TY - JOUR
T1 - Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel
AU - Al-Refai, Mohammed
AU - Abdeljawad, Thabet
N1 - Funding Information:
The first author acknowledges the support of the United Arab Emirates University under the Fund No. 31S239. The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Publisher Copyright:
© 2017, The Author(s).
PY - 2017/12/1
Y1 - 2017/12/1
N2 - In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.
AB - In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.
KW - fractional derivatives
KW - fractional diffusion equations
KW - maximum principle
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U2 - 10.1186/s13662-017-1356-2
DO - 10.1186/s13662-017-1356-2
M3 - Article
AN - SCOPUS:85031489729
SN - 1687-1839
VL - 2017
JO - Advances in Difference Equations
JF - Advances in Difference Equations
IS - 1
M1 - 315
ER -