Search Results (11)

This paper presents a targeted study on the dynamic stress concentration (DSC) in piezoelectric materials induced by SH waves, focusing on the impact of elliptical openings. By using the elliptic cylindrical coordinate system and Mathieu functions, the wave fields are decomposed into functional series. Through the establishment of a set of infinite equations with mode coefficients based on the boundary conditions, the distribution of the dynamic stress concentration coefficient is visualized via numerical simulation. Furthermore, the impact of incident wave frequency, incident angle, and elliptic eccentricity on the stress concentration coefficient is analyzed. The results demonstrate a strong correlation between these parameters and the dynamic stress concentration coefficient. These findings hold significant implications for enhancing the strength and fatigue life of piezo-electric materials, as well as for selecting appropriate nondestructive testing methods. Full article

For low-permeability reservoirs, water-flooding development is usually adopted, which leads to induced fractures near the wellbore, increasing reservoir heterogeneity, and making water-flooding development more complex. This paper focuses on low-permeability reservoirs, considering the characteristics of induced fractures and elliptic-flow composite, and the well-test model for injection wells is established. The mathematical model in Laplace space is obtained through dimensionless transformation and Laplace transformation. Subsequently, the Mathieu function is introduced to obtain the bottom hole pressure, and the pressure response curve is drawn. The six flow stages of the curve are defined, and the sensitivity of parameters such as half-length of induced fractures, range of lateral-swept area, permeability in unswept area, and outer boundary distance at constant pressure are analyzed. The results show that the half-length of the fracture mainly affects the linear flow of the fracture, the range of the lateral wave-affected area mainly affects the radial flow of the swept area, the permeability of the unswept area mainly affects the radial flow of the unswept area, and the outer boundary distance at constant pressure mainly affects the boundary flow. Based on the production performance of a certain injection well in J Oilfield, a series of key parameters are obtained through analytical solution model inversion, including the induced-fracture half-length of 10.32 m, the lateral-swept range of elliptic partition flow of 128.95 m, the permeability of the swept area of 6.87 mD, and the mobility ratio of 119.92, which show the superiority of the analytical solution model. Full article

For functions of the form $\varphi \left(\xi \right)=\xi +{\sum }_{n=2}^{\infty }{c}_n{\xi }^n$, we identified two new subclasses of bi-starlike functions and bi-convex functions by using Mathieu-type series defined in the disc $\Delta =\{\xi \in \mathbb{C}:|\xi |<1\}$. We derived constraints for $|c_2|$ and $|c_3|$, and the subclasses are connected to the shell-shaped area. The Fekete–Szegö functional properties for the aforementioned function subclasses were also investigated. Additionally, a number of related corollaries are shown. Full article

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis’s infinite product formula for $\pi$, Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums. Full article

Integral form expressions are obtained for the Mathieu-type series and for their associated alternating versions, the terms of which contain a $(p,\nu )$-extended Gauss hypergeometric function. Contiguous recurrence relations are found for the Mathieu-type series with respect to two parameters, and finally, particular cases and related bounding inequalities are established. Full article

For the first time, we attempted to define two new sub-classes of bi-univalent functions in the open unit disc of the complex order involving Mathieu-type series, associated with generalized telephone numbers. The initial coefficients of functions in these classes were obtained. Moreover, we also determined the Fekete–Szegö inequalities for function in these and several related corollaries. Full article

The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are the focus of a recent survey article. The sharp version of double series inequality result is extended in the case of a not exhaustively investigated non-homogeneous kernel, which mutually covers the homogeneous kernel cases as well. Particularly, novel Hilbert’s double series inequality results are presented, which include the upper bounds built exclusively with non-weighted ${\ell }_p$–norms. The main mathematical tools are the integral expression of Mathieu $(\mathit{a},\mathit{\lambda })$-series, the Hölder inequality and a generalization of the double series theorem by Yang. Full article

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejér and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent. One of our proofs is assisted by symbolic computation. Full article

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new subclass of analytic (or invariant) functions. Our defined function class is symmetric under rotation. Some useful results like Fekete-Szegö functional, a number of sufficient conditions, radius problems, and results related to partial sums are derived. Full article

Fractional calculus image formulas involving various special functions are important for evaluation of generalized integrals and to obtain the solution of differential and integral equations. In this paper, the Saigo’s fractional integral operators involving hypergeometric function in the kernel are applied to the product of Srivastava’s polynomials and the generalized Mathieu series, containing the factor $x^{\lambda }{(x^k+c^k)}^{-\rho }$ in its argument. The results are expressed in terms of the generalized hypergeometric function and Hadamard product of the generalized Mathieu series. Corresponding special cases related to the Riemann–Liouville and Erdélyi–Kober fractional integral operators are also considered. Full article

We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series $S_{\mu }\left(r\right)$ , which are expressed in terms of the Hadamard product of the generalized Mathieu series $S_{\mu }\left(r\right)$ and the Fox–Wright function ${}_p{\mathsf{\Psi }}_q\left(z\right)$ . Corresponding assertions for the classical Riemann–Liouville and Erdélyi–Kober fractional integral and differential operators are deduced. Further, it is emphasized that the results presented here, which are for a seemingly complicated series, can reveal their involved properties via the series of the two known functions. Full article