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Fractional integration and differentiation of variable order: an overview

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Abstract

We give an overview of a selection of studies on fractional operations of integration and differentiation of variable order, when this order may vary from point to point. We touch on both the Euclidean setting and also the general setting within the framework of quasimetric measure spaces.

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Abbreviations

ℝn :

is the n-dimensional Euclidean space, \(|x|=\sqrt {x_{1}^{2}+\cdots+ x_{n}^{2}}\);

\(\mathbb{S}^{n-1}\) :

is the unit sphere in ℝn centered at the origin, \(|\mathbb{S}^{n-1}|\) is its surface area;

Δ:

is the Laplace operator;

(X,ϱ,μ):

denotes a quasimetric measure space with quasidistance ϱ and measure μ;

δ(x,Ω)=inf y∈Ω ϱ(x,y):

is the distance of a point x∈X to a set Ω⊂X.

References

  1. Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15(2), 195–208 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Almeida, A., Samko, S.: Pointwise inequalities in variable Sobolev spaces and applications. Z. Anal. Anwend. 26(2), 179–193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balakrishnan, A.V.: Fractional powers of closed operators and the semigroups generated by them. Pac. J. Math. 10(2), 419–437 (1960)

    Article  MATH  Google Scholar 

  4. Coimbra, C.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cooper, G.R.J., Cowan, D.R.: Filtering using variable order vertical derivatives. Comput. Geosci. 30, 455–459 (2004)

    Article  Google Scholar 

  6. Diening, L., Harjulehto, P., Hástó, Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  7. Diening, L., Hästö, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2. Math. Inst. Acad. Sci., Praha (2004)

    Google Scholar 

  8. Diening, L., Samko, S.: On potentials in generalized Ḧolder spaces over uniform domains in ℝn. Rev. Mat. Complut. 24(2), 357–373 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diethelm, K.: The Analysis of Fractional Differential Equations: an Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  10. Edmunds, D.E., Kokilashvili, V., Meskhi, A.: Two-weight estimates for singular integrals defined on spaces of homogeneous type. Can. J. Math. 52(3), 468–502 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gatto, A.E.: On fractional calculus associated to doubling and non-doubling measures. In: Marshall, J., Ash, et al. (eds.) Harmonic Analysis. Calderon–Zygmund and Beyond. A Conference in Honor of Stephen Vagi’s Retirement, Chicago, IL, USA, December 6–8, 2002. Contemporary Mathematics, vol. 411, pp. 15–37. Am. Math. Soc., Providence (2006)

    Google Scholar 

  12. Gatto, A.E., Garcia-Cuerva, J.: Boundedness properties of fractional integral operators associated to non-doubling measures. Stud. Math. 162(3), 245–261 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gatto, A.E., Segovia, C., Vagi, S.: On fractional differentiation on spaces of homogeneous type. Rev. Mat. Iberoam. 12(1), 1–35 (1996)

    MathSciNet  Google Scholar 

  14. Gatto, A.E., Vagi, S.: Fractional integrals on spaces of homogeneous type. In: Sadosky, C. (ed.) Analysis and Partial Differential Equations. Lecture Notes in Pure and Appl. Math., vol. 122, pp. 171–216. Dekker, New York (1990)

    Google Scholar 

  15. Guliev, V., Hasanov, J., Samko, S.: Boundedness of the maximal, potential and singular integral operators in the generalized variable exponent Morrey spaces M p(⋅),θ(⋅),ω(⋅)(Ω). J. Math. Sci. 170(4), 1–21 (2010)

    MathSciNet  Google Scholar 

  16. Guliev, V., Hasanov, J., Samko, S.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107, 285–304 (2010)

    MathSciNet  Google Scholar 

  17. Hilfer, R. (ed.): Fractional Calculus in Physics. World Scientific, Singapore (2000). 463 pp.

    MATH  Google Scholar 

  18. Karapetyants, N.K., Ginzburg, A.I.: Fractional integrals and singular integrals in the Hölder classes of variable order. Integral Transforms Spec. Funct. 2(2), 91–106 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karapetyants, N.K., Ginzburg, A.I.: Fractional integro-differentiation in Hölder classes of variable order. Dokl. Akad. Nauk SSSR 339(4), 439–441 (1994)

    Google Scholar 

  20. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). 523 pp.

    MATH  Google Scholar 

  21. Kiryakova, V.: Generalized Fractional Calculus and Applications. Longman, Harlow (1994). 388 pp.

    MATH  Google Scholar 

  22. Kobelev, L.Ya., Kobelev, Ya.L., Klimontovich, Yu.L.: Equilibrium statistical physics in fractal media with constant and variable memory. Dokl. Akad. Nauk SSSR 391(1), 35–39 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Kobelev, Ya.L., Kobelev, L.Ya., Klimontovich, Yu.L.: Statistical physics of dynamic systems with variable memory. Dokl. Phys. 48(6), 285–289 (2003)

    Article  MathSciNet  Google Scholar 

  24. Kokilashvili, V.: On a progress in the theory of integral operators in weighted Banach function spaces. In: Function Spaces, Differential Operators and Nonlinear Analysis, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2. Math. Inst. Acad. Sci., Praha (2004)

    Google Scholar 

  25. Kokilashvili, V., Meskhi, A.: On some weighted inequalities for fractional integrals on nonhomogeneous spaces. Z. Anal. Ihre Anwend. 24(4), 871–885 (2005)

    Article  MathSciNet  Google Scholar 

  26. Kokilashvili, V., Meskhi, A.: Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent. Integral Transforms Spec. Funct. 18(9), 609–628 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kokilashvili, V., Meskhi, A.: Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure. Complex Var. Elliptic Equ. 55(8–10), 923–936 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Kokilashvili, V., Samko, S.: Weighted boundedness of the maximal, singular and potential operators in variable exponent spaces. In: Kilbas, A.A., Rogosin, S.V. (eds.) Analytic Methods of Analysis and Differential Equations, pp. 139–164. Cambridge Scientific, Bethesda (2008)

    Google Scholar 

  29. Kokilashvili, V., Samko, S.: Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent. Acta Math. Sin. 24(11), 1775–1800 (2009)

    Article  MathSciNet  Google Scholar 

  30. Macías, R.A., Segovia, C.: Lipshitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)

    Article  MATH  Google Scholar 

  31. Marchaud, A.: Sur les derivees et sur les differences des fonctions des variables reelles. J. Math. Pures Appl. 6(4), 337–425 (1927)

    Google Scholar 

  32. Martinez, C., Sanz, M.: The Theory of Fractional Powers of Operators. Elsevier, Amsterdam (2001)

    MATH  Google Scholar 

  33. McBride, A.C.: Fractional Calculus and Integral Transforms of Generalized Functions. Pitman, London (1979). 179 pp.

    MATH  Google Scholar 

  34. Mikhlin, S.G.: Multi-Dimensional Singular Integrals and Integral Equations. Pergamon, Elmsford (1965)

    Google Scholar 

  35. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  36. Oldham, K.B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, San Diego (1974)

    MATH  Google Scholar 

  37. Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999). 368 pp.

    Google Scholar 

  38. Rafeiro, H., Samko, S.: On multidimensional analogue of Marchaud formula for fractional Riesz-type derivatives in domains in R n. Fract. Calc. Appl. Anal. 8(4), 393–401 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Riesz, M.: L’intégrales de Riemann–Liouville et potentiels. Acta Litt. Sci. Szeged 9, 1–42 (1938)

    MathSciNet  Google Scholar 

  40. Ross, B., Samko, S.G.: Fractional integration operator of variable order in the Hölder spaces H λ(x). Int. J. Math. Math. Sci. 18(4), 777–788 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rubin, B.S.: Fractional Integrals and Potentials. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 82. Longman, Harlow (1996). 409 pp.

    MATH  Google Scholar 

  42. Samko, N., Samko, S., Vakulov, B.: Weighted Sobolev theorem in Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 335, 560–583 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Samko, N., Samko, S., Vakulov, B.: Weighted Sobolev theorem in Lebesgue spaces with variable exponent; corrigendum. Armen. J. Math. 3(2), 92–97 (2009)

    MathSciNet  Google Scholar 

  44. Samko, N., Samko, S., Vakulov, B.: Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces. J. Funct. Spaces Appl. 8(3), 215–244 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Samko, N., Vakulov, B.: Spherical fractional and hypersingular integrals in generalized hölder spaces with variable characteristic. Math. Nachr. 284, 355–369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Samko, S.: Differentiation and integration of variable (fractional) order. Dokl. Akad. Nauk, Ross. Akad. Nauk 342(4), 458–460 (1995)

    MathSciNet  Google Scholar 

  47. Samko, S.: Fractional integration and differentiation of variable order. Anal. Math. 21(3), 213–236 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. Samko, S.: Convolution and potential type operators in L p(x). Integral Transforms Spec. Funct. 7(3–4), 261–284 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. Samko, S.: Differentiation and integration of variable order and the spaces L p(x). In: Proceeding of International Conference on Operator Theory and Complex and Hypercomplex Analysis, Mexico City, Mexico, 12–17 December 1994. Contemp. Math., vol. 212, pp. 203–219 (1998)

    Chapter  Google Scholar 

  50. Samko, S.: Hypersingular Integrals and Their Applications. Analytical Methods and Special Functions, vol. 5. Taylor & Francis, London (2002). 358 + xvii pp.

    MATH  Google Scholar 

  51. Samko, S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct. 16(5–6), 461–482 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, New York (1993). 1012 pp. (Russian edition: Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Tekhnika, Minsk (1987))

    MATH  Google Scholar 

  53. Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transforms Spec. Funct. 1(4), 277–300 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  54. Samko, S.G., Shargorodsky, E., Vakulov, B.: Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. II. J. Math. Anal. Appl. 325(1), 745–751 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. Samko, S.G., Vakulov, B.G.: Weighted Sobolev theorem with variable exponent. J. Math. Anal. Appl. 310, 229–246 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. Stein, E.M.: The characterization of functions arising as potentials. I. Bull. Am. Math. Soc. 67(1), 102–104 (1961)

    Article  Google Scholar 

  57. Vakulov, B.G.: Spherical potentials of complex order in the variable order Holder spaces. Integral Transforms Spec. Funct. 16(5–6), 489–497 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. Vakulov, B.G.: Spherical convolution operators in Hölder spaces of variable order. Mat. Zametki 80(5), 683–695 (2006). Transl. in Math. Notes 80(5), 645–657 (2006)

    Article  MathSciNet  Google Scholar 

  59. Vakulov, B.G., Karapetiants, N.K., Shankishvili, L.D.: Spherical hypersingular operators of imaginary order and their multipliers. Fract. Calc. Appl. Anal. 4(1), 101–112 (2001)

    MathSciNet  MATH  Google Scholar 

  60. Valério, D., Sá da Costa, J.: Variable order fractional derivatives and their numerical approximations. Signal Process. 91(3), 470–483 (2011)

    Article  MATH  Google Scholar 

  61. Yosida, K.: Functional Analysis. Springer, Berlin (1967)

    MATH  Google Scholar 

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  1. University of Algarve, Faro, Portugal

    Stefan Samko

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Samko, S. Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn 71, 653–662 (2013). https://doi.org/10.1007/s11071-012-0485-0

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