Distributed and variable order differential-operator equations

  • Sabir Umarov
Part of the Developments in Mathematics book series (DEVM, volume 41)


In Section 5.6 we studied the existence of a solution to the multi-point value problem for a fractional order pseudo-differential equation with m fractional derivatives of the unknown function. This is an example of fractional distributed order differential equations. Our main purpose in this chapter is the mathematical treatment of boundary value problems for general distributed and variable order fractional differential-operator equations. We will study the existence and uniqueness of a solution to initial and multi-point value problems in different function spaces.


Variable-order Differential Equations Order Distribution Fractional Derivative Cauchy Type Problem Duhamel Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [Agr04]
    Agraval, Om.: Analytical solution for stochastic response of a fractionally damped beam. J. Vibration and Acoustics, 126, 561–566 (2004)Google Scholar
  2. [AUS06]
    Andries, E., Umarov, S.R., Steinberg, St.: Monte Carlo random walk simulations based on distributed order differential equations with applications to cell biology. Frac. Calc. Appl. Anal., 9 (4), 351–369 (2006)MathSciNetzbMATHGoogle Scholar
  3. [AB92]
    Antipko, I.I., Borok, V.M.: The Neumann boundary value problem in an infinite layer. J. Sov. Math. 58, 541–547 (1992) (Translation from: Theoret. Funktion. Anal. Prilozh.53, 71–78 (1990))Google Scholar
  4. [BT00]
    Bagley, R.L., Torvic P.J.: On the existence of the order domain and the solution of distributed order equations I, II. Int. J. Appl. Math. 2, 865–882, 965–987 (2000)zbMATHGoogle Scholar
  5. [Bor71]
    Borok, V.M.: Correctly solvable boundary-value problems in an infinite layer for systems of linear partial differential equations. Math. USSR. Izv. 5, 193–210 (1971)CrossRefGoogle Scholar
  6. [Cap67]
    Caputo M.: Linear models of dissipation whose Q is almost frequency independent, II. Geophys. J. R. Astr. Soc., 13, 529–539 (1967)CrossRefGoogle Scholar
  7. [Cap01]
    Caputo, M.: Distributed order differential equations modeling dielectric induction and diffusion. Fract. Calc. Appl. Anal., 4, 421–442 (2001)MathSciNetzbMATHGoogle Scholar
  8. [CGSG03]
    Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Gonchar V.Yu.: Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal., 6, 259–279 (2003)MathSciNetzbMATHGoogle Scholar
  9. [CGS05]
    Chechkin, A.V., Gorenflo, R., Sokolov I.M.: Fractional diffusion in inhomogeneous media. J. Physics. A: Math. Gen., 38, 679–684 (2005)MathSciNetCrossRefGoogle Scholar
  10. [CSK11]
    Chechkin, A.V., Sokolov, I.M., Klafter, J.: Natural and modified forms of distributed order fractional diffusion equations. J. Klafter, S.C. Lim, R. Metzler (eds): Fractional Dynamics: Recent Advances. Singapore: World Scientific, Ch. 5, 107–127 (2011)Google Scholar
  11. [DF01]
    Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal., 4, 531–542 (2001)MathSciNetzbMATHGoogle Scholar
  12. [DF09]
    Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comp. Appl. Math. 225 (1), 96–104 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [DP65]
    Ditkin, V.A., Prudnikov A.P.: Integral Transforms and Operational Calculus. Oxford, Pergamon (1965)zbMATHGoogle Scholar
  14. [Dub81]
    Dubinskii, Yu. A.: On a method of solving partial differential equations. Sov. Math. Dokl. 23, 583–587 (1981)Google Scholar
  15. [GV91]
    Gorenflo, R., Vessella, S.: Abel Integral Equations: Analysis and Applications. Lecture Notes in Mathematics, 1461, Springer Verlag, Berlin (1991)Google Scholar
  16. [GLU00]
    Gorenflo, R., Luchko, Yu., Umarov, S.R.: The Cauchy and multi-point partial pseudo-differential equations of fractional order. Fract. Calc. Appl. Anal., 3 (3), 249–275 (2000)MathSciNetzbMATHGoogle Scholar
  17. [HKU10]
    Hahn, M.G., Kobayashi, K., Umarov, S.R.: SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations. J. Theoret. Prob., 25 (1), 262–279 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [JCP12]
    Jiao, Zh., Chen, Y.-Q., Podlubny, I.: Distributed-Order Dynamic Systems. Springer, London (2012)CrossRefzbMATHGoogle Scholar
  19. [Kat12]
    Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comp. Phys. 259, 11–22 (2012)MathSciNetCrossRefGoogle Scholar
  20. [KAN10]
    Kazemipour, S.A., Ansari, A., Neyrameh, A.: Explicit solution of space-time fractional Klein-Gordon equation of distributed order via the Fox H-functions. M. East J. Sci. Res. 6 (6), 647–656 (2010)Google Scholar
  21. [Koc08]
    Kochubey, A. Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. and Appl. 340 (1), 252–281 (2008)MathSciNetCrossRefGoogle Scholar
  22. [Lim06]
    Lim S. C.: Fractional derivative quantum fields at positive temperature. Physica A: Statistical Mechanics and its Applications 363, 269–281 (2006)MathSciNetCrossRefGoogle Scholar
  23. [LH02]
    Lorenzo, C.F., Hartley T.T.: Variable order and distributed order fractional operators. Nonlinear Dynamics 29, 57–98 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [MS06]
    Meerschaert, M.M., Scheffler, H.-P.: Stochastic model for ultraslow diffusion. Stochastic professes and their applications, 116 (9), 1215–1235 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Pod99]
    Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, V 198. Academic Press, San Diego, Boston (1999)Google Scholar
  26. [Pta84]
    Ptashnik, B.I.: Ill-Posed Boundary Value Problems for Partial Differential Equations. Kiev (1984) (in Russian)Google Scholar
  27. [RC10]
    Ramirez, L.E.S., Coimbra, C.F.M.: On the selection and meaning of variable order operators for dynamic modeling. Intern. J. Diff. Equ., 16 pp. (2010)Google Scholar
  28. [SKM87]
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, New York and London (1993)zbMATHGoogle Scholar
  29. [SCK05]
    Soon, C.M., Coimbra, C.F.M., Kobayashi, M.H.: The variable viscoelasticity oscillator. Ann. Phys. (Leipzig) 14, 378–389 (2005)Google Scholar
  30. [Ton28]
    Tonelli, L.: Su un problema di Abel.] Math. Ann. 99, 183–199 (1928)Google Scholar
  31. [Uma86]
    Umarov, S.R.: Boundary value problems for differential operator and pseudo-differential equations. Izv. Acad. Sci., RU, 4, 38–42 (1986) (in Russian)MathSciNetGoogle Scholar
  32. [Uma97]
    Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations I. Differ. Equations, 33, 831–840 (1997)MathSciNetzbMATHGoogle Scholar
  33. [Uma98]
    Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations II. Differ. Equations, 34, 374–381 (1988)MathSciNetGoogle Scholar
  34. [Uma12]
    Umarov, S.R.: On fractional Duhamel’s principle and its applications. J. Differential Equations 252 (10), 5217–5234 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [UG05-2]
    Umarov, S.R., Gorenflo, R. The Cauchy and multipoint problem for distributed order fractional differential equations. ZAA, 24, 449–466 (2005)MathSciNetzbMATHGoogle Scholar
  36. [Van89]
    Van, T. D.: On the pseudo-differential operators with real analytic symbol and their applications. J. Fac. Sci. Univ. Tokyo, IA, Math. 36, 803–825 (1989)Google Scholar
  37. [Wid46]
    Widder, D.V.: The Laplace transform. Princeton University Press (1946)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sabir Umarov
    • 1
  1. 1.Department of MathematicsUniversity of New HavenWest HavenUSA

Personalised recommendations