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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


A Study on Langevin Equations with ψ-Hilfer Fractional Derivative

Discontinuity, Nonlinearity, and Complexity 8(3) (2019) 261--269 | DOI:10.5890/DNC.2019.09.002

S. Harikrishnan, K. Kanagarajan, D. Vivek

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore-641020, India

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In this paper, we discuss the existence, uniqueness and four types of Ulam stability results for a general class of Langevin equations. An illustrate example is given tocheck the applicable of this results.


  1. [1]  Ahmad, B., Nieto, J.J., Alsaedi, A., and El-Shahed, M. (2012), A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Analysis: Real World Applications, 13, 599-606.
  2. [2]  Chen, A. and Chen, Y. (2011), Existence of solutions to nonlinear langevin equation involving two fractional orders with boundary value conditions, Boundary Value Problems, DOI:10.1155/2011/516481.
  3. [3]  Fa, K.S. (2006), Generalized Langevin equation with fractional derivative and long-time correlation function, Physical Review E, 73, DOI: 10.1103/PhysRevE.73.061104.
  4. [4]  Fa, K.S. (2007), Fractional Langevin equation and Riemann-Liouville fractional derivative, European Physical Journal E, 24, 139-143.
  5. [5]  Omid, B. (2016), On fractional Langevin equation involving two fractional orders, Communications in Nonlinear Science and Numerical Simulation, DOI: 10.1016/j.cnsns.2016.05.023.
  6. [6]  Wang, J.R. and Li, X. (2015) Ulam-Hyers stability of fractional Langevin equations, Applied Mathematics and Computation, 258, 72-83.
  7. [7]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Amsterdam: Elsevier.
  8. [8]  Vanterler da, C. Sousa, J., and Capelas de Oliveira, E. (2017), On the ψ-Hilfer fractional derivative, arXiv:1704.08186.
  9. [9]  Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.