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The short memory principle for solving Abel differential equation of fractional order. (English) Zbl 1236.34008
Summary: The short memory principle (SMP) is applied for solving the Abel differential equation with fractional order. We evaluate the approximate solution at the end of required interval, and construct a suitable iteration scheme employing this end point as initial value. Numerical experiments show that our iteration method is simple and efficient, and that a proper length of memory could maintain the validity of the short memory principle.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
65L05 Numerical methods for initial value problems
45J05 Integro-ordinary differential equations
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[1] Diethelm, K., The analysis of fractional differential equations, (2010), Springer-Verlag Berlin Heidelberg
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA · Zbl 0918.34010
[3] Sabatier, J.; Agrawal, O.P.; Tenreiro Machado, J.A., Advances in fractional calculus: theoretical developments and applications in physics and engineering, (2007), Springer The Netherlands · Zbl 1116.00014
[4] García, I.A.; Giacomini, H.; Giné, J., Generalized nonlinear superposition principles for polynomial planar vector fields, J. Lie theory, 15, 89-104, (2005) · Zbl 1073.34003
[5] El-Sayed, A.M.A.; Gaafar, F.M., Fractional order differential equations with memory and fractional-order relaxation – oscillation model, Pure math. appl., 12, 296-310, (2001)
[6] Lewandowski, R.; Chorazyczewski, B., Identification of the parameters of the kelvin – voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers, Comput. struct., 88, 1-17, (2010)
[7] Sorkun, H.H.; Yalcinbas, S., Approximate solutions of linear Volterra integral equation systems with variable coefficients, Appl. math. model., 34, 3451-3464, (2010) · Zbl 1201.45001
[8] Maleknejad, K.; Hashemizadeh, E.; Ezzati, R., A new approach to the numerical solution of Volterra integral equations by using bernstein’s approximation, Commun. nonlinear sci. numer. simul., 16, 647-655, (2011) · Zbl 1221.65334
[9] Tahmasbi, A.; Fard, O.S., Numerical solution of linear Volterra integral equations system of the second kind, Appl. math. comput., 201, 547-552, (2008) · Zbl 1148.65101
[10] Izzo, G.; Jackiewicz, Z.; Messina, E.; Vecchio, A., General linear methods for Volterra integral equations, J. comput. appl. math., 234, 2768-2782, (2010) · Zbl 1196.65200
[11] Gnana Bhaskar, T.; Lakshmikantham, V.; Leela, S., Fractional differential equations with a krasnoselskii – krein type condition, Nonlinear anal. hybrid syst., 3, 734-737, (2009) · Zbl 1181.34008
[12] McRae, F.A., Monotone iterative technique and existence results for fractional differential equations, Nonlinear anal., 71, 6093-6096, (2009) · Zbl 1260.34014
[13] Muslim, M., Existence and approximation of solutions to fractional equations, Math. comput. modelling, 49, 1164-1172, (2009) · Zbl 1165.34304
[14] Diethelm, K., An improvement of a noncalssical numerical method for the computation of fractional derivatives, J. vib. acoust., 131, 014502, (2009)
[15] Giné, J.; Santallusia, X., Abel differential equations admitting a certain first integral, J. math. anal. appl., 370, 187-199, (2010) · Zbl 1202.34006
[16] Momani, S.; Odibat, Z., Numerical approach to differential equations of fractional order, J. comput. appl. math., 207, 96-110, (2007) · Zbl 1119.65127
[17] Ertürk, V.S.; Momani, S.; Odibat, Z., Application of generalized differential tranform method to multi-order fractional differential equations, Commun. nonlinear sci. numer. simul., 13, 1642-1654, (2008) · Zbl 1221.34022
[18] Odibat, Z., Analytic study on linear systems of fractional differential equations, Comput. math. appl., 59, 1171-1183, (2010) · Zbl 1189.34017
[19] Odibat, Z.; Momani, S.; Xu, H., A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. math. model., 34, 593-600, (2010) · Zbl 1185.65139
[20] Ertürk, V.S.; Momani, S., Solving systems of fractional differential equations using differential transform method, J. comput. appl. math., 215, 142-151, (2008) · Zbl 1141.65088
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