publication . Article . Preprint . 2017

Weyl and Marchaud derivatives: a forgotten history

Ferrari, Fausto;
Open Access English
  • Published: 21 Nov 2017 Journal: Mathematics (issn: 2227-7390, Copyright policy)
  • Publisher: MDPI AG
Abstract
In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics.
Subjects
free text keywords: fractional derivatives, Weyl derivative, Mathematics, extension operator, Mathematics - Analysis of PDEs, Grünwald–Letnikov derivative, fractional Laplace operator, Marchaud derivative, QA1-939
46 references, page 1 of 4

2 log 2 0

2f (x) − f (x + ξ) − f (x − ξ)

dξ = [1] N.H. Abel Solution de quelques problèmes à l'aide d'integrales definies. Gesammelte mathematische

Werke Leipzig: Teubner, 1 11-27 (First pub. in Mag. Naturvidenkaberne, Aurgang, 1, 2, Christiania

(1823) ) [2] M. Allen, L. Caffarelli, A. Vasseur. Porous medium flow with both a fractional potential pressure and

fractional time derivative. Chin. Ann. Math. Ser. B 38 (2017), no. 1, 45-82. [3] A. Bernardis, F. J. Martín-Reyes, P. R. Stinga, and J. L Torrea. Maximum principles, extension problem

and inversion for nonlocal one-sided equations. Journal of Differential Equations 260, No 7 (2016),

6333-6362. [4] C. Bucur. Local density of Caputo-stationary functions in the space of smooth functions. ESAIM:

COCV 23 (2017), 1361-1380. [5] C. Bucur, F. Ferrari. An extension problem for the fractional derivative defined by Marchaud, Fract.

Calc. Appl. Anal.,19 (2016), no. 4, 867-887. [6] D. Baleanu, K. Diethelm, E. Scalas, J. Trujillo. Fractional calculus. Models and numerical methods. [OpenAIRE]

Second edition. Series on Complexity, Nonlinearity and Chaos, 5. World Scientific Publishing Co. Pte.

Ltd., Hackensack, NJ, 2017. [7] P.L. Butzer, U. Westphal. An introduction to fractional calculus. Applications of fractional calculus in

physics, 1-85, World Sci. Publ., River Edge, NJ, 2000. [8] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial

Differential Equations, 32 (2007), 1245-1260. [9] J.L. Bell, and H. Korté. "Hermann Weyl". The Stanford Encyclopedia of Philosophy (Winter 2016 Edi-

tion), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2016/entries/weyl/>. [10] J-F. Condette. Marchaud André Paul [note bibliographique] Histoire biographique de l'enseignement

46 references, page 1 of 4
Abstract
In this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics.
Subjects
free text keywords: fractional derivatives, Weyl derivative, Mathematics, extension operator, Mathematics - Analysis of PDEs, Grünwald–Letnikov derivative, fractional Laplace operator, Marchaud derivative, QA1-939
46 references, page 1 of 4

2 log 2 0

2f (x) − f (x + ξ) − f (x − ξ)

dξ = [1] N.H. Abel Solution de quelques problèmes à l'aide d'integrales definies. Gesammelte mathematische

Werke Leipzig: Teubner, 1 11-27 (First pub. in Mag. Naturvidenkaberne, Aurgang, 1, 2, Christiania

(1823) ) [2] M. Allen, L. Caffarelli, A. Vasseur. Porous medium flow with both a fractional potential pressure and

fractional time derivative. Chin. Ann. Math. Ser. B 38 (2017), no. 1, 45-82. [3] A. Bernardis, F. J. Martín-Reyes, P. R. Stinga, and J. L Torrea. Maximum principles, extension problem

and inversion for nonlocal one-sided equations. Journal of Differential Equations 260, No 7 (2016),

6333-6362. [4] C. Bucur. Local density of Caputo-stationary functions in the space of smooth functions. ESAIM:

COCV 23 (2017), 1361-1380. [5] C. Bucur, F. Ferrari. An extension problem for the fractional derivative defined by Marchaud, Fract.

Calc. Appl. Anal.,19 (2016), no. 4, 867-887. [6] D. Baleanu, K. Diethelm, E. Scalas, J. Trujillo. Fractional calculus. Models and numerical methods. [OpenAIRE]

Second edition. Series on Complexity, Nonlinearity and Chaos, 5. World Scientific Publishing Co. Pte.

Ltd., Hackensack, NJ, 2017. [7] P.L. Butzer, U. Westphal. An introduction to fractional calculus. Applications of fractional calculus in

physics, 1-85, World Sci. Publ., River Edge, NJ, 2000. [8] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial

Differential Equations, 32 (2007), 1245-1260. [9] J.L. Bell, and H. Korté. "Hermann Weyl". The Stanford Encyclopedia of Philosophy (Winter 2016 Edi-

tion), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2016/entries/weyl/>. [10] J-F. Condette. Marchaud André Paul [note bibliographique] Histoire biographique de l'enseignement

46 references, page 1 of 4
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publication . Article . Preprint . 2017

Weyl and Marchaud derivatives: a forgotten history

Ferrari, Fausto;