Existence and numerical simulation of solutions for nonlinear fractional pantograph equations

References

• R.P. Agarwal and B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl. 62 (2011), pp. 1200–1214. doi: 10.1016/j.camwa.2011.03.001
   Web of Science ®Google Scholar
• R.P. Agarwal, M. Benchohra, and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), pp. 973–1033. doi: 10.1007/s10440-008-9356-6
   Web of Science ®Google Scholar
• A. Aghajani, Y. Jalilian, and J.J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal. 15(1) (2012), pp. 44–69. doi: 10.2478/s13540-012-0005-4
   Web of Science ®Google Scholar
• W.G. Ajello, H.I. Freedman, and J. Wu, A model of stage structured population growth with density depended time delay, SIAM J. Appl. Math. 52 (1992), pp. 855–869. doi: 10.1137/0152048
   Web of Science ®Google Scholar
• A. Anguraj, P. Karthikeyan, M. Rivero, and J.J. Trujillo, On new existence results for fractional integro-differential equations with impulsive and integral conditions, Comput. Math. Appl. 66(12) (2014), pp. 2587–2594. doi: 10.1016/j.camwa.2013.01.034
   Web of Science ®Google Scholar
• K. Balachandran, S. Kiruthika, and J.J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci. 33(3) (2013), pp. 712–720. doi: 10.1016/S0252-9602(13)60032-6
   Web of Science ®Google Scholar
• M. Benchohra, F. Berhoun, and G. Nguerekata, Bounded solutions for fractional order differential equations on the half-line, Bull. Math. Anal. Appl. 4 (2012), pp. 62–71.
   Google Scholar
• D.A. Benson, The fractional advection-dispersion equation: Development and application, Ph.D. thesis, University of Nevada at Reno, 1998.
   Google Scholar
• A.H. Bhrawy, A.A. AL-Zahrani, Y.A. Alhamed, and D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional pantograph equations, Romanian J. Phys. 59 (2014), pp. 646–657.
   Web of Science ®Google Scholar
• K.L. Bowers and J. Lund, Numerical solution of singular Poisson problems via the Sinc-Galerkin method, SIAM J. Numer. Anal. 24(1) (1987), pp. 36–51. doi: 10.1137/0724004
   Web of Science ®Google Scholar
• M. Buhamann and A. Iserles, Stability of the discretized pantograph differential equation, J. Math. Comput. 60 (1993), pp. 575–589. doi: 10.1090/S0025-5718-1993-1176707-2
   Web of Science ®Google Scholar
• S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Berlin, Heidelberg, 2008.
   Google Scholar
• S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg, 2011.
   Google Scholar
• K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations, Fract. Calc. Appl. Anal. 14 (2011), pp. 475–490. doi: 10.2478/s13540-011-0029-1
   Web of Science ®Google Scholar
• K. Diethelm, N.J. Ford, and A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam. 29 (2002), pp. 3–22. doi: 10.1023/A:1016592219341
   Web of Science ®Google Scholar
• K. Diethelm, N.J. Ford, and A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms 36 (2004), pp. 31–52. doi: 10.1023/B:NUMA.0000027736.85078.be
   Web of Science ®Google Scholar
• Y. Jalilian and R. Jalilian, Existence of solution for delay fractional differential equations, Mediterr. J. Math. 10 (2013), pp. 1731–1747. doi: 10.1007/s00009-013-0281-1
   Web of Science ®Google Scholar
• A. Khastan, J.J. Nieto, and R. Rodríguez-López, Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty, Fixed Point Theory Appl. 2014 (2014), p. 21. doi: 10.1186/1687-1812-2014-21
   Google Scholar
• A.A. Kilbas, H. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
   Google Scholar
• J. Klafter, S.C. Lim, and R. Metzler, Fractional Dynamics in Physics: Recent Advances, World Scientific, Singapore, 2011.
   Google Scholar
• P. Kumar and O.P. Agrawal, An approximate method for numerical solution of fractional differential equations, Signal Process. 86 (2006), pp. 2602–2610. doi: 10.1016/j.sigpro.2006.02.007
   Web of Science ®Google Scholar
• D.L. Lewis, J. Lund, and K.L. Bowers, The space-time Sinc-Gallerkin method for parabolic problems, Internat. J. Numer. Methods Engrg. 24 (1987), pp. 1629–1644. doi: 10.1002/nme.1620240903
   Web of Science ®Google Scholar
• C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Numerical Analysis and Scientific Computing Series, Chapman & Hall/CRC, Boca Raton, 2015.
   Google Scholar
• R. Magin, M.D. Ortigueira, I. Podlubny, and J. Trujillo, On the fractional signals and systems, Signal Process. 91 (2011), pp. 350–371. doi: 10.1016/j.sigpro.2010.08.003
   Web of Science ®Google Scholar
• R.L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010), pp. 1586–1593. doi: 10.1016/j.camwa.2009.08.039
   Web of Science ®Google Scholar
• F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
   Google Scholar
• F.C. Merala, T.J. Roystona, and R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), pp. 939–945. doi: 10.1016/j.cnsns.2009.05.004
   Web of Science ®Google Scholar
• M. Mori, A. NurMuhammad, and T. Mori, Numerical solution of Volterra integral equation with weakly singular kernel based on the DE-Sinc method, Japan. J. Indast. Appl. Math. 25 (2008), pp. 165–183. doi: 10.1007/BF03167518
   Web of Science ®Google Scholar
• J.R. Ockendon and A.B. Taylor, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. Ser. A 322 (1971), pp. 447–468. doi: 10.1098/rspa.1971.0078
   Web of Science ®Google Scholar
• Z.M. Odibat and S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform. 26 (2008), pp. 15–27.
   Google Scholar
• T. Okayama, T. Matsuo, and M. Sugihara, Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration, Mathematical Engineering Technical Reports 2009-01, The University of Tokyo, 2009.
   Google Scholar
• T. Okayama, T. Matsuo, and M. Sugihara, Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind, J. Comput. Appl. Math. 234 (2010), pp. 1211–1227. doi: 10.1016/j.cam.2009.07.049
   Web of Science ®Google Scholar
• A. Pedas and E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods, J. Comput. Appl. Math. 255 (2014), pp. 216–230. doi: 10.1016/j.cam.2013.04.049
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
   Google Scholar
• B.V. Riley, The numerical solution of Volterra integral equations with nonsmooth solutions based on Sinc approximation, Appl. Numer. Math. 9 (1992), pp. 249–257. doi: 10.1016/0168-9274(92)90019-A
   Web of Science ®Google Scholar
• J. Sabatier, H.C. Nguyen, C. Farges, J.Y. Deletage, X. Moreau, F. Guillemard, and B. Bavoux, Fractional models for thermal modeling and temperature estimation of a transistor junction, Adv. Difference Equ. 2011 (2011), p. 12. Article ID 687363. doi: 10.1155/2011/687363
   Web of Science ®Google Scholar
• S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
   Google Scholar
• D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980.
   Google Scholar
• F. Stenger, A Sinc-Galerkin method of solution of boundary value problems, Math. Comput. 33(145) (1979), pp. 85–109.
   Web of Science ®Google Scholar
• F. Stenger, Numerical methods based on Whittaker cardinal, or Sinc functions, SIAM Rev. 23(2) (1981), pp. 165–224. doi: 10.1137/1023037
   Web of Science ®Google Scholar
• V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer-Verlag, Berlin Heidelberg, 2010.
   Google Scholar
• J. Tenreiro Machado, V. Kiryakova, and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 1140–1153. doi: 10.1016/j.cnsns.2010.05.027
   Web of Science ®Google Scholar
• L. Zhang, B. Ahmad, and G. Wang, The existence of an extremal solution to a nonlinear system with the right-handed Riemann–Liouville fractional derivative, Appl. Math. Lett. 31 (2014), pp. 1–6. doi: 10.1016/j.aml.2013.12.014
   Web of Science ®Google Scholar