Toward a new understanding of cohomological method for fractional partial differential equations

Document Type : Research Paper


Department of Mathematics, Iran University of Science and Technology, P.O.Box, 16846-13114, Narmak, Tehran, Iran.


One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation ψ αf = g, on T n. We prove that if f is not analytic, then fractional integro-differential equation I 1−α t Dα x u(x, t) + iI1−α x Dα t u(x, t) = f(t) has no solution in C1 (B) with 0 < α ≤ 1. We also obtain solutions for the space-time fractional heat equations on S 1 and T n. At the end of this article, there are examples of fractional partial differential equations and a fractional integral equation together with their solutions.


  • [1]           E. K. Alaoui, H. Hmili, and et al., Cohomological equations and invariant distributions on a compact lie group, Hokkaido Mathematical Journal, 2 (2014), 151–173.
  • [2]           E. Bazhlekovaand and I. Bazhlekov, Viscoelastic flows with fractional derivative models: com- putational approach by convolutional calculus of dimovski, Fractional Calculus and Applied Analysis, 4 (2014 ), 954–976.
  • [3]           A. Dehghan-Nezhad and A. E. K. ALAOUI, Equations cohomologiques de flots riemanniens et de diffeomorphismes danosov, Journal of the Mathematical Society of Japan, 4 (2007), 1105– 1134.
  • [4]           F. F. Di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math, 1 (1857), 359–360.
  • [5]           B. Hasselblatt and A. Katok, Handbook of dynamical systems (2002), 287-289.
  • [6]           R. Hilfer and et al., Applications of fractional calculus in physics, World scientific Singapore, 12 (2000), 1-87.
  • [7]           A. B. Katok, Combinatorial constructions in ergodic theory and dynamics, 30 (2003), 53-117.
  • [8]           A. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 204 (2006), 449-469.
  • [9]           A. Livsic, Homology properties of y-systems, Mathematical notes of the Academy of Sciences of the USSR, 5 (1971), 69-135.
  • [10]         A. Livsic, Cohomology of dynamical systems, Mathematics of the USSR-Izvestiya, 6 (1972).
  • [11]         R. L. Magin, Fractional calculus in bioengineering, Begell House Redding, 6 (2006), 1278-1301.
  • [12]         K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, (1993), 1-104.
  • [13]         K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, (1974), 87-93.
  • [14]         I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, frac- tional differential equations, to methods of their solution and some of their applications, Else- vier, (1998), 21-105.
  • [15]         Y. A. Rossikhin and M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Applied Mechanics Reviews, 1 (2010), 25-45.
  • [16]         H. Rudolf, Applications of fractional calculus in physics, World Scientific, (2000), 41-121.
  • [17]         S. G. Samko, A. A. Kilbas, O. I. Marichev, and et al., Fractional integrals and derivatives, Gordon and Breach Science Publishers, (1993), 1-52 .
  • [18]         W. M. Schmidt, Diophantine approximation, Springer Science and Business Media, (1996), 331-429.
  • [19]         V. E. Tarasov, On history of mathematical economics: Application of fractional calculus, Math- ematics, 7 (2019), 26-35.
  • [20]         V. E. Tarasov, Applications in physics, Walter de Gruyter GmbH and Co KG, 2019.
  • [21]         M. S. Tavazoei, Fractional order chaotic systems: history, achievements, applications, and future challenges, The European Physical Journal Special Topics, 229 (2020), 1-28.
  • [22]         M. Unser, S. Horbelt, and T. Blu, Fractional derivatives, splines and tomography, 2000 10th European Signal Processing Conference, (2000), 1-4.