Subclasses of Analytic Functions of Complex Order Involving Generalized Jackson's (p, q)-derivative

This paper aims at introducing a new generalized differential operator and new subclass of analytic functions to obtain some interesting properties like coefficient estimates and fractional derivatives.

Multivalent Prestarlike Functions with Respect to Symmetric Points

A class of p-valent functions of complex order is defined with the primary motive of unifying the concept of prestarlike functions with various other classes of multivalent functions. Interesting properties such as inclusion relations, integral representation, coefficient estimates and the solution to the Fekete–Szegő problem are obtained for the defined function class. Further, we extended the results using quantum calculus. Several consequences of our main results are pointed out.

Certain Class of Analytic Functions with respect to Symmetric Points Defined by Q-Calculus

In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard to j , k -symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to j , k -symmetric points, we successfully obtained the coefficient bounds for functions in the newly specified class. We also quantified few applications as special cases which are new (or known).

From D = 3 to D = 2 dimensions: A note on topological order

We construct, by a procedure involving a dimensional reduction from a Chern–Simons theory with borders, an effective theory for (1 + 1)-dimensional superconductor. That system can be in either an ordinary phase or a topological one, depending on the value of two phases, associated to complex order parameters.

The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.