Wiman’s Type Inequality in Multiple-Circular Domain

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions f∈A0p(G) in an arbitrary complete Reinhard domain G⊂Cp, p∈N represented by the power series of the form f(z)=f(z1,⋯,zp)=∑‖n‖=0+∞anzn with the domain of convergence G. We have proven the following statement: If f∈Ap(G) and h∈Hp, then for a given ε=(ε1,…,εp)∈R+p and arbitrary δ>0 there exists a set E⊂|G| such that ∫E∩Δεh(r)dr1⋯drpr1⋯rp<+∞ and for all r∈Δε∖E we have Mf(r)≤μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}∏j=1p(lnerjεj)p−12+δ. Note, that this assertion at p=1,G=C,h(r)≡const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={z∈C:|z|<1},h(r)≡1/(1−r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,k∈Z+,n+k∈N.

Properties of Certain Classes of Holomorphic Functions Related to Strongly Janowski Type Function

Most subclasses of univalent functions are characterized with functions that map open unit disc ∇ onto the right-half plane. This concept was later modified in the literature with those mappings that conformally map ∇ onto a circular domain. Many researchers were inspired with this modification, and as such, several articles were written in this direction. On this note, we further modify this idea by relating certain subclasses of univalent functions with those that map ∇ onto a sector in the circular domain. As a result, conditions for univalence, radius results, growth rate, and several inclusion relations are obtained for these novel classes. Overall, many consequences of findings show the validity of our investigation.

Large-Eddy Simulation of Flow Past a Circular Cylinder Using OpenFOAM and Nektar++

Steady incoming flow past a circular cylinder has been a classical problem in fluid mechanics owing to its extensive practical applications in e.g. offshore engineering. In this study, large-eddy simulations are performed for flow past a circular cylinder at the Reynolds number (Re) of 3900. Particular focuses are on the comparisons of different numerical methods and computational domain patterns. The case Re = 3900 is computed with both OpenFOAM and Nektar++, which are based on the conventional finite volume method and the highorder spectral/hp element method, respectively. It is found that the computational cost for the Nektar++ model is only less than 10% of that for the OpenFOAM model. In addition, both circular and C-shaped domains are tested for the OpenFOAM and Nektar++ models. It is found that a circular domain is required for the OpenFOAM model to minimise the footprint of mesh non-orthogonality on the simulated flow, while the Nektar++ model does not have strict requirements for the orthogonality of the mesh. The present findings regarding the computational cost and the domain/mesh patterns are expected to be applicable to the numerical modelling of bluff-body flows in general. Based on Nektar++ and the circular domain, additional simulations are performed at Re = 1000 and 7000. For the three Re values investigated, the Strouhal number, hydrodynamic forces and the streamwise and spanwise vorticity fields are examined and compared.

An Efficient Finite Element Method and Error Analysis for Schrödinger Equation with Inverse Square Singular Potential

We provide in this study an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates’ transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.

Thermoelastic Characteristics of Functionally Graded Circular Disk Models under the Loading of Contact Forces

A rotating functionally graded circular disk undergoing a contact load is taken into account to investigate the thermoelastic characteristics. By considering contact force, a pair of partial differential equations is induced as the governing equations based on Hooke’s law. The behavior of circular disk modes is described with the variations of contact force and homogeneous thickness. A finite volume model is introduced to obtain approximate solutions for the governing equations because of the complexity of the equations. Contact force is highly influential in the radial direction compared to the circumferential direction in the displacement distribution, while a large radial stress appears near the area of the contact point. In the strain distribution, the magnitude increases as the angle grows near part of the outer boundary in the circular domain. The radial distribution profiles are susceptible to the growth of contact force in nearby area of the outer boundary, whereas the influences on the circumferential direction profiles are trivial. The increase of homogeneous thickness dwindles the radial magnitude of displacement, stress, and strain distribution profiles over nearby area of the outer boundary of the circular domain. As a result, numerical approach demonstrates that contact force and homogeneous thickness are indispensable parameters and provide deep influence on the thermoelastic movements of a rotating circular disk. Thus, the results obtained may be useful to design an appropriate FGM circular disk model for the industrial area by controlling the above parameters.

On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domain

ABSTRACT In this paper, we proposed two ways to understand the rank deficiency in the continuous system (boundary integral equation method, BIEM) and discrete system (boundary element method, BEM) for a circular case. The infinite-dimensional degree of freedom for the continuous system can be reduced to finite-dimensional space using the generalized Fourier coordinates. The property of the second-order tensor for the influence matrix under different observers is also examined. On the other hand, the discrete system in the BEM can be analytically studied, thanks to the spectral property of circulant matrix. We adopt the circulant matrix of odd dimension, (2N + 1) by (2N + 1), instead of the previous even one of 2N by 2N to connect the continuous system by using the Fourier bases. Finally, the linkage of influence matrix in the continuous system (BIE) and discrete system (BEM) is constructed. The equivalence of the influence matrix derived by using the generalized coordinates and the circulant matrix are proved by using the eigen systems (eigenvalue and eigenvector). The mechanism of degenerate scale for a circular domain can be analytically explained in the discrete system.

On q-analogue of Janowski-type starlike functions with respect to symmetric points

The main objective of the present paper is to define a class of q q -starlike functions with respect to symmetric points in circular domain. Some interesting results of these functions have been evaluated in this article. The sufficiency criteria in the form of convolutions are evaluated. Furthermore, other geometric properties such as coefficient bounds, distortion theorem, closure theorem and extreme point theorem are also obtained for these newly defined functions.

A calculus for flows in periodic domains

Purpose: We present a constructive procedure for the calculation of 2-D potential flows in periodic domains with multiple boundaries per period window.Methods: The solution requires two steps: (i) a conformal mapping from a canonical circular domain to the physical target domain, and (ii) the construction of the complex potential inside the circular domain. All singly periodic domains may be classified into three distinct types: unbounded in two directions, unbounded in one direction, and bounded. In each case, we use conformal mappings to relate the target periodic domain to a canonical circular domain with an appropriate branch structure.Results: We then present solutions for a range of potential flow phenomena including flow singularities, moving boundaries, uniform flows, straining flows and circulatory flows.Conclusion: By using the transcendental Schottky-Klein prime function, the ensuing solutions are valid for an arbitrary number of obstacles per period window. Moreover, our solutions are exact and do not require any asymptotic approximations.

On Janowski Analytic p,q-Starlike Functions in Symmetric Circular Domain

The main object of the present paper is to apply the concepts of p,q-derivative by establishing a new subclass of analytic functions connected with symmetric circular domain. Further, we investigate necessary and sufficient conditions for functions belonging to this class. Convex combination, weighted mean, arithmetic mean, growth theorem, and convolution property are also determined.