Initial and boundary value problems for fractional order differential equations

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


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


  1. [A26]
    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. [Baz98]
    Bazhlekova E.: The abstract Cauchy problem for the fractional evolution equation. Frac. Calc. Appl. Anal., 1, 255–270 (1998)MathSciNetzbMATHGoogle Scholar
  3. [Baz01]
    Bazhlekova, E.: Fractional evolution equations in Banach spaces. Dissertation, Technische Universiteit Eindhoven, 117 pp (2001)Google Scholar
  4. [CKS03]
    Chechkin, A.V., Klafter, J., Sokolov I.M.: Fractional Fokker-Planck equation for ultraslow diffusion. EPL, 63 (3), 326–334 (2003)CrossRefGoogle Scholar
  5. [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
  6. [CGKS8]
    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)MathSciNetCrossRefGoogle Scholar
  7. [ES95]
    El-Sayed, A.M. Fractional order evolution equations. J. of Frac. Calc., 7, 89–100 (1995)MathSciNetzbMATHGoogle Scholar
  8. [EK04]
    Eidelman, S.D., Kochubei, A.N.: Cauchy problem for fractional diffusion equations. Journal of Differential Equations, 199, 211–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Fel52]
    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. [Fuj90]
    Fujita, Y.: Integrodifferential equation which interpolates the heat and the wave equations. Osaka J. Math. 27, 309–321, 797–804 (1990)MathSciNetzbMATHGoogle Scholar
  11. [Ger48]
    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)MathSciNetzbMATHGoogle Scholar
  12. [Goo10]
    Goodrich, C.S.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [GAR04]
    Gorenflo, R., Abdel-Rehim, E.: Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. J. Comp. and Appl. Math. 205 871–881 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [GM98-1]
    Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal., 1, 167–191 (1998)MathSciNetzbMATHGoogle Scholar
  15. [GM98-2]
    Gorenflo, R., Mainardi, F.: Fractional calculus and stable probability distributions. Archives of Mechanics, 50, 377–388 (1998)MathSciNetzbMATHGoogle Scholar
  16. [GM99]
    Gorenflo, R., Mainardi, F.: Approximation of Lévy-Feller diffusion by random walk. ZAA, 18 (2) 231–246 (1999)MathSciNetzbMATHGoogle Scholar
  17. [GM01]
    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. [GV03]
    Gorenflo, R., Vivoli, A,: Fully discrete random walks for space-time fractional diffusion equations. Signal Processing, 83, 2411–2420 (2003)CrossRefzbMATHGoogle Scholar
  19. [GLU00a]
    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)MathSciNetGoogle Scholar
  20. [GMM02]
    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)CrossRefGoogle Scholar
  21. [GK10]
    Guezane-Lakoud, A., Kelaiaia, S.: Solvability of a three-point nonlinear boundary value problem. Electron. J. Differ. Equat. 139, 1–9 (2010)MathSciNetGoogle Scholar
  22. [Hil00]
    Hilfer R. (ed): Applications Of Fractional Calculus In Physics. World Scientific (2000)Google Scholar
  23. [Hor83]
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, I - IV. Springer-Verlag, Berlin-Heidelberg-New-York (1983)Google Scholar
  24. [Ibr14]
    Ibrahim, R.W.: Solutions to systems of arbitrary-order differential equations in complex domains. Electronic Journal of Differential Equations, 46, 2014, 1–13 (2014)CrossRefGoogle Scholar
  25. [KO14]
    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)MathSciNetCrossRefGoogle Scholar
  26. [KJ11]
    Kehue, L., Jigen, P.: Fractional abstract Cauchy problems. Integr. Equ. Oper. Theory, 70, 333–361 (2011)CrossRefGoogle Scholar
  27. [KMSL13]
    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)CrossRefzbMATHGoogle Scholar
  28. [KST06]
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory And Applications of Fractional Differential Equations. Elsevier (2006)Google Scholar
  29. [Koc89]
    Kochubei, A.: Parabolic pseudo-differential equations, hypersingular integrals and Markov processes. Math. USSR, Izvestija 33, 233–259 (1989)MathSciNetCrossRefGoogle Scholar
  30. [Koc08]
    Kochubey, A. Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. and Appl. 340 (1), 252–281 (2008)MathSciNetCrossRefGoogle Scholar
  31. [Kos93]
    Kostin, V.A.: The Cauchy problem for an abstract differential equation with fractional derivatives. Russ. Dokl. Math. 46, 316–319 (1993)MathSciNetGoogle Scholar
  32. [Lim06]
    Lim S. C.: Fractional derivative quantum fields at positive temperature. Physica A: Statistical Mechanics and its Applications 363, 269–281 (2006)MathSciNetCrossRefGoogle Scholar
  33. [LH02]
    Lorenzo, C.F., Hartley T.T.: Variable order and distributed order fractional operators. Nonlinear Dynamics 29, 57–98 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Mag06]
    Magin R.: Fractional Calculus in Bioengineering. Begell House Publishers Inc. (2006)Google Scholar
  35. [MPG99]
    Mainardi, F., Paradisi, P., Gorenflo, R.: Probability distributions generated by fractional diffusion equations. Fracalmo Center publ., 46 pp. (1999) (available online:
  36. [MPG07]
    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. [Mai10]
    Mainardi, F.: Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models. Imperial College Press (2010)Google Scholar
  38. [MLP01]
    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. [McC96]
    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. [MS01]
    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)zbMATHGoogle Scholar
  41. [MK00]
    Metzler, R,. Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339, 1–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. [MK04]
    Metzler, R., Klafter, J.: The restaurant in the end of random walk. Physics A: Mathematical and General. 37 (31), 161–208 (2004)MathSciNetCrossRefGoogle Scholar
  43. [MN14]
    Mijena, J.B., Nane, N.: Space-time fractional stochastic partial differential equations. (2014) (available online:
  44. [MR93]
    Miller, K.C., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, Inc., New York (1993)zbMATHGoogle Scholar
  45. [Nak75]
    Nakhushev, A.M.: A mixed problem for degenerate elliptic equations. Differ. Equations, 11, 152–155 (1975)Google Scholar
  46. [Nat86]
    Natterer, F.: The mathematics of computerized tomography. Chichester, UK, Wiley (1986)zbMATHGoogle Scholar
  47. [Naz97]
    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)MathSciNetGoogle Scholar
  48. [Nig86]
    Nigmatullin, R.R.: The realization of generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133, 425–430 (1986)CrossRefGoogle Scholar
  49. [OS74]
    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. [PR92]
    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. [Pod99]
    Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, V 198. Academic Press, San Diego, Boston (1999)Google Scholar
  52. [Rad82]
    Radyno, Ya.V.: Linear equations and bornology. BSU, Minsk (1982) (in Russian)Google Scholar
  53. [Ros75]
    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. [RSh10]
    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. [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
  56. [Sax01]
    Saxton, M.J.: Anomalous Subdiffusion in Fluorescence Photobleaching Recovery: A Monte Carlo Study. Biophys. J., 81(4), 2226–2240 (2001)CrossRefGoogle Scholar
  57. [SJ97]
    Saxton, M.J., Jacobson, K.: Single-particle tracking: applications to membrane dynamics. Ann. Rev. Biophys. Biomol. Struct., 26, 373–399 (1997)CrossRefGoogle Scholar
  58. [SGM00]
    Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)MathSciNetCrossRefGoogle Scholar
  59. [ScS01]
    Schmitt, F.G., Seuront, L.: Multifractal random walk in copepod behavior. Physica A, 301, 375–396 (2001)CrossRefzbMATHGoogle Scholar
  60. [Sch90]
    Schneider, W.R.: Fractional diffusion. Lect. Notes Phys. 355, Heidelberg, Springer, 276–286 (1990)Google Scholar
  61. [SW89]
    Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. Journal of Mathematical Physics, 30, 134–144 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  62. [Sto13]
    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)zbMATHGoogle Scholar
  63. [Tat14]
    Tatar, S.: Existence and uniqueness in an inverse source problem for a one-dimensional time-fractional diffusion equation. (available online:
  64. [TU94]
    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. [Uma94]
    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. [Uma98]
    Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations II. Differ. Equations, 34, 374–381 (1988)MathSciNetGoogle Scholar
  67. [Uma12]
    Umarov, S.R.: On fractional Duhamel’s principle and its applications. J. Differential Equations 252 (10), 5217–5234 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  68. [US06]
    Umarov, S.R., Saydamatov, E.M.: A fractional analog of the Duhamel principle. Frac. Calc. Appl. Anal, 9 (1), 57–70 (2006)MathSciNetzbMATHGoogle Scholar
  69. [US07]
    Umarov, S.R., Saydamatov E.M.: A generalization of the Duhamel principle for fractional order differential equations. Doklady Mathematics, 75 (1), 94–96 (2007)MathSciNetCrossRefGoogle Scholar
  70. [UZ99]
    Uchaykin, V.V., Zolotarev, V.M.: Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht (1999)CrossRefGoogle Scholar
  71. [WZh14]
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  72. [Wis86]
    Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  73. [Zas02]
    Zaslavsky, G.: Chaos, fractional kinetics, and anomalous transport. Physics Reports, 371, 461–580 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  74. [ZhX11]
    Zhang, Y., Xiang Xu.: Inverse source problem for a fractional diffusion equation. Inverse problems, 27, 035010, 12 pp. (2011)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