On a fractional differential equation with fractional boundary conditions

Document Type : Original Article


1 Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, 578 Sari, Iran,

2 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47416- 95447, Iran


In this article, we study a new nonlinear Langevin equation of two fractional orders with fractional boundary value conditions which is a generalization of previous Langevin equations. Based on Banach and Schauder fixed point theorems, the existence and uniqueness of solutions of this equation are investigated. Moreover, our hypotheses are simpler than similar works.


Main Subjects

[1] B. Ahmad, J.J. Nieto, A. Alsaedi, M. El Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal, RWA,13 2012, 599-606.[2] B. Ahmad, J.J. Nieto, Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions, Int. J. Difference Equ., 2010.
[3] B. Ahmad, J.J. Nieto, A. Alsaedi, A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders, Adv. Diff. Equ., 2012(1) 2012,1-16.
[4] O. Baghani, On fractional Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul., 42 2017, 675-681.
[5] E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana{Baleanu fractional derivative, Chaos, Solitons & Fractals, 116 2018, 121-125.
[6] G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Analysis, 2019.
[7] W.P. do Carmo, M.K. Lenzi, E.K. Lenzi, M. Fortuny, A.F. Santos, A fractional model to relative viscosity prediction of water-in-crude oil emulsions, Journal of Petroleum Science and Engineering, 172 2019, 493-501.
[8] M. D'Ovidio, P. Loreti, S.S. Ahrabi, Modi ed fractional logistic equation, Physica A: Statistical Mechanics and its Applications, 505 2018, 818-824.
[9] C.H. Eab, S.C. Lim, Fractional generalized Langevin equation approach to single- le diffusion, Physica A, 389 2010, 2510-2521.
[10] H. Eslamizadeh, H. Raanaei, Dynamical study of fission process at low excitation energies in the framework of the four-dimensional Langevin equations, Physics Letters B, 783 2018, 163-168.
[11] H. Fazli, J.J. Nieto, Fractional Langevin equation with anti-periodic boundary conditions, Chaos, Solitons & Fractals, 114 2018, 332-337.
[12] J.H. Jeon, R. Metzler, Fractional Brownian motion and motion governed by the fractional Langevin equation in con ned geometries, Physical Review, 81 2010, 021103.
[13] N. Kadkhoda, H. Jafari, Application of fractional sub-equation method to the space-time fractional differential equations, Int. J. Adv. Appl. Math. Mech, 4(2) 2017, 1-6.
[14] N. Kadkhoda, A numerical approach for solving variable-order differential equations using Bernstein polynomials, Alexandria Engineering Journal, 59(5) 2020, 3041-3047.
[15] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and application of fractional differential equations, Elsevier B.V, Netherlands 2006.
[16] R. Kubo, The uctuation{dissipation theorem, Rep. Prog. Phys., 29 1966, 255-284.
[17] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, second ed., Springer Verlag, Berlin, 1991.
[18] B. Li, S. Sun, Y. Sun, Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, J. Appl. Math. Comput., 2017, 683-692.
[19] S.C. Lim, M. Li, L.P. Teo, Langevin equation with two fractional orders, Phys. Lett. A 372 2008, 6309-6320.
[20] J. Long, R. Xiao, W. Chen, Fractional viscoelastic models with nonsingular kernels, Mechanics of Materials, 127 2018, 55-64.
[21] J.A.T. Machado, A.M. Lopes, Fractional-order modeling of a diode, Commun. Nonlinear Sci. Numer. Simul., 70 2019, 343-353.
[22] A. Ortega, J.J. Rosales, J.M. Cruz-Duarte, M. Guia, Fractional model of the dielectric dispersion, Optik, 180 2019, 754-759.
[23] I. Podlubny, Fractional differential equations, Academic Press, San Diego, CA, 1999.
[24] J.V.C. Sousa, K.D. Kucche, E.C. de Oliveira, Stability of ψ-Hilfer impulsive fractional differential equations, Applied Mathematics Letters, 88 2019, 73-80.
[25] S. Ullah, M.A. Khan, M. Farooq, A fractional model for the dynamics of TB virus, Chaos, Solitons & Fractals, 116 2018, 63-71.
[26] T. Yu, K. Deng, M. Luo, Existence and uniqueness of solutions of initial value problems for nonlinear Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul, 19 2014, 1661-1668.
[27] F.S. Zafarghandi, M. Mohammadi, E. Babolian, S. Javadi, Radial basis functions method for solving the fractional diffusion equations, Appl. Math. Comput. 342 2019, 224-246.