Dynamics of Partial Differential Equations

Volume 17 (2020)

Number 4

Equivalent definitions of Caputo derivatives and applications to subdiffusion equations

Pages: 383 – 402

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n4.a4


Mykola Krasnoschok (Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Sloviansk, Ukraine)

Vittorino Pata (Dipartimento di Matematica, Politecnico di Milano, Italy)

Sergii V. Siryk (CONCEPT Lab, Istituto Italiano di Tecnologia, Genova, Italy; and National Technical University of Ukraine, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine)

Nataliya Vasylyeva (Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Sloviansk, Ukraine)


An equivalent definition of the fractional Caputo derivative $D^\nu_t g$, for $\nu \in (0, 1)$, is found, within suitable assumptions on $g$. Some applications to the fractional calculus and to the theory of fractional partial differential equations are then discussed. In particular, this alternative definition is used to prove the maximum principle for the classical solutions to the linear subdiffusion equation subject to nonhomogeneous boundary conditions. This approach also allows to construct numerical solutions to the initial-boundary value problem for the subdiffusion equation with memory.


Caputo derivative, subdiffusion equations, maximum principle, numerical solutions

2010 Mathematics Subject Classification

Primary 35B50, 35R11. Secondary 26A33, 35B30, 65M06.

This work is partially supported by the Grant H2020-MSCA-RISE-2014 project number 645672 (AMMODIT: Approximation Methods for Molecular Modelling and Diagnosis Tools).

Received 24 January 2020

Published 16 November 2020