## A generalized definition of caputo derivatives and its application to fractional odes

Publication ,  Journal Article
Li, L; Liu, JG
Published in: SIAM Journal on Mathematical Analysis
January 1, 2018

We propose a generalized definition of Caputo derivatives from t = 0 of order \gamma \in (0, 1) using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann-Liouville fractional calculus which agrees with the traditional Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions.

## Published In

SIAM Journal on Mathematical Analysis

1095-7154

0036-1410

January 1, 2018

50

3

## Start / End Page

2867 / 2900

• Applied Mathematics
• 4904 Pure mathematics
• 4901 Applied mathematics
• 0102 Applied Mathematics
• 0101 Pure Mathematics

### Citation

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Li, L., & Liu, J. G. (2018). A generalized definition of caputo derivatives and its application to fractional odes. SIAM Journal on Mathematical Analysis, 50(3), 2867–2900. https://doi.org/10.1137/17M1160318
Li, L., and J. G. Liu. “A generalized definition of caputo derivatives and its application to fractional odes.” SIAM Journal on Mathematical Analysis 50, no. 3 (January 1, 2018): 2867–2900. https://doi.org/10.1137/17M1160318.
Li L, Liu JG. A generalized definition of caputo derivatives and its application to fractional odes. SIAM Journal on Mathematical Analysis. 2018 Jan 1;50(3):2867–900.
Li, L., and J. G. Liu. “A generalized definition of caputo derivatives and its application to fractional odes.” SIAM Journal on Mathematical Analysis, vol. 50, no. 3, Jan. 2018, pp. 2867–900. Scopus, doi:10.1137/17M1160318.
Li L, Liu JG. A generalized definition of caputo derivatives and its application to fractional odes. SIAM Journal on Mathematical Analysis. 2018 Jan 1;50(3):2867–2900.

## Published In

SIAM Journal on Mathematical Analysis

1095-7154

0036-1410

January 1, 2018

50

3

2867 / 2900