Transient Propagation of Spherical Waves in Porous Material: Application of Fractional Calculus

An Efficient Numerical Scheme for Solving a Fractional-Order System of Delay Differential Equations

Fractional order systems of delay differential equations are very advantageous in analyzing the dynamics of various fields such as population dynamics, neural networking, ecology, and physiology. The aim of this paper is to present an implicit numerical scheme along with its error analysis to solve a fractional-order system of delay differential equations. The proposed method is an extension of the L1 numerical scheme and has the error estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^2)$$\end{document}O(h2), where h denotes the step size. Further, we solve various non-trivial examples using the proposed method and compare the results with those obtained by some other established methods such as the fractional Adams method and the three-term new predictor–corrector method. We observe that the proposed method is more accurate as compared to the fractional Adams method and the new predictor–corrector method. Moreover, it converges for very small values of the order of fractional derivative.

Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not.