New discretization of Caputo–Fabrizio derivative

Abstract

We derive a numerical approximation, namely L1–2 formula, to the Caputo–Fabrizio derivative by using a quadratic interpolation. Quadratic and cubic convergence rates are achieved for L1 and L1–2 formulas using Lagrange interpolation, respectively. We compute Caputo–Fabrizio derivatives of some known functions both theoretically and numerically. In addition, we solve non/linear sub-diffusion equations to test theoretical findings. Numerical results confirm the theoretically observed convergence rates.

Appendix

Proof (Proof of Theorem 4)

$$\begin{aligned} ^{CF} _0\mathcal {D}^{\alpha }_t t^n&= \frac{M(\alpha )}{1 - \alpha } \exp (-ct)\int _{0}^{t}ns^{n-1}\exp (cs)\mathrm{d}s \\&=\frac{M(\alpha )}{1 - \alpha } \exp (-ct)\left( ns^{n-1}\frac{\exp (cs)}{c}|_0^t-\int _{0}^{t}n(n-1)s^{n-2}\frac{\exp (cs)}{c}\mathrm{d}s \right) \\&=\frac{M(\alpha )}{1 - \alpha } \exp (-ct)\left( ns^{n-1}\frac{\exp (cs)}{c}|_0^t - \left( n(n-1)s^{n-2} \frac{\exp (cs)}{c^2} |_0^t \right. \right. \\&\quad \left. \left. -\, \int _{0}^{t} n(n-1)(n-2)s^{n-3}\frac{\exp (cs)}{c^2}\mathrm{d}s \right) \right) . \end{aligned}$$

The result follows inductively. $\square$

Proof (Proof of Theorem 5)

$$\begin{aligned} ^{CF} _0\mathcal {D}^{\alpha }_t \cos (wt)&= \frac{M(\alpha )}{1 - \alpha } \exp (-ct)\int _{0}^{t}-w\sin (ws)\exp (cs)\mathrm{d}s\\&=\frac{M(\alpha )}{1 - \alpha } \exp (-ct)(-w)\left( \sin (ws)\frac{\exp (cs)}{c}|_0^t-w\int _{0}^{t}\cos (ws)\frac{\exp (cs)}{c}\mathrm{d}s\right) \\&=\frac{M(\alpha )}{1 - \alpha } \exp (-ct)(-w) \left( \sin (ws)\frac{\exp (cs)}{c}|_0^t -w \left( \cos (ws)\frac{\exp (cs)}{c^2}|_0^t \right. \right. \\&\quad + \left. \left. w\int _{0}^{t}\sin (ws)\frac{\exp (cs)}{c^2}\mathrm{d}s\right) \right) . \end{aligned}$$

By arranging the terms, the final result is obtained.

Proof (Proof of Theorem 6)

$$\begin{aligned} ^{CF} _0\mathcal {D}^{\alpha }_t \sin (wt)&= \frac{M(\alpha )}{1 - \alpha } \exp (-ct)\int _{0}^{t} w\cos (ws)\exp (cs)\mathrm{d}s\\&=\frac{M(\alpha )}{1 - \alpha } \exp (-ct)w \left( \cos (ws)\frac{\exp (cs)}{c}|_0^t + w\int _{0}^{t}\sin (ws)\frac{\exp (cs)}{c}\mathrm{d}s\right) \\&=\frac{M(\alpha )}{1 - \alpha } \exp (-ct)w \left( \cos (ws)\frac{\exp (cs)}{c}|_0^t + w \left( \sin (ws)\frac{\exp (cs)}{c^2}|_0^t \right. \right. \\&\quad \left. \left. - w\int _{0}^{t}\cos (ws)\frac{\exp (cs)}{c^2}\mathrm{d}s \right) \right) . \end{aligned}$$

By arranging the terms, the final result is obtained.

Proof (Proof of Theorem 7)

$$\begin{aligned} ^{CF} _0\mathcal {D}^{\alpha }_t \exp (wt)&= \frac{M(\alpha )}{1 - \alpha } \exp (-ct) \int _{0}^{t} w\exp ((w+c)s)\mathrm{d}s\\&= \frac{M(\alpha )}{1 - \alpha } \exp (-ct) w \left( \frac{\exp ((w+c)s)}{w+c} |_0^t \right) .\end{aligned}$$