Skip to main content
SpringerLink
Log in
Book cover

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols pp 249–283Cite as

Distributed and variable order differential-operator equations

  • Chapter

  • 1411 Accesses

Part of the book series: Developments in Mathematics ((DEVM,volume 41))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We assume that such \(\eta _{0}\) exists.

References

  1. Agraval, Om.: Analytical solution for stochastic response of a fractionally damped beam. J. Vibration and Acoustics, 126, 561–566 (2004)

    Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. 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. 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)

    MATH  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. Caputo M.: Linear models of dissipation whose Q is almost frequency independent, II. Geophys. J. R. Astr. Soc., 13, 529–539 (1967)

    Article  Google Scholar 

  7. Caputo, M.: Distributed order differential equations modeling dielectric induction and diffusion. Fract. Calc. Appl. Anal., 4, 421–442 (2001)

    MathSciNet  MATH  Google Scholar 

  8. 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)

    MathSciNet  MATH  Google Scholar 

  9. Chechkin, A.V., Gorenflo, R., Sokolov I.M.: Fractional diffusion in inhomogeneous media. J. Physics. A: Math. Gen., 38, 679–684 (2005)

    Article  MathSciNet  Google Scholar 

  10. 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. Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal., 4, 531–542 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comp. Appl. Math. 225 (1), 96–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ditkin, V.A., Prudnikov A.P.: Integral Transforms and Operational Calculus. Oxford, Pergamon (1965)

    MATH  Google Scholar 

  14. Dubinskii, Yu. A.: On a method of solving partial differential equations. Sov. Math. Dokl. 23, 583–587 (1981)

    Google Scholar 

  15. Gorenflo, R., Vessella, S.: Abel Integral Equations: Analysis and Applications. Lecture Notes in Mathematics, 1461, Springer Verlag, Berlin (1991)

    Google Scholar 

  16. 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)

    MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiao, Zh., Chen, Y.-Q., Podlubny, I.: Distributed-Order Dynamic Systems. Springer, London (2012)

    Book  MATH  Google Scholar 

  19. Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comp. Phys. 259, 11–22 (2012)

    Article  MathSciNet  Google Scholar 

  20. 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. Kochubey, A. Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. and Appl. 340 (1), 252–281 (2008)

    Article  MathSciNet  Google Scholar 

  22. Lim S. C.: Fractional derivative quantum fields at positive temperature. Physica A: Statistical Mechanics and its Applications 363, 269–281 (2006)

    Article  MathSciNet  Google Scholar 

  23. Lorenzo, C.F., Hartley T.T.: Variable order and distributed order fractional operators. Nonlinear Dynamics 29, 57–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Meerschaert, M.M., Scheffler, H.-P.: Stochastic model for ultraslow diffusion. Stochastic professes and their applications, 116 (9), 1215–1235 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, V 198. Academic Press, San Diego, Boston (1999)

    Google Scholar 

  26. Ptashnik, B.I.: Ill-Posed Boundary Value Problems for Partial Differential Equations. Kiev (1984) (in Russian)

    Google Scholar 

  27. 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. 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)

    MATH  Google Scholar 

  29. Soon, C.M., Coimbra, C.F.M., Kobayashi, M.H.: The variable viscoelasticity oscillator. Ann. Phys. (Leipzig) 14, 378–389 (2005)

    Google Scholar 

  30. Tonelli, L.: Su un problema di Abel.] Math. Ann. 99, 183–199 (1928)

    Google Scholar 

  31. Umarov, S.R.: Boundary value problems for differential operator and pseudo-differential equations. Izv. Acad. Sci., RU, 4, 38–42 (1986) (in Russian)

    MathSciNet  Google Scholar 

  32. Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations I. Differ. Equations, 33, 831–840 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations II. Differ. Equations, 34, 374–381 (1988)

    MathSciNet  Google Scholar 

  34. Umarov, S.R.: On fractional Duhamel’s principle and its applications. J. Differential Equations 252 (10), 5217–5234 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Umarov, S.R., Gorenflo, R. The Cauchy and multipoint problem for distributed order fractional differential equations. ZAA, 24, 449–466 (2005)

    MathSciNet  MATH  Google Scholar 

  36. 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. Widder, D.V.: The Laplace transform. Princeton University Press (1946)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of New Haven, West Haven, CT, USA

    Sabir Umarov

Authors
  1. Sabir Umarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Umarov, S. (2015). Distributed and variable order differential-operator equations. In: Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols. Developments in Mathematics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-20771-1_6

Download citation

Publish with us

Policies and ethics

Search

Navigation