Local density of Caputo-stationary functions of any order

ABSTRACT

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation. In addition, our result applies also to the ψ-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions.

COMMUNICATED BY:

• D. Repovš

KEYWORDS:

• Density results
• approximation
• Caputo derivative
• higher order operators

AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 34A08

Acknowledgements

The authors would like to thank the Istituto Nazionale di Alta Matematica (INdAM) and Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Australian Research Council Discovery Project 170104880 NEW “Nonlocal Equations at Work”.