Distributed-order wave equations with composite time fractional derivative

ABSTRACT

In this paper we investigate the solution of generalized distributed-order wave equations with composite time fractional derivative and external force, by using the Fourier–Laplace transform method. We represent the corresponding solutions in terms of infinite series in three parameter (Prabhakar), Mittag–Leffler and Fox H-functions, as well as in terms of the so-called Prabhakar integral operator. Generalized uniformly distributed-order wave equation is analysed by using the Tauberian theorem, and the mean square displacement is graphically represented by applying a numerical Laplace inversion algorithm. The numerical results and asymptotic behaviors are in good agreement. Some interesting examples of distributed-order wave equations with special external forces by using the Dirac delta function are also considered.

KEYWORDS:

• Distributed-order equations
• composite derivative
• wave equation
• Mittag–Leffler functions
• Fox H-function
• numerical Laplace inversion
• Tauberian theorem

2010 MSC CLASSIFICATIONS:

• 26A33
• 33E12
• 40E05

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Trifce Sandev  http://orcid.org/0000-0001-9120-3847

Notes

1. Three parameter Mittag–Leffler function is defined by $E_{\alpha ,\beta }^{\delta }\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\left((\delta {)}_k/\mathrm{\Gamma }(\alpha k+\beta )\right)\left(z^k/k!\right)$, $(\delta {)}_k=\mathrm{\Gamma }(\delta +k\left)/\mathrm{\Gamma }\right(\delta )$, and $E_{\alpha ,\beta }^1\left(z\right)=E_{\alpha ,\beta }\left(z\right)$ is the two parameter Mittag–Leffler function. Its Laplace transform is given by $\mathcal{L}\left[t^{\beta -1}{E}_{\alpha ,\beta }^{\delta }\right(\pm a{t}^{\alpha }\left)\right]\left(s\right)=s^{\alpha \delta -\beta }/(s^{\alpha }\pm a{)}^{\delta }$, $\mathrm{\Re }\left(s\right)>|a{|}^{1/\alpha }$.

2. The Fox H-function is defined by the following Mellin–Barnes integrable $H_{p,q}^{m,n}\left(z\right)=H_{p,q}^{m,n}\left[z\left|\begin{array}{c}(a_p,A_p)\\ (b_q,B_q)\end{array}\right.\right]=\left(1/2\pi \mathrm{i}\right){\int }_{\mathrm{\Omega }}\theta \left(s\right)z^{s}ds$, where $\theta \left(s\right)=\prod _{j=1}^m\mathrm{\Gamma }(b_j-B_{j}s)\prod _{j=1}^n\mathrm{\Gamma }(1-a_j+A_{j}s)/\prod _{j=m+1}^q\mathrm{\Gamma }(1-b_j+B_{j}s)\prod _{j=n+1}^p\mathrm{\Gamma }(a_j-A_{j}s)$, $0\le n\le p$, $1\le m\le q$, $a_i,b_j\in C$, $A_i,B_j\in R^{+}$, $i=1,\dots ,p$, $j=1,\dots ,q$.

3. For calculation of the second moment, we apply the Mellin transform of the Fox H-function [24] ${\int }_0^{\mathrm{\infty }}{x}^{\xi -1}{H}_{p,q}^{m,n}\left[ax\left|\begin{array}{l}(a_p,A_p)\\ (b_q,B_q)\end{array}\right.\right]\mathrm{d}x=a^{-\xi }\theta (-\xi ),$ where $\theta (-\xi )$ is given by the definition of the Fox H-function.

4. The Tauberian theorem states that for a slowly varying function $L\left(t\right)$ at infinity, i.e. $\underset{t\to \mathrm{\infty }}{lim}\left(L\left(at\right)/L\left(t\right)\right)=1$, a>0, if $\hat{R}\left(s\right)\simeq s^{-\rho }L\left(1/s\right)$, $s\to 0$, $\rho \ge 0,$ then $r\left(t\right)={\mathcal{L}}^{-1}\left[\hat{R}\right(s\left)\right]\simeq \left(1/\mathrm{\Gamma }\left(\rho \right)\right)t^{\rho -1}L\left(t\right)$, $t\to \mathrm{\infty }$.

Additional information

Funding

Zivorad Tomovski was supported by NWO [grant number 040.11.629], Department of Applied Mathematics, TU Delft.