Publication Cover
Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 14
228
Views
7
CrossRef citations to date
0
Altmetric
Articles

The forward and inverse problems for a fractional boundary value problem

, , &
Pages 2474-2484 | Received 31 May 2017, Accepted 19 Aug 2017, Published online: 18 Sep 2017

References

  • Bai C. Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem. Electron J Differ Equ. 2013;136:1–12.
  • Ahmad B, Ntouyas S. A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract Calc Appl Anal. 2014;17:348–360.
  • Bagley RL, Torvik PJ. On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech. 1984;51:294–298.
  • Cabada A, Hamdi Z. Nonlinear fractional differential equations with integral boundary value conditions. Appl Math Comput. 2014;228:251–257.
  • Graef JR, Kong L, Kong Q, et al . Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition. Electron J Qual Theory Differ Equ. 2013;55:1–11.
  • Henderson J, Luca R. Positive solutions for a system of nonlocal fractional boundary value problems. Fract Calc Appl Anal. 2013;16:985–1008.
  • Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific; 2000.
  • Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. London: Elsevier; 2006.
  • Tarasov V. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. New York (NY): Springer-Verlag; 2011.
  • Agarwal R, O’Regan D, Staněk S. Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J Math Anal Appl. 2010;371:57–68.
  • Ahmad B, Nieto JJ. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput Math Appl. 2009;58:1838–1843.
  • Bai Z, Lü H. Positive solutions for boundary value problem of nonlinear fractional differential equation. J Math Anal Appl. 2005;311:495–505.
  • Feng M, Zhang X, Ge W. New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound Value Probl. 2011;2011. Article ID: 720702, 20p.
  • Goodrich C. Existence of a positive solution to a class of fractional differential equations. Appl Math Lett. 2010;23:1050–1055.
  • Graef JR, Kong L, Kong Q, et al . Fractional boundary value problems with integral boundary conditions. Appl Anal. 2013;92:2008–2020.
  • Graef JR, Kong L, Kong Q, Wang M. Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions. Fract Calc Appl Anal. 2012;15:509–528.
  • Graef JR, Kong L, Kong Q, et al . A fractional boundary value problem with Dirichlet boundary condition. Commun Appl Anal. 2015;19:497–504.
  • Graef JR, Kong L, Kong Q, et al . On a fractional boundary value problem with a perturbation term. J Appl Anal Comp. 2017;7:57–66.
  • Graef JR, Kong L, Wang M. A Chebyshev spectral method for solving Riemann-Liouville fractional boundary value problems. Appl Math Comput. 2014;241:140–150.
  • Henderson J, Luca R. Existence and multiplicity of positive solutions for a system of fractional boundary value problems. Bound Value Probl. 2014;2014:1–17.
  • Henderson J, Luca R. Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Bound Value Probl. 2015;2015:1–12.
  • Jiang D, Yuan C. The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 2010;72:710–719.
  • Kong L, Kong Q, Wang M. Existence and uniqueness of solutions for a fractional boundary value problem with a separated boundary condition. Dyn Syst Appl. 2014;23:691–698.
  • Kong Q, Wang M. Positive solutions of nonlinear fractional boundary value problems with Dirichlet boundary conditions. Electron J Qual Theory Differ Equ. 2012;17:1–13.
  • Yang L, Chen H. Unique positive solutions for fractional differential equation boundary value problems. Appl Math Lett. 2010;23:1095–1098.
  • Zhang S. Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput Math Appl. 2010;59:1300–1309.
  • Kunze HE, Hicken JE, Vrscay ER. Inverse problems for ODEs using contraction maps and suboptimality of the ‘collage method’. Inverse Prob. 2004;20:977–991.
  • Kunze H, La Torre D, Levere K, et al . Inverse problems via the (generalized collage theorem) for vector-valued lax-milgram-based variational problems. 2015;2015. Article ID: 764643, 8p.
  • Kunze HE, Vrscay ER. Solving inverse problems for ordinary differential equations using the Picard contraction mapping. Inverse Probl. 1999;15:745–770.
  • Zeidler E. Nonlinear functional analysis and its applications I: fixed-point theorems. New York (NY): Springer-Verlag; 1986.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.