A regularization method for Caputo fractional derivatives in the Banach space $L^\infty [0, T]$

Abstract

This work is dedicated to the investigation of a regularization method for the problem of determining Caputo fractional derivatives of a function in the Banach space $L^\infty [0, T]$. This regularization method is based on the approximation of the first-order derivative of the function by the solution of a well-posed problem depending on a regularization parameter. We then discuss the Hölder type stability results for the method according to two choice rules for the regularization parameter, which are an a priori parameter choice rule and an a posteriori parameter choice rule. Some numerical examples are provided.

Appendix A: The uniqueness theorem

Theorem A.1

(The uniqueness theorem) Problem (3) admits a unique solution in $L^\infty [0, T]$. In addition, the solution is given by

$$\begin{aligned} u_\beta (t)=-\dfrac{1}{\beta ^2}e^{-t/\beta }\int _0^t e^{z/\beta }q^\delta (z)dz+\dfrac{q^\delta (t)}{\beta },~t\in [0, T]. \end{aligned}$$

Proof

Multiplying both side of (3) by $\dfrac{e^{t/\beta }}{\beta }$, we get

$$\begin{aligned} \dfrac{e^{t/\beta }}{\beta }\left( \beta u_\beta (t)+\int _0^t u_\beta (z)dz\right) =\dfrac{e^{t/\beta }}{\beta }q^\delta (t), \end{aligned}$$

which is equivalent to

$$\begin{aligned} \dfrac{d}{dt}\left( e^{t/\beta }\int _0^t u_\beta (z)dz\right) =\dfrac{e^{t/\beta }}{\beta }q^\delta (t). \end{aligned}$$

(48)

Integrating both sides of (48) from 0 to t, we get

$$\begin{aligned} e^{t/\beta }\int _0^t u_\beta (z)dz=\dfrac{1}{\beta }\int _0^t e^{z/\beta }q^\delta (z)dz \end{aligned}$$

or

$$\begin{aligned} \int _0^t u_\beta (z)dz=\dfrac{1}{\beta }e^{-t/\beta }\int _0^t e^{z/\beta }q^\delta (z)dz. \end{aligned}$$

(49)

Differentiating both sides of (49) with respect to t, we obtain (4). This formula also guarantees the uniqueness of the solution. $\square$