×
Wang, Jinrong; Lin, Zeng

On the impulsive fractional anti-periodic BVP modelling with constant coefficients. (English) Zbl 1296.34037

J. Appl. Math. Comput. 46, No. 1-2, 107-121 (2014).
Summary: This paper is inspired by the recent papers on the BVP for impulsive fractional differential equations. We present a general framework to find the solutions of anti-periodic BVP for linear impulsive fractional differential equations with constant coefficients. Some sufficient conditions for existence of the solutions for nonlinear problem are established. Finally, numerical examples are given to illustrate the results.

Cited in 12 Documents

MSC:

34A08 Fractional ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations

Keywords:

impulsive; fractional anti-periodic BVP; Mittag-Leffler functions; solutions
Cite Review PDF
Full Text: DOI

References:

[1] Ahmad, B., Sivasundaram, S.: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. 3, 251-258 (2009) · Zbl 1193.34056
[2] Ahmad, B., Sivasundaram, S.: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. 4, 134-141 (2010) · Zbl 1187.34038
[3] Ahmad, B., Wang, G.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 59, 1341-1349 (2010) · Zbl 1228.34012
[4] Tian, Y., Bai, Z.: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59, 2601-2609 (2010) · Zbl 1193.34007 · doi:10.1016/j.camwa.2010.01.028
[5] Cao, J., Chen, H.: Some results on impulsive boundary value problem for fractional differential inclusions. Electron. J. Qual. Theory Differ. Equ. 2010(11), 1-24 (2010) · Zbl 1261.34003
[6] Wang, G., Ahmad, B., Zhang, L.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 74, 792-804 (2011) · Zbl 1214.34009 · doi:10.1016/j.na.2010.09.030
[7] Wang, G., Zhang, L., Song, G.: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. Nonlinear Anal. 74, 974-982 (2011) · Zbl 1223.34091 · doi:10.1016/j.na.2010.09.054
[8] Wang, G., Ahmad, B., Zhang, L.: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput. Math. Appl. 59, 1389-1397 (2010) · Zbl 1189.94049 · doi:10.1016/j.camwa.2009.08.031
[9] Wang, X.: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383-2391 (2011) · Zbl 1231.34009 · doi:10.1016/j.camwa.2011.07.026
[10] Cao, J., Chen, H.: Impulsive fractional differential equations with nonlinear boundary conditions. Math. Comput. Model. 55, 303-311 (2012) · Zbl 1255.34006 · doi:10.1016/j.mcm.2011.07.037
[11] Yang, L., Chen, H.: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011, 404917 (2011) · Zbl 1219.34011
[12] Wang, J., Zhou, Y., Fečkan, M.: On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl. 64, 3008-3020 (2012) · Zbl 1268.34032 · doi:10.1016/j.camwa.2011.12.064
[13] Fečkan, M., Zhou, Y., Wang, J.: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050-3060 (2011) · Zbl 1252.35277
[14] Kosmatov, N.: Initial value problems of fractional order with fractional impulsive conditions. Results Math. 63, 1289-1310 (2013) · Zbl 1286.34011 · doi:10.1007/s00025-012-0269-3
[15] Wang, J., Fečkan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 345-361 (2011) · Zbl 1264.34014 · doi:10.4310/DPDE.2011.v8.n4.a3
[16] Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389-3405 (2012) · Zbl 1268.34033 · doi:10.1016/j.camwa.2012.02.021
[17] Wang, J., Li, X.Z., Wei, W.: On the natural solution of an impulsive fractional differential equation of order q∈(1,2). Commun. Nonlinear Sci. Numer. Simul. 17, 4384-4394 (2012) · Zbl 1248.35226 · doi:10.1016/j.cnsns.2012.03.011
[18] Li, X.P., Chen, F.L., Li, X.Z.: Generalized anti-periodic boundary value problems of impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 18, 28-41 (2013) · Zbl 1253.35207 · doi:10.1016/j.cnsns.2012.06.014
[19] Stamova, I., Stamov, G.: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simul. 19, 702-709 (2014) · Zbl 1470.34202 · doi:10.1016/j.cnsns.2013.07.005
[20] Liu, Z., Lu, L., Szántó, I.: Existence of solutions for fractional impulsive differential equations with p-Laplacian operator. Acta Math. Hung. 141, 203-219 (2013). doi:10.1007/s10474-013-0305-0 · Zbl 1313.34242 · doi:10.1007/s10474-013-0305-0
[21] Baleanu, D., Machado, J.A.T., Luo, A.C.J.: Fractional Dynamics and Control. Springer, Berlin (2012) · Zbl 1231.93003 · doi:10.1007/978-1-4614-0457-6
[22] Diethelm, K.: The analysis of fractional differential equations. Lecture Notes in Mathematics (2010) · Zbl 1215.34001
[23] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006) · Zbl 1092.45003
[24] Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge (2009) · Zbl 1188.37002
[25] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[26] Michalski, M.W.: Derivatives of noninteger order and their applications. Dissertationes Mathematicae, CCCXXVIII, Inst. Math., Polish Acad. Sci. (1993) · Zbl 0880.26007
[27] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[28] Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles. Fields and Media. Springer, Berlin (2011). HEP
[29] Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465-4475 (2010) · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
[30] Wang, J., Fec̆kan, M., Zhou, Y.: Presentation of solutions of impulsive fractional Langevin equations and existence results. Eur. Phys. J. Spec. Top. 222, 1855-1872 (2013)
[31] Atkinson, C., Osseiran, A.: Rational solutions for the time-fractional diffusion equation. SIAM J. Appl. Math. 71, 92-106 (2011) · Zbl 1222.26011 · doi:10.1137/100799307
[32] Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities, vol. 1. Academic Press, New York (1969) · Zbl 0177.12403
[33] Wei, W., Xiang, X., Peng, Y.: Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55, 141-156 (2006) · Zbl 1101.45002 · doi:10.1080/02331930500530401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.