Generalized Sturm-Liouville and Langevin equations via Hadamard fractional derivatives with anti-periodic boundary conditions

Springer Science and Business Media LLC - Tập 2016 - Trang 1-13 - 2016
Chanakarn Kiataramkul1, Sotiris K Ntouyas2,3, Jessada Tariboon1, Atthapol Kijjathanakorn1
1Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand
2Department of Mathematics , University of Ioannina , Ioannina , GREECE
3Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Tóm tắt

In this paper, we introduce a new class of anti-periodic boundary value problems by combining Sturm-Liouville and Langevin fractional differential equations of Hadamard type. Existence and uniqueness results are proved via fixed point theorems. Examples illustrating the obtained results are also presented.

Tài liệu tham khảo

Rivero, M, Trujillo, JJ, Velasco, MP: A fractional approach to the Sturm-Liouville problem. Cent. Eur. J. Phys. (2013). doi:10.2478/s11534-013-0216-2

Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)

Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston (2012)

Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095-1100 (2010)

Ahmad, B, Ntouyas, SK, Tariboon, J: Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential equations. Adv. Differ. Equ. 2015, 293 (2015)

Ahmad, B, Nieto, JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011, 36 (2011)

Ahmad, B, Ntouyas, SK, Alsaedi, A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011, Article ID 107384 (2011)

Tariboon, J, Ntouyas, SK, Thiramanus, P: Riemann-Liouville fractional differential equations with Hadamard fractional integral conditions. Int. J. Appl. Math. Stat. 54, 119-134 (2016)

Ahmad, B, Ntouyas, SK, Alsaedi, A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013, Article ID 320415 (2013)

Ahmad, B, Nieto, JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013, Article ID 149659 (2013)

Zhang, L, Ahmad, B, Wang, G, Agarwal, RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249, 51-56 (2013)

Liu, X, Jia, M, Ge, W: Multiple solutions of a p-Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013, 126 (2013)

Hadamard, J: Essai sur l’etude des fonctions donnees par leur developpement de Taylor. J. Math. Pures Appl. 8, 101-186 (1892)

Butzer, PL, Kilbas, AA, Trujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387-400 (2002)

Butzer, PL, Kilbas, AA, Trujillo, JJ: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1-27 (2002)

Butzer, PL, Kilbas, AA, Trujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1-15 (2002)

Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191-1204 (2001)

Kilbas, AA, Trujillo, JJ: Hadamard-type integrals as G-transforms. Integral Transforms Spec. Funct. 14, 413-427 (2003)

Coffey, WT, Kalmykov, YP, Waldron, JT: The Langevin Equation, 2nd edn. World Scientific, Singapore (2004)

Lim, SC, Li, M, Teo, LP: Langevin equation with two fractional orders. Phys. Lett. A 372, 6309-6320 (2008)

Lim, SC, Teo, LP: The fractional oscillator process with two indices. J. Phys. A, Math. Theor. 42, Article ID 065208 (2009)

Uranagase, M, Munakata, T: Generalized Langevin equation revisited: mechanical random force and self-consistent structure. J. Phys. A, Math. Theor. 43, Article ID 455003 (2010)

Denisov, SI, Kantz, H, Hänggi, P: Langevin equation with super-heavy-tailed noise. J. Phys. A, Math. Theor. 43, Article ID 285004 (2010)

Lozinski, A, Owens, RG, Phillips, TN: The Langevin and Fokker-Planck equations in polymer rheology. In: Handbook of Numerical Analysis, vol. 16, pp. 211-303 (2011)

Lizana, L, Ambjörnsson, T, Taloni, A, Barkai, E, Lomholt, MA: Foundation of fractional Langevin equation: harmonization of a many-body problem. Phys. Rev. E 81, Article ID 051118 (2010)

Ahmad, B, Eloe, PW: A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices. Commun. Appl. Nonlinear Anal. 17, 69-80 (2010)

Ahmad, B, Nieto, JJ, Alsaedi, A, El-Shahed, M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 13, 599-606 (2012)

Tariboon, J, Ntouyas, SK, Thaiprayoon, C: Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. Adv. Math. Phys. 2014, Article ID 372749 (2014)

Jarad, F, Abdeljawad, T, Baleanu, D: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012)

Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)

Krasnoselskii, MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10, 123-127 (1955)

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Tập 2016

1-13

Thông tin tác giả