Mathematics

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen's theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen's theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.

Recently, Ali et al defined the class $\mathcal{W}_{\beta}(\alpha, \gamma)$ consisting of functions $f$ which satisfy $$\Re e^{i\phi}\left((1-\alpha+2\gamma)\frac{f(z)}{z}+(\alpha-2\gamma)f'(z)+\gamma zf''(z)-\beta\right)>0,$$ for all $z\in E=\left\{z : |z|<1\right\}$ and for $\alpha, \gamma\geq0$ and $\beta<1$, $\phi\in \mathbb{R}$ (the set of reals). For $f\in{\mathcal{W}_{\beta}(\alpha, \gamma)}$, they discussed the convexity of the integral transform $$V_{\lambda}(f)(z):=\int_{0}^{1}\lambda(t)\frac{f(tz)}{t}dt,$$ where $\lambda$ is a non-negative real-valued integrable function satisfying the condition $\displaystyle\int_{0}^{1}\lambda(t)dt=1$. The aim of present paper is to find conditions on $\lambda(t)$ such that $V_{\lambda}(f)$ is convex of order $\delta$ ($0\leq\delta\leq1/2$) whenever $f\in{\mathcal{W}}_{\beta}(\alpha, \gamma)$. As applications, we study various choices of $\lambda(t)$, related to classical integral transforms.

We introduce a new class of harmonic univalent functions by using a generalized differential operator and investigate some of its geometric properties, like, coefficient estimates, extreme points and inclusion relations. Finally, we show that this class is invariant under Bernandi-Libera-Livingston integral for harmonic functions.

For α, γ ≥ 0, sufficient conditions are obtained that ensure the normalized analytic functions f satisfying a differential inequality (1 − α + 2γ) f (z) z + (α − 2γ)f (z) + γzf (z) − 1 < λ, to be starlike of order β in the open unit disc. As an application, we construct new starlike function f of order β which can be expressed in terms of double integral f (z) = 1 0 1 0 g(r, s, z)drds, of some suitable analytic function in the open unit disc.