Substantial, tempered, and shifted fractional derivatives: Three faces of a tetrahedron

Manuel D. Ortigueira, Gabriel Bengochea, J. Tenreiro Machado

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)
58 Downloads (Pure)


The substantial, tempered, and shifted fractional derivatives, useful in medium range systems, are reviewed and highlighted in a unified framework. Their historical evolution is described and their properties studied. Moreover, they are characterized and assessed under the light of the strict sense criterion for the definition of derivatives. The relationship between these derivatives and several formulations developed in fractional calculus are explored. Additionally, the interpretations in the time and frequency domains are included. In the scope of the framework, new tempered linear systems and transfer functions are introduced. As particular applications, several systems for compensation and control purposes are revisited.

Original languageEnglish
Pages (from-to)9191-9209
Number of pages19
JournalMathematical Methods in the Applied Sciences
Issue number11
Publication statusPublished - 30 Jul 2021


  • shifted fractional derivative
  • substantial fractional derivative
  • tempered fractional derivative
  • tempered fractional system


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