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Monotone iterative sequences for nonlinear boundary value problems of fractional order. (English) Zbl 1219.34005
Summary: We extend the maximum principle and the method of upper and lower solutions to boundary value problems with the Caputo fractional derivative. We establish positivity and uniqueness results for the problem. We then introduce two well-defined monotone sequences of upper and lower solutions which converge uniformly to the actual solution of the problem. A numerical iterative scheme is introduced to obtain an accurate approximate solution for the problem. The accuracy and efficiency of the new approach are tested through two examples.
34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
34A45Theoretical approximation of solutions of ODE
[1]Caputo, M.: Linear models of dissipation whose q is almost frequency independent, part II geophys, JR. astr. Soc. 13, 529-539 (1967)
[2]Kilbas, A. A.; Srivastava, H. M.; Trujjllo, J. J.: T heory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[3]Podlubny, I.: Fractional differential equations, mathematics in science and engineering, (1993)
[4]Miller, K.; Ross, B.: An introduction to the fractional calculus fractional differential equations, (1993)
[5]Agarwal, R. P.; Benchohra, N.; Hamani, S.: A survey on existing results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl. Math 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[6]Cang, J.; Tan, Y.; Xu, H.; Liao, S.: Series solution of non-linear Riccati differential equations with fractional order, Chaos, solitons fractals 40, 1-9 (2009) · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[7]Furati, K.; Tatar, N.: An existence result for a nonlocal fractional differential problem, J. fract. Calc. 26, 43-51 (2004) · Zbl 1101.34001
[8]Momani, S.; Shawagfeh, N.: Decomposition method for solving fractional Riccati differential equations, Appl. math. Comput. 182, 1083-1092 (2006) · Zbl 1107.65121 · doi:10.1016/j.amc.2006.05.008
[9]Shi, A.; Zhang, S.: Upper and lower solutions method and a fractional differential equation boundary value problem, Electron. J. Qual. theory differ. Equ. 30, 1-13 (2009) · Zbl 1183.34009 · doi:emis:journals/EJQTDE/2009/200930.html
[10]Al-Mdallal, Q. M.: On the numerical solution of fractional Sturm–Liouville problems, J. comput. Math. 87, 2837-2845 (2010) · Zbl 1202.65100 · doi:10.1080/00207160802562549
[11]Al-Mdallal, Q. M.; Syam, M. I.; Anwar, M. N.: A collocation-shooting method for solving fractional boundary value problems, Commun. nonlinear sci. Numer. simul. 15, 3814-3822 (2010) · Zbl 1222.65078 · doi:10.1016/j.cnsns.2010.01.020
[12]Bataineh, A.; Alomari, A.; Noorani, M.; Hashim, I.; Nazar, R.: Series solutions of systems of nonlinear fractional differential equations, Acta. appl. Math. 105, 189-198 (2009) · Zbl 1187.34007 · doi:10.1007/s10440-008-9271-x
[13]Momani, S.; Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006)
[14]Abbas, S.; Benchohra, M.: Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear anal. Hybrid syst. 4, 406-413 (2010) · Zbl 1202.35340 · doi:10.1016/j.nahs.2009.10.004
[15]Benchohra, M.; Hamani, S.: The method of upper and lower solutions and impulsive fractional differential inclusions, Nonlinear anal. Hybrid syst. 3, 433-440 (2009) · Zbl 1221.49060 · doi:10.1016/j.nahs.2009.02.009
[16]Pao, C. V.: Nonlinear parabolic and elliptic equations, (1992)
[17]H.L. Royden, Real analysis, 3rd ed. New York, Macmillan, London, Collier Macmillan, 1988. · Zbl 0704.26006