International Journal of Differential Equations

Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions

Bashir Ahmad and Juan J. Nieto

Full-text: Open access

Abstract

We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 649486, 10 pages.

Dates
Received: 8 August 2009
Accepted: 14 November 2009
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399876

Digital Object Identifier
doi:10.1155/2010/649486

Mathematical Reviews number (MathSciNet)
MR2575288

Zentralblatt MATH identifier
1207.34007

Citation

Ahmad, Bashir; Nieto, Juan J. Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 649486, 10 pages. doi:10.1155/2010/649486. https://projecteuclid.org/euclid.ijde/1485399876


Export citation

References

  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, The Netherlands, 2006.MR2218073
  • V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif., USA, 1999.
  • B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009.MR2525567
  • B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009.MR2516016
  • B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.
  • B. Ahmad and J. J. Nieto, “Existence of solutions for anti-periodic čommentComment on ref. [4?]: Please update the information of these references [4, 5, 6, 7?], if possible. boundary value problems involving fractional differential equations via Leray-Schauder degree theory,” to appear in Topological Methods in Nonlinear Analysis.
  • B. Ahmad, “Existence of solutions for irregular boundary value problems involving nonlinear fractional differential equations,” Applied Mathematics Letters, 2009.
  • B. Ahmad and J. J. Nieto, “Existence of solutions for impulsive anti-periodic boundary value problems of fractional order,” to appear in Taiwanese Journal of Mathematics.
  • Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.MR2483665
  • V. Daftardar-Gejji and S. Bhalekar, “Boundary value problems for multi-term fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 754–765, 2008.MR2429175
  • R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.MR1890104
  • S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553–558, 2008.MR2378723
  • A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis, vol. 72, pp. 580–586, 2010.
  • D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “Fractional Nambu mechanics,” International Journal of Theoretical Physics, vol. 48, no. 4, pp. 1044–1052, 2009.MR2491296
  • M. R. Ubriaco, “Entropies based on fractional calculus,” Physics Letters A, vol. 373, no. 30, pp. 2516–2519, 2009.MR2542685
  • W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, vol. 14 of World Scientific Series in Contemporary Chemical Physics, World Scientific, River Edge, NJ, USA, 2nd edition, 2004.MR2053912
  • S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309–6320, 2008.MR2462401
  • S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A, vol. 42, no. 6, Article ID 065208, 34 pages, 2009.MR2525432
  • D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, UK, 1980.