On Starlike Functions of Negative Order Defined by q-Fractional Derivative

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α∈(−3,1], q∈(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α∈[−1,1], q∈(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included.

Negative Order KdV Equation with No Solitary Traveling Waves

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions.

Elliptic problems in Besov and Sobolev—Triebel—Lizorkin spaces of low regularity

Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev–Triebel–Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.

Multifunctional Gratings for Multiband Spatial Filtering, Retroreflection, Splitting and Demultiplexing Based on C2 Symmetric Photonic Crystals

The concept of multifunctional reflection-mode gratings based on rod-type photonic crystals with C2 symmetry is introduced and examined. The specific modal properties lead to the vanishing dependence of the first-negative-order maximum on the angle of incidence within a wide range, and the nearly sinusoidal redistribution of the incident-wave energy between zero order (specular reflection) and first negative diffraction order (deflection) at frequency variation that are the key features enabling various functionalities in one structure and functionality merging. The elementary functionalities offered by the studied structures, of which multifunctional scenarios can be designed, include but are not restricted to multiband spatial filtering, multiband splitting, and demultiplexing. The proposed structures are shown to be capable in multifunctional operation in case of an obliquely incident polychromatic wave. The generalized demultiplexing is demonstrated for the case when several polychromatic wavesare incident at different angles. The same deflection properties yield multiband splitting, and merging demultiplexing and splitting functionalties in one functionality, which may contribute to various multifunctional scenarios. The proposed gratings arealso studied in transmissive configuration.

Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism.

Electro-Optical Parameters with Adverse Order of 10CB Liquid Crystal Molecules Studied under the Influence of an External High Electric Field: A Theoretical Approach

The 10CB liquid crystal (LC) at the higher electric field has a negative-order parameter. The 10CB LC has a definite order that maintains smectic phase stability. 10CB LC has supported the isotropic phase stability at an elevated external electric field and exhibited an unfavorable orientation of the order parameter. At the more upper electric field, the director angle (θ) fluctuated and contained θ less than 90°. The negative orientation of the order parameter and the large absorption anisotropy enhanced the optical birefringence with a reversal of sign. The negative-order parameter responsible for the IR absorbance has increased instead of reduction. The 10CB liquid crystal is a member of the cyano biphenyl series; therefore, the dipolar strength is maximum contributed to by the C-H atom asymmetric stretching of the alkyl chain length. The 10CB LC has a large bandgap (3.60 eV). Keywords: 10CB, Electric field, Electro-optical effect, Spectroscopy, Density functional theory (DFT).

Two new Painlev´e integrable KdV–Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation and new negative-order KdV-CBS equation

In this work, we develop two new (3+1)-dimensional KdV–Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation and (3+1)-dimensional negative-order KdV-CBS (nKdV-nCBS) equation. The newly developed equations pass the Painlev´e integrability test via examining the compatibility conditions for each developed model. We examine the dispersion relation and derive multiple soliton solutions for each new equation.