jiang, L.-S. Liu, Existence of solutions for a sequential fractional differential system with coupled boundary conditions, Bound. Value Probl., 2016 (2016), 15 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] D.-Q. Jiang, C.-J. Yuan, The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal., 72 (2010), 710–719.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
  • MathSciNet


• [14] R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A., 278 (2000), 107–125.
  • View Article
  • Google Scholar


• [15] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
  • Google Scholar


• [16] X.-W. Su, S.-Q. Zhang, Unbounded solutions to a boundary value problem of fractional order on the half-line, Comput. Math. Appl., 61 (2011), 1079–1087.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] Y.-Q. Wang, L.-S. Liu, Necessary and sufficient condition for the existence of positive solution to singular fractional differential equations, Adv. Difference Equ., 2015 (2015), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] Y.-Q. Wang, L.-S. Liu, Positive solutions for a class of fractional 3-point boundary value problems at resonance, Adv. Difference Equ., 2017 (2017), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Anal., 74 (2011), 6434–6441.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74 (2011), 3599–3605.
  • View Article
  • Google Scholar


• [21] Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Existence and uniqueness of a positive solution to singular fractional differential equations, Bound. Value Probl., 2012 (2012), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] Y.-Q. Wang, L.-S. Liu, Y.-H. Wu, Positive solutions for a fractional boundary value problem with changing sign nonlinearity, Abstr. Appl. Anal., 2012 (2012), 17 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] Y. Wang, L.-S. Liu, X.-U. Zhang, Y.-H. Wu, Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection, Appl. Math. Comput., 258 (2015), 312–324.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] X.-J. Xu, X.-L. Fei, The positive properties of Green’s function for three point boundary value problems of nonlinear fractional differential equations and its applications, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1555–1565.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] X.-J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Therm. Sci., 2017 (2017), 12 pages.
  • Google Scholar


• [26] X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional integral transforms and their applications, Elsevier/Academic Press, Amsterdam (2016)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] X.-J. Yang, F. Gao, J. A. Tenreiro Machado, D. Baleanu, A new fractional derivative involving the normalized sinc function without singular kernel, ArXiv, 2017 (2017), 11 pages.
  • Google Scholar


• [28] X.-J. Yang, H. M. Srivastava, J. A. Tenreiro Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753–756.
  • Google Scholar


• [29] X.-J. Yang, J. A. Tenreiro Machado, A new fractional operator of variable order: Application in the description of anomalous diffusion, Phys. A, (2017), (In press).
  • View Article
  • Google Scholar


• [30] X.-J. Yang, Z.-Z. Zhang, J. A. Tenreiro Machado, D. Baleanu, On local factional operators view of computational complexity: Diffusion and relaxation defined on cantor sets, Therm. Sci., 20 (2016), S755–S767.
  • Google Scholar


• [31] C.-B. Zhai, L. Xu, Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2820–2827.
  • View Article
  • MathSciNet
  • Google Scholar


• [32] Y.-H. Zhang, Z.-B. Bai, T.-T. Feng, Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance, Comput. Math. Appl., 61 (2011), 1032–1047.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] X.-U. Zhang, L.-S. Liu, B. Wiwatanapataphee, Y.-H. Wu, The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition, Appl. Math. Comput., 235 (2014), 412–422.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] X.-U. Zhang, L.-S. Liu, Y.-H. Wu, The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives, Appl. Math. Comput., 218 (2012), 8526–8536.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] X.-U. Zhang, L.-S. Liu, Y.-H. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 27 (2014), 26–33.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH