Skip to main content
SpringerLink
Log in

Initial and boundary value problems for fractional order differential equations

  • Chapter
Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

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

  • 1430 Accesses

Abstract

In this chapter we will discuss boundary value problems for fractional order differential and pseudo-differential equations. For methodological clarity we first consider in detail the Cauchy problem for pseudo-differential equations of time-fractional order β, \(m - 1 <\beta <m,\) (\(m \in \mathbb{N}\))

$$\displaystyle\begin{array}{rcl} D_{{\ast}}^{\beta }u(t,x) = A(D)u(t,x) + h(t,x),\quad t> 0,\ x \in \mathbb{R}^{n},& &{}\end{array}$$
(5.1)
$$\displaystyle\begin{array}{rcl} \frac{\partial ^{k}u(0,x)} {\partial t^{k}} =\varphi _{k}(x),\quad x \in \mathbb{R}^{n},\ k = 0,\ldots,m - 1,& &{}\end{array}$$
(5.2)

where h(t, x) and \(\varphi _{k},\ k = 0,\ldots,m - 1,\) are given functions in certain spaces described later, \(D = (D_{1},\ldots,D_{n})\), \(D_{j} = -i \frac{\partial } {\partial x_{j}},\ j = 1,\ldots,n\), A(D) is a ΨDOSS with a symbol A(ξ) ∈ XS p (G) defined in an open domain \(G \subset \mathbb{R}^{n}\), and \(D_{{\ast}}^{\beta }\) is the fractional derivative of order β > 0 in the sense of Caputo-Djrbashian (see Section 3.5)

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

Access this chapter

Log in via an institution
eBook
USD 16.99
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

Institutional subscriptions

Notes

  1. 1.

    K β (r) > 0 for all r > 0 if 0 < β < 1. 

  2. 2.

    Regarding the convergence of this series see Remark 5.4.

  3. 3.

    T is an arbitrary positive finite number.

  4. 4.

    With the sign correction effected by the definition of \(\mathbf{D}_{-}^{\alpha }\).

  5. 5.

    Weyl’s lemma [Hor83] states that a distribution f(x), satisfying the equation Δ f = 0 on an open set \(\varOmega \subset \mathbb{R}^{n}\) in the weak sense, is an ordinary harmonic function.

References

  1. Abel, N.H.: Solution of a mechanical problem. (Translated from the German) In: D. E. Smith (ed) A Source Book in Mathematics, Dover Publications, New York, 656–662 (1959)

    Google Scholar 

  2. Bazhlekova E.: The abstract Cauchy problem for the fractional evolution equation. Frac. Calc. Appl. Anal., 1, 255–270 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Bazhlekova, E.: Fractional evolution equations in Banach spaces. Dissertation, Technische Universiteit Eindhoven, 117 pp (2001)

    Google Scholar 

  4. Chechkin, A.V., Klafter, J., Sokolov I.M.: Fractional Fokker-Planck equation for ultraslow diffusion. EPL, 63 (3), 326–334 (2003)

    Article  Google Scholar 

  5. 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 

  6. Chechkin, A.V., Gonchar, V.Yu., Gorenflo, R., Korabel, N., Sokolov, I.M.: Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights. Phys. Rev. E, 78, 021111(13) (2008)

    Article  MathSciNet  Google Scholar 

  7. El-Sayed, A.M. Fractional order evolution equations. J. of Frac. Calc., 7, 89–100 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Eidelman, S.D., Kochubei, A.N.: Cauchy problem for fractional diffusion equations. Journal of Differential Equations, 199, 211–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feller, W.: On a generalization of Marcel Riesz potentials and the semi-groups generated by them. Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome suppl. dédié a M. Riesz. Lund, 73–81 (1952)

    Google Scholar 

  10. Fujita, Y.: Integrodifferential equation which interpolates the heat and the wave equations. Osaka J. Math. 27, 309–321, 797–804 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Gerasimov, A.: A generalization of linear laws of deformation and its applications to problems of internal friction, Prikl. Matem. i Mekh. 12 (3), 251–260 (1948) (in Russian)

    MathSciNet  MATH  Google Scholar 

  12. Goodrich, C.S.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gorenflo, R., Abdel-Rehim, E.: Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. J. Comp. and Appl. Math. 205 871–881 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal., 1, 167–191 (1998)

    MathSciNet  MATH  Google Scholar 

  15. Gorenflo, R., Mainardi, F.: Fractional calculus and stable probability distributions. Archives of Mechanics, 50, 377–388 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Gorenflo, R., Mainardi, F.: Approximation of Lévy-Feller diffusion by random walk. ZAA, 18 (2) 231–246 (1999)

    MathSciNet  MATH  Google Scholar 

  17. Gorenflo, R., Mainardi, F.: Random walk models approximating symmetric space-fractional diffusion processes. In Elschner, Gohberg and Silbermann (eds): Problems in Mathematical Physics (Siegfried Prössdorf Memorial Volume). Birkhäuser Verlag, Boston-Basel-Berlin, 120–145 (2001)

    Google Scholar 

  18. Gorenflo, R., Vivoli, A,: Fully discrete random walks for space-time fractional diffusion equations. Signal Processing, 83, 2411–2420 (2003)

    Article  MATH  Google Scholar 

  19. Gorenflo, R., Luchko, Yu., Umarov, S.R.: On boundary value problems for pseudo-differential equations with boundary operators of fractional order. Fract. Calc. Appl. Anal., 3 (4), 454–468 (2000)

    MathSciNet  Google Scholar 

  20. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion, Chemical Physics, 284, 521–541 (2002)

    Article  Google Scholar 

  21. Guezane-Lakoud, A., Kelaiaia, S.: Solvability of a three-point nonlinear boundary value problem. Electron. J. Differ. Equat. 139, 1–9 (2010)

    MathSciNet  Google Scholar 

  22. Hilfer R. (ed): Applications Of Fractional Calculus In Physics. World Scientific (2000)

    Google Scholar 

  23. Hörmander, L.: The Analysis of Linear Partial Differential Operators, I - IV. Springer-Verlag, Berlin-Heidelberg-New-York (1983)

    Google Scholar 

  24. Ibrahim, R.W.: Solutions to systems of arbitrary-order differential equations in complex domains. Electronic Journal of Differential Equations, 46, 2014, 1–13 (2014)

    Article  Google Scholar 

  25. Kadem, A., Kirane, M., Kirk, C.M., Olmstead, W.E.: Blowing-up solutions to systems of fractional differential and integral equations with exponential non-linearities. IMA Journal of Applied Mathematics, 79, 1077–1088 (2014)

    Article  MathSciNet  Google Scholar 

  26. Kehue, L., Jigen, P.: Fractional abstract Cauchy problems. Integr. Equ. Oper. Theory, 70, 333–361 (2011)

    Article  Google Scholar 

  27. Keyantuo, V., Miana, P.J., Sánches-Lajusticia, L.: Sharp extensions for convoluted solutions of abstract Cauchy problems. Integr. Equat. Oper. Theory, 77, 211–241 (2013)

    Article  MATH  Google Scholar 

  28. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory And Applications of Fractional Differential Equations. Elsevier (2006)

    Google Scholar 

  29. Kochubei, A.: Parabolic pseudo-differential equations, hypersingular integrals and Markov processes. Math. USSR, Izvestija 33, 233–259 (1989)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Kostin, V.A.: The Cauchy problem for an abstract differential equation with fractional derivatives. Russ. Dokl. Math. 46, 316–319 (1993)

    MathSciNet  Google Scholar 

  32. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Magin R.: Fractional Calculus in Bioengineering. Begell House Publishers Inc. (2006)

    Google Scholar 

  35. Mainardi, F., Paradisi, P., Gorenflo, R.: Probability distributions generated by fractional diffusion equations. Fracalmo Center publ., 46 pp. (1999) (available online: http://arxiv.org/pdf/0704.0320v1.pdf)

  36. Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Sub-diffusion equations of fractional order and their fundamental solutions. In “Mathematical methods in engineering”, Springer, 23–55 (2007)

    Google Scholar 

  37. Mainardi, F.: Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models. Imperial College Press (2010)

    Google Scholar 

  38. Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Frac. Calc. Appl. Anal., 4 (2), 153–192 (2001)

    Google Scholar 

  39. McCulloch, J.: Financial applications of stable distributions. In Statistical Methods in Finance: Handbook of Statistics 14, Madfala, G., Rao, C.R. (eds). Elsevier, Amsterdam, 393–425 (1996)

    Google Scholar 

  40. Meerschaert, M.M., Scheffler, H.-P.: Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. John Wiley and Sons, Inc. (2001)

    MATH  Google Scholar 

  41. Metzler, R,. Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  42. Metzler, R., Klafter, J.: The restaurant in the end of random walk. Physics A: Mathematical and General. 37 (31), 161–208 (2004)

    Article  MathSciNet  Google Scholar 

  43. Mijena, J.B., Nane, N.: Space-time fractional stochastic partial differential equations. (2014) (available online: http://arxiv.org/pdf/1409.7366v1.pdf)

  44. Miller, K.C., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, Inc., New York (1993)

    MATH  Google Scholar 

  45. Nakhushev, A.M.: A mixed problem for degenerate elliptic equations. Differ. Equations, 11, 152–155 (1975)

    Google Scholar 

  46. Natterer, F.: The mathematics of computerized tomography. Chichester, UK, Wiley (1986)

    MATH  Google Scholar 

  47. Nazarova, M. Kh.: On weak well-posedness of certain boundary value problems generated by a singular Bessel’s operator. Uzb. Math. J. 3, 63–70 (1997) (in Russian)

    MathSciNet  Google Scholar 

  48. Nigmatullin, R.R.: The realization of generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133, 425–430 (1986)

    Article  Google Scholar 

  49. Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Acad. Press, Dover Publications, New York - London (1974)

    Google Scholar 

  50. Päivärinta, L., Rempel, S.: Corner singularities of solutions to Δ ±1∕2 u = f in two dimensions. Asymptotic analysis, 5, 429–460 (1992)

    Google Scholar 

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

    Google Scholar 

  52. Radyno, Ya.V.: Linear equations and bornology. BSU, Minsk (1982) (in Russian)

    Google Scholar 

  53. Ross, B. (ed): Proceedings of the first international conference “Fractional Calculus and Its Applications. University of New Haven, June 1974”, Springer, Berlin-Heidelberg-New-York (1975)

    Google Scholar 

  54. Rossikhin, Yu.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Applied Mechanics Reviews, 63, 52 pp. (2010)

    Google Scholar 

  55. 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 

  56. Saxton, M.J.: Anomalous Subdiffusion in Fluorescence Photobleaching Recovery: A Monte Carlo Study. Biophys. J., 81(4), 2226–2240 (2001)

    Article  Google Scholar 

  57. Saxton, M.J., Jacobson, K.: Single-particle tracking: applications to membrane dynamics. Ann. Rev. Biophys. Biomol. Struct., 26, 373–399 (1997)

    Article  Google Scholar 

  58. Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)

    Article  MathSciNet  Google Scholar 

  59. Schmitt, F.G., Seuront, L.: Multifractal random walk in copepod behavior. Physica A, 301, 375–396 (2001)

    Article  MATH  Google Scholar 

  60. Schneider, W.R.: Fractional diffusion. Lect. Notes Phys. 355, Heidelberg, Springer, 276–286 (1990)

    Google Scholar 

  61. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. Journal of Mathematical Physics, 30, 134–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  62. Stojanović, M.: Well-Posedness of diffusion-wave problem with arbitrary finite number of time fractional derivatives in Sobolev spaces H s. Frac. Calc. Appl. Anal. 13 (1), 21–22 (2010)

    MATH  Google Scholar 

  63. Tatar, S.: Existence and uniqueness in an inverse source problem for a one-dimensional time-fractional diffusion equation. (available online: http://person.zirve.edu.tr/statar/s12.pdf)

  64. Turmetov, B. Kh., Umarov, S.R.: On a boundary value problem for an equation with the fractional derivative. Russ. Acad. Sci., Dokl., Math., 48, 579–582 (1994)

    Google Scholar 

  65. Umarov, S.R.: On some boundary value problems for elliptic equations with a boundary operator of fractional order. Russ. Acad. Sci., Dokl., Math. 48, 655–658 (1994)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  68. Umarov, S.R., Saydamatov, E.M.: A fractional analog of the Duhamel principle. Frac. Calc. Appl. Anal, 9 (1), 57–70 (2006)

    MathSciNet  MATH  Google Scholar 

  69. Umarov, S.R., Saydamatov E.M.: A generalization of the Duhamel principle for fractional order differential equations. Doklady Mathematics, 75 (1), 94–96 (2007)

    Article  MathSciNet  Google Scholar 

  70. Uchaykin, V.V., Zolotarev, V.M.: Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht (1999)

    Book  Google Scholar 

  71. Wei, T., Zhang, Z.Q.: Stable numerical solution to a Cauchy problem for a time fractional diffusion equation. Engineering analysis with boundary elements, 40, 128–137 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  72. Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  73. Zaslavsky, G.: Chaos, fractional kinetics, and anomalous transport. Physics Reports, 371, 461–580 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  74. Zhang, Y., Xiang Xu.: Inverse source problem for a fractional diffusion equation. Inverse problems, 27, 035010, 12 pp. (2011)

    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). Initial and boundary value problems for fractional order differential 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_5

Download citation

Publish with us

Policies and ethics

Search

Navigation