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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 16, 2015

The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)

  • Xianmin Zhang , Praveen Agarwal , Zuohua Liu and Hui Peng
From the journal Open Mathematics

Abstract

In this paper we consider the generalized impulsive system with Riemann-Liouville fractional-order q ∈ (1,2) and obtained the error of the approximate solution for this impulsive system by analyzing of the limit case (as impulses approach zero), as well as find the formula for a general solution. Furthermore, an example is given to illustrate the importance of our results.

References

[1] Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) Search in Google Scholar

[2] Kilbas, AA, Srivastava, HH, CityplaceTrujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) Search in Google Scholar

[3] Baleanu, D, Diethelm, K, Scalas, E, CityplaceTrujillo, JJ: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore (2012) 10.1142/8180Search in Google Scholar

[4] Diethelm, K, Ford, N, J: Analysis of fractional differential equations, J. Math. Anal. Appl. 265, 229–248 (2002) Search in Google Scholar

[5] Ye, H, Gao, J, Ding, Y: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007) Search in Google Scholar

[6] Benchohra, M, CityHenderson, J, CityplaceNtouyas, StateSK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008) Search in Google Scholar

[7] Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl Math. 109, 973–1033 (2010) 10.1007/s10440-008-9356-6Search in Google Scholar

[8] Zaid M, Odibat: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010) Search in Google Scholar

[9] Ahmad, B, Nieto, JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder degree theory. Topol. Methods Nonlinear Anal. 35(2), 295–304 (2010) Search in Google Scholar

[10] Bai, Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal.: TMA. 72, 916–24 (2010) Search in Google Scholar

[11] Mophou, GM, N’Guérékata, GM: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61–69 (2010) Search in Google Scholar

[12] Deng, W: Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Anal.: TMA. 72, 1768- 1777 (2010) Search in Google Scholar

[13] Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001) Search in Google Scholar

[14] Butzer, PL, Kilbas, AA, placeCityTrujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387-400 (2002) Search in Google Scholar

[15] Butzer, PL, Kilbas, AA, CityplaceTrujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1–15 (2002) Search in Google Scholar

[16] Thiramanus, P, Ntouyas, StateSK, Tariboon, J: Existence and Uniqueness Results for Hadamard-Type Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions. Abstr. Appl. Anal. (2014). Doi:10.1155/2014/902054 10.1155/2014/902054Search in Google Scholar

[17] Klimek, M: Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16, 4689-4697 (2011) Search in Google Scholar

[18] Ahmad, B, placeCityNtouyas, StateSK: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17, 348–360 (2014) Search in Google Scholar

[19] Jarad, F, Abdeljawad, T, Baleanu, D: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012) 10.1186/1687-1847-2012-142Search in Google Scholar

[20] Gambo, YY, Jarad, F, Baleanu, D, Abdeljawad, T: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014, 10 (2014) 10.1186/1687-1847-2014-10Search in Google Scholar

[21] Zhang, X: The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect. Adv. Differ. Equ. 2015, 215 (2015) 10.1186/s13662-015-0552-1Search in Google Scholar

[22] Liu, K, Hu, RJ, Cattani, C, Xie, GN, Yang, XJ, Zhao, Y: Local fractional Z transforms with applications to signals on Cantor sets.Abstr. Appl. Anal. 638–648,(2014) 10.1155/2014/638648Search in Google Scholar

[23] Bhrawy,AH, Zaky, MA: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895(2015) Search in Google Scholar

[24] Yang, AM, Cattani, C, Zhang, C, Xie, GN, Yang, XJ: Local fractional Fourier series solutions for nonhomogeneous heat equations arising in fractal heat flow with local fractional derivative. Adv. Mech. Eng. (2014) 10.1155/2014/514639Search in Google Scholar

[25] Bhrawy, AH, Abdelkawy, MA: A fully spectral collocation approximation for multi-dimensional fractional Schrodinger equations. J. Comput. Phys. 281 (15), 876–895(2015) Search in Google Scholar

[26] Bhrawy, AH, Doha, EH, Ezz-Eldien, SS, Abdelkawy, MA: A numerical technique based on the shifted Legendre polynomials for solving the timefractional coupled KdV equation. Calcolo (2015) 10.1007/s10092-014-0132-x . 10.1007/s10092-014-0132-xSearch in Google Scholar

[27] Bhrawy, AH: A highly accurate collocation algorithm for 1+1 and 2+1 fractional percolation equations. J. Vib. Control (2015) DOI: 10.1177/1077546315597815 10.1177/1077546315597815Search in Google Scholar

[28] Bhrawy, AH, Zaky, MA: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation.Nonlinear Dynam. 80 (1), 101–116 (2015) 10.1007/s11071-014-1854-7Search in Google Scholar

[29] Ahmad, B, Sivasundaram, S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal.: HS. 3, 251–258 (2009) Search in Google Scholar

[30] Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal.: HS. 4, 134–141 (2010) Search in Google Scholar

[31] 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) Search in Google Scholar

[32] 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) 10.14232/ejqtde.2011.1.11Search in Google Scholar

[33] Wang, X: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383– 2391(2011) Search in Google Scholar

[34] Stamova, I, Stamov, G: Stability analysis of impulsive functional systems of fractional order. Commun Nonlinear Sci Numer Simulat. 19, 702–709 (2014) 10.1016/j.cnsns.2013.07.005Search in Google Scholar

[35] Abbas, S, Benchohra, M: Impulsive hyperbolic functional differential equations of fractional order with state-dependent delay. Fract. Calc. Appl. Anal. 13, 225–242 (2010) Search in Google Scholar

[36] Abbas, S, Benchohra, M: Upper and lower solutions method for impulsive hyperbolic differential equations with fractional order. Nonlinear Anal.: HS.4, 406–413 (2010) 10.1016/j.nahs.2009.10.004Search in Google Scholar

[37] Abbas, S, Agarwal, RP, Benchohra, M: Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay. Nonlinear Anal.: HS. 4, 818–829 (2010) Search in Google Scholar

[38] Abbas, S, Benchohra, M, Gorniewicz, L: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn. 72 (1), 49–60 (2010) Search in Google Scholar

[39] Benchohra, M, Seba, D: Impulsive partial hyperbolic fractional order differential equations in banach spaces. J. Fract. Calc. Appl. 1 (4), 1–12 (2011) Search in Google Scholar

[40] Guo, T, Zhang, K: Impulsive fractional partial differential equations. Appl. Math. Comput. 257, 581–590 (2015) Search in Google Scholar

[41] Zhang, X, Zhang, X, Zhang, M: On the concept of general solution for impulsive differential equations of fractional order q 2 (0,1). Appl. Math. Comput. 247, 72–89 (2014) Search in Google Scholar

[42] Zhang, X: On the concept of general solutions for impulsive differential equations of fractional order q 2 (1, 2). Appl. Math. Comput. 268, 103–120 (2015) Search in Google Scholar

Received: 2015-8-15
Accepted: 2015-9-29
Published Online: 2015-12-16

©2015 Zhang et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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