Effect of Chemical Reaction towards MHD Marginal Layer Movement of Casson Nanofluid through Porous Media above a Moving Plate with an Adaptable Thickness

In the current workflow and heat exchange of a Casson nanoliquid across a penetrable media above a moving plate with variable thermal conductivity, adaptive thickness and chemical reaction are analyzed. First, the governing nonlinear equations of partial derivative terms with proper extreme conditions are changed into equations of ordinary derivative terms with suitable similarity conversions. Then the resulting equations are worked out using the Keller box method. The effects of various appropriate parameters are analyzed by constructing the visual representations of velocity, thermal, and fluid concentration. The velocity profile increased for shape parameter, and the opposite trend is observed for magnetic, Casson, porosity parameters. Temperature profile increases for magnetic, Casson, Brownian motion parameter, and thermophoresis parameters. Concentration profiles show a decreasing trend for wall thickness, Brownian movement, chemical reaction parameters. Also, skin friction values and calculated and matched with previous literature found in accordance. Also, local parameters Nusselt and Sherwood numbers are calculated and analyzed in detail.

Теория Титчмарша - Вейля сингулярного уравнения Хана - Штурма - Лиувилля

In this work, we will consider the singular Hahn--Sturm--Liouville difference equation defined by $-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)$, $x\in (\omega _{0},\infty),$ where $\lambda$ is a complex parameter, $v$ is a real-valued continuous function at $\omega _{0}$ defined on $[\omega _{0},\infty)$. These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the $\omega,q$-Hahn difference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classical Titchmarsh--Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn--Sturm--Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson--N\"{o}rlund integral and then we study families of regular Hahn--Sturm--Liouville problems on $[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.

20. The derivative claim and the rule in Foss v Harbottle

This chapter discusses further aspects of shareholder remedies, namely the common law multiple derivative claim; derivative claims under Companies Act 2006 (CA 2006), Part 11; the reflective loss principle; personal actions at common law; and specific statutory rights under the CA 2006. At common law, a shareholder aggrieved by a breach of duty by a director could bring a derivative claim on behalf of the company, as an exception to the rule in Foss v Harbottle. That common law claim now remains as a common law multiple derivative claim whereas the ‘ordinary’ derivative claim now is a statutory claim under CA 2006, Part 11. This chapter explores both types of derivative claim and assesses their value to shareholders. An important constraint on shareholder recovery is the principle governing reflective loss which has recently been restated by the Supreme Court. This chapter considers the current position in the light of that development.

ON ZYGMUND–TYPE INEQUALITIES CONCERNING POLAR DERIVATIVE OF POLYNOMIALS

Let \(P(z)\) be a polynomial of degree \(n\), then concerning the estimate for maximum of \(|P'(z)|\) on the unit circle, it was proved by S. Bernstein that \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension of this inequality. The polar derivative \(D_{\alpha}[P](z)\) of \(P(z)\), with respect to a point \(\alpha \in \mathbb{C}\), generalizes the ordinary derivative in the sense that \(\lim_{\alpha\to\infty} D_{\alpha}[P](z)/{\alpha} = P'(z).\) Recently, for polynomials of the form \(P(z) = a_0 + \sum_{j=\mu}^n a_jz^j,\) \(1\leq\mu\leq n\) and having no zero in \(|z| < k\) where \(k > 1\), the following Zygmund-type inequality for polar derivative of \(P(z)\) was obtained: $$\|D_{\alpha}[P]\|_p\leq n \Big(\dfrac{|\alpha|+k^{\mu}}{\|k^{\mu}+z\|_p}\Big)\|P\|_p, \quad \text{where}\quad |\alpha|\geq1,\quad p>0.$$In this paper, we obtained a refinement of this inequality by involving minimum modulus of \(|P(z)|\) on \(|z| = k\), which also includes improvements of some inequalities, for the derivative of a polynomial with restricted zeros as well.

A variety of wave amplitudes for the conformable fractional (2 + 1)-dimensional Ito equation

The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed.

Certain subclass of analytic functions with respect to symmetric points associated with conic region

<abstract><p>The purpose of this paper is to introduce and study a new subclass of analytic functions with respect to symmetric points associated to a conic region impacted by Janowski functions. Also, the study has been extended to quantum calculus by replacing the ordinary derivative with a $ q $-derivative in the defined function class. Interesting results such as initial coefficients of inverse functions and Fekete-Szegö inequalities are obtained for the defined function classes. Several applications, known or new of the main results are also presented.</p></abstract>

De Sitter scalar–spinor interaction in Minkowski limit

The scalar–spinor interaction Lagrangian is presented by the Yukawa potential. In dS ambient space formalism, the interaction Lagrangian of scalar–spinor fields was obtained from a new transformation which is very similar to the gauge theory. The interaction of massless minimally coupled (mmc) scalar and spinor fields was investigated. The Minkowski limit of the mmc scalar field and massive spinor field interaction in the ambient space formalism of de Sitter spacetime is calculated. The interaction Lagrangian and mmc scalar field in the null curvature limit become zero and the local transformation in the null curvature limit become a constant phase transformation and the interaction in this limit become zero. The covariant derivative reduces to ordinary derivative too. Then, we conclude that this interaction is due to the curvature of spacetime and then the mmc scalar field may be a part of a gravitational field.

Reflection properties of zeta related functions in terms of fractional derivatives

We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ζ(n, a) of integer first argument can be extended to the more general case of ζ(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ζ(s, a) with ℜ(s) > 2 is essentially the Weyl (s − 2)-derivative of ζ(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch’s transcendent.

Modeling Alcohol Concentration in Blood via a Fractional Context

We use a conformable fractional derivative G T α through two kernels T ( t , α ) = e ( α − 1 ) t and T ( t , α ) = t 1 − α in order to model the alcohol concentration in blood; we also work with the conformable Gaussian differential equation (CGDE) of this model, to evaluate how the curve associated with such a system adjusts to the data corresponding to the blood alcohol concentration. As a practical application, using the symmetry of the solution associated with the CGDE, we show the advantage of our conformable approaches with respect to the usual ordinary derivative.