A computationally efficient method for tempered fractional differential equations with application

Abstract

This paper investigates the numerical approximation of the tempered fractional integral by using the Sinc-collocation scheme. The algorithm is extended to solve a class of tempered fractional differential equations that converges to the solution with exponential rate. Several numerical examples compare the numerical approximations with the exact solutions. The behavioral responses of the lumped capacitance model with tempered fractional order for transient conduction are investigated. The efficiency and accuracy of the proposed scheme are analyzed in the perspective of the $L_2$-norm error and convergence order

Appendix A

Proposition 1

If $0<\alpha \le 1$ and $\lambda >0$, then the solution of the TFDE

$$\begin{aligned} \left\{ \begin{array}{ll} _{0+}{\mathbb {D}}_t^{\alpha ,\lambda }[y](t)+\kappa {y(t)}=f(t), &{} {\quad 0\le {t}<\infty } \\ y(0)=y_0 \end{array} \right. , \end{aligned}$$

(42)

is given by

$$\begin{aligned} y(t)=y_0e^{-\lambda {t}}E_{\alpha ,1}(-\kappa {t}^\alpha ) +\int _{0}^{t}f(t-\zeta )e^{-\lambda \zeta }\zeta ^{\alpha -1} E_{\alpha ,\alpha }(-\kappa \zeta ^\alpha ) \mathrm{d}\zeta , \end{aligned}$$

(43)

where $E_{\nu ,\beta }(\cdot )$ is the two-parameter Mittag–Leffler function defined as follows:

$$\begin{aligned} E_{\nu ,\beta }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (\nu {k}+\beta )}, \quad \nu ,\beta \in {\mathbb {C}}, \quad \ Re(\nu )>0, \quad \ Re(\beta )>0, \end{aligned}$$

(44)

and ${\mathbb {C}}$ denotes the complex plane.

Proof

We assume that $\mathcal {L}\{y(t);s\}=Y(s)$ and $\mathcal {L}\{f(t);s\}=F(s)$. By applying the Laplace transform to both sides of (42) and using the expression

$$\begin{aligned} \displaystyle {\mathcal {L}\{_{0+}{\mathbb {D}}_t^{\alpha ,\lambda }[y](t);s\}=(s+\lambda )^\alpha {Y(s)} -(s+\lambda )^{\alpha -1}(e^{\lambda {t}}y(t))|_{t=0}}, \end{aligned}$$

we obtain

$$\begin{aligned} \displaystyle {(s+\lambda )^\alpha {Y(s)}-(s+\lambda )^{\alpha -1}y_0 +\kappa {Y(s)}=F(s)}. \end{aligned}$$

Then, it follows

$$\begin{aligned} \displaystyle {Y(s)=\frac{(s+\lambda )^{\alpha -1}}{(s+\lambda )^\alpha +\kappa }y_0+\frac{F(s)}{(s+\lambda )^\alpha +\kappa }}, \end{aligned}$$

by virtue of the relation

$$\begin{aligned} \displaystyle {\mathcal {L}\{t^{\beta -1}E_{\alpha ,\beta }(-at^\alpha )\} =\frac{s^{\alpha -\beta }}{s^\alpha +a},\quad Re(s)>|a|^{\frac{1}{\alpha }}}, \end{aligned}$$

and

$$\begin{aligned} \displaystyle {y(t)=y_0e^{-\lambda {t}}E_{\alpha ,1}(-\kappa {t}^\alpha ) +\int _{0}^{t}f(t-\zeta )e^{-\kappa \zeta }\zeta ^{\alpha -1}E_{\alpha ,\alpha }(-\kappa \zeta ^\alpha )\text {d}\zeta .} \end{aligned}$$

This completes the proof of Proposition 1. $\square$