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Original Articles

On the existence and uniqueness of solutions to a new class of fractional boundary value problems

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Pages 1611-1627 | Received 12 Aug 2014, Accepted 16 Jun 2015, Published online: 24 Jul 2015

References

  • R.P. Agarwal, M. Benchohara, and B.A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Diff. Equ. Math. Phys. 44 (2008), pp. 1–21.
  • Z. Bai and H. Lu, Positive solutions for a boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), pp. 495–505. doi: 10.1016/j.jmaa.2005.02.052
  • D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2012.
  • R. Darzi, B. Mohammadzadeh, A. Neamaty, and D. Baleanu, On the existence and uniqueness of solution of a nonlinear fractional differential equations, J. Comput. Anal. Appl. 15(1) (2013), pp. 152–162.
  • M. El-Shahed and J.J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problems of fractional order, Comput. Math. Appl. 59 (2010), pp. 3438–3443. doi: 10.1016/j.camwa.2010.03.031
  • M. El-Shahed and J. Zhang, Positive solutions for boundary value problem of nolinear fractional differential equation, Abstr. Appl. Anal. (2007), p. 8. Article ID 10368, doi:10.1155/2007/10368.
  • A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  • X. Li, Operational method for solving fractional differential equations using cubic B-spline approximation, Int. J. Comput. Math. 91(12) (2014), pp. 2584–2602. doi: 10.1080/00207160.2014.884792
  • S. Liang and J. Zhang, Positive solutions for boundary value problem of nonlinear fractional differential equation, Nonlinear Anal 71 (2009), pp. 5545–5550. doi: 10.1016/j.na.2009.04.045
  • K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, John Wiley and Sons, New York, 1993.
  • T.F. Nonnenmacher and R. Metzler, On the Riemann–Liouville fractional calculus and some recent applications, Fractals 3 (1995), pp. 557–566. doi: 10.1142/S0218348X95000497
  • K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
  • G. Oturanc, A. Kurnaz, and Y. Keskin, A new analytical approximate method for the solution of fractional differential equations, Int. J. Comput. Math. 85(1) (2008), pp. 131–142. doi: 10.1080/00207160701405477
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
  • M.U. Rehman and R.A. Khan, Existence and uniqueness solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett. 23 (2010), pp. 1038–1044. doi: 10.1016/j.aml.2010.04.033
  • B. Ross (Ed.), The Fractional Calculus and its Application, Lecture Notes in Mathematics, vol. 475, Springer-Verlag, Berlin, 1975.
  • S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integral and Derivatives (Theory and Application), Gordon and Breach, Yverdon, 1993.
  • S. Stanek, The existence of positive solutions of singular fractional boundary value problems, Comput. Math. Appl. 62 (2011), pp. 1379–1388. doi: 10.1016/j.camwa.2011.04.048
  • D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980.
  • X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009), pp. 64–69. doi: 10.1016/j.aml.2008.03.001
  • F.B. Tatom, The relationship between fractional calculus and fractals, Fractals 3 (1995), pp. 217–229. doi: 10.1142/S0218348X95000175
  • Y. Xu, Z. He and Q. Xu, Numerical solutions of fractional advection-diffusion equations with a kind of new generalized fractional derivative, Int. J. Comput. Math. 91(3) (2014), pp. 588–600. doi: 10.1080/00207160.2013.799277
  • C. Yu and G.Z. Gao, On the solution of nonlinear fractional order differential equation, Nonlinear. Anal. Theor. Meth. Appl. 63 (1998), pp. 971–976. doi: 10.1016/j.na.2005.01.008

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