Trigonometric Type Research Articles

Correction to: Effect of trigonometric sine, square and triangular wavetype time-periodic gravity-aligned oscillations on Rayleigh–Bénard convection in Newtonian liquids and Newtonian Nanoliquids

The influence of trigonometric sine, square and triangular wave-types of time-periodic gravity-aligned oscillations on Rayleigh–Benard convection in Newtonian liquids and in Newtonian nanoliquids is studied in the paper using the generalized Buongiorno two-phase model. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion and some slip mechanisms. Using the method of multiscales, the Lorenz model is transformed to a Ginzburg–Landau equation the solution of which helps in quantifying the heat transport through the Nusselt number. Enhancement of heat transport in Newtonian liquids due to the presence of nanoparticles/nanotubes is clearly explained. The study reveals that all the three wave types of gravity modulation delay the onset of convection and thereby to a diminishment of heat transport. It is also found that in the case of trigonometric sine type of gravity modulation heat transport is intermediate to that of the cases of triangular and square types. The paper is the first such work that attempts to theoretically explain the effect of three different wave-types of gravity modulation on onset of convection and heat transport in the presence/absence of nanoparticles/nanotubes. Comparing the heat transport by the single-phase and by the generalized two-phase models, the conclusion is that the single-phase model under-predicts heat transport in nanoliquids irrespective of the type of gravity modulation being effected on the system. The results of the present study reiterate the findings of related experimental and numerical studies.

Thermo-Elastic Study of Sandwich Plates by Alternative Hierarchical Finite Element Method Based on Reddy’s C1HSDT

The dynamic behavior of a structure is influenced by the environment in which it is located it among the community seeking more structure, we have the thermal loading, this work investigates a plate sandwich subjected to thermal stress, the modeling of the plate is made by a third order model developed by Reddy TSDT (Third Order Shear Deformation Theory), while the TLT theory (Theory Thermal Layers) is used to transform the three-dimensional problem to a two-dimensional thermal problem. Next, a rectangular-p element with four nodes at the vertices and four sides is used to model the structure, and the thermal conduction. In the structure part, the forms used functions are trigonometric family C0 type for membrane displacements and rotations and type C1 for inflected movements, the thermal portion is modeled by C0 types of shape functions where the degrees of freedom to the nodes are the temperature, the temperature gradient and the temperature curve, the thermoelastic study to determine the displacements of the submerged plate by the method of integration time PTIM (Precis Time Integration Method). Finally, a study of convergence of the developed numerical code is made, the found results are validated with those found in the literature, and different parametric studies are made for the sandwich plates in different situations, structure, and thermo- elastic.

On the application of the Ritz method to free vibration and buckling analysis of highly anisotropic plates

In this paper insights are provided into the implementation and use of the Ritz method for free vibration and buckling analysis of composite plates. Focus is given on the choice of the trial functions in relation to the degree and the kind of anisotropy exhibited by the plates. The Ritz approximation is applied to models based both on the classical lamination theory and a more advanced variable-kinematic formulation, capable of dealing with several higher order plate theories within an unified framework. A very efficient computation of the Ritz integrals is proposed, which allows to handle hundreds of admissible functions. In this way, accurate upper bound solutions can be obtained even for problems where the convergence rate of the Ritz method is low due to extreme levels of anisotropy. The effect of different forms of elastic couplings, boundary conditions and amount of material anisotropy on the convergence and accuracy of the solution is investigated when different sets of admissible functions – Legendre and Chebyshev polynomials, as well as of trigonometric type – are adopted. Important remarks about the completeness and numerical efficiency of the selected basis are also provided.

Exact solutions for a coupled discrete nonlinear Schrödinger system with a saturation nonlinearity

Deriving exact solutions of nonlinear evolution equations has recently received tremendous attention in mathematics and physics. In this paper, using the ( G ′ ∕ G ) -expansion method, we obtain some exact solutions for a coupled discrete nonlinear Schrödinger system with a saturation nonlinearity. These exact solutions include the hyperbolic function type, trigonometric function type and rational function type, of which the hyperbolic function type solutions may generate kink and anti-kink solitons.

δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials

Abstract In this paper, with reference to the previous work [WITUŁA, R.—SŁOTA, D.: δ-Fibonacci numbers, Appl. Anal. Discrete Math. 3 (2009), 310–329] concerning the, so called, δ-Fibonacci numbers, the concepts of δ-Lucas numbers, δ-Fibonacci and δ-Lucas polynomials are introduced. There are discussed the basic properties of such objects, as well as their applications, especially for description of certain polynomials and identities of algebraic and trigonometric type. Many from among these identities describe the binomial transformations of the respective integer sequences and polynomials. Similarly as for δ-Fibonacci numbers, also for δ-Lucas numbers some attractive identities–bridges are obtained, connecting these numbers in practice with every sequence of integer numbers.

The closed-loop kinematics modeling and numerical calculation of the parallel hexapod robot in space

The Lie group Denavit–Hartenberg method and application in the parallel hexapod robot are researched. The hexapod robot and the ground form opened and closed loop alternately; the complete kinematics equation cannot be derived by Denavit–Hartenberg method with a trigonometric function type. The complete kinematics model of the hexapod robot is constructed with the improved Denavit–Hartenberg method. The numerical solving strategy is schemed. The simulation results provide the positive and athwart kinematics response relation during the lift and forward patterns in space. The research shows that the complete kinematics model can be constructed by the Lie group Denavit–Hartenberg method; the numerical calculation method can solve the pose–attitude response correctly.

A functional equation motivated by some trigonometric identities

We deal with the following functional equation:f(xy)+λf(x)f(y)=ϕ(x,y), where λ is a complex number and f and ϕ are complex mappings defined on a semigroup. Moreover, we assume an addition formula of trigonometric type for ϕ. Our research is motivated by some earlier results related to a problem posed by S. Butler in 2003.

Complex solitons with real energies

Using Hirota’s direct method and Bäcklund transformations we construct explicit complex one and two-soliton solutions to the complex Korteweg–de Vries (KdV) equation, the complex modified KdV (mKdV) equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be -symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex -symmetric solutions to the KdV equation may also be constructed alternatively from real solutions to the mKdV by means of Miura transformations. The multi-soliton solutions obtained from Hirota’s method break the -symmetric, whereas those obtained from Bäcklund transformations are -invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are -symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space–time and the fact that any of our N-soliton solutions may be decomposed into N separate -symmetrizable one-soliton solutions.

Integral functional equations on locally compact groups with involution

Our main goal is to introduce some integral-type generalizations of the cosine and sine equations for complex-valued functions defined on a group G that need not be abelian. These equations provide a joint generalization of many trigonometric type functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s and Van Vleck’s equations. We prove that the continuous solutions for the first type and the central continuous solutions for the second one of these equations can be expressed in terms of characters, additive maps and matrix elements of irreducible, 2-dimensional representations of the group G. So the theory is part of harmonic analysis on groups.

On a functional equation of trigonometric type

In this paper, we study the functional equation, f(x+y)-f(x)f(y)=dsinxsiny. Some generalizations of the above functional equation are also considered.

Analysis of the enhanced coupling efficiency for different profile curves of special optical taper

For enhancing the coupling efficiency between the beam and the photodiode, a special optical taper is proposed for receiving optical signal. Based on the circular symmetric structure of special optical taper, the profile curve equations of it are deduced, including the trigonometric function type, parabolic type and exponential type. Moreover, the coupling efficiencies for special optical tapers with different profile curves are studied. The relationships of incident position, incident angle and coupling efficiency are analyzed. Finally, the comparison of coupling efficiency analytical results is also given.

On the Superstability of Trigonometric Type Functional Equations

The aim of this paper is to study the superstability for the mixed trigonometric functional equation: , , ), ( ) ( 2 )) ( ( ) ( G y x y g x f y x f xy f ∈ = − σ ) ( ,g f E and the stability of the Pexider type functional equation: , , ), ( ) ( 2 )) ( ( ) ( G y x y h x g y x f xy f ∈ = − σ ) ( , , h g f E where G is any group, not necessarily abelian, g f , and h are unknown complex valued functions and σ is an involution of G . As a consequence we prove that if f satisfies the inequality δ σ ≤ − − ) ( ) ( 2 )) ( ( ) ( y f x f y x f xy f for all G y x ∈ , then f is bounded.

The fractional coupled KdV equations: Exact solutions and white noise functional approach

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the modified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.

Unified Bernstein and Bleimann-Butzer-Hahn basis and its properties

In this paper we introduce the unification of Bernstein and Bleimann-Butzer-Hahn basis via the generating function. We give the representation of this unified family in terms of Apostol-type polynomials and Stirling numbers of the second kind. More generating functions of trigonometric type are also obtained to this unification.

A quantum affine algebra for the deformed Hubbard chain

The integrable structure of the one-dimensional Hubbard model is based on Shastry's R-matrix and the Yangian of a centrally extended superalgebra. Alcaraz and Bariev have shown that the model admits an integrable deformation whose R-matrix has recently been found. This R-matrix is of trigonometric type and here we derive its underlying exceptional quantum affine algebra. We also show how the algebra reduces to the above-mentioned Yangian and to the conventional quantum affine algebra in two special limits.

Group analysis of an ideal plasticity model

A group analysis of a system describing an ideal plastic flow is made in order to obtain analytical solutions. The complete Lie algebra of point symmetries of this system are given. Two of the infinitesimal generators that span the Lie algebra are original to this paper. A classification into conjugacy classes of all one- and two-dimensional subalgebras is performed. Invariant and partially invariant solutions corresponding to certain conjugacy classes are obtained using the symmetry reduction method. Solutions of algebraic, trigonometric, inverse trigonometric and elliptic type are provided as illustrations and other solutions expressed in terms of one or two arbitrary functions have also been found. For some of these solutions, a physical interpretation allows one to determine the shape of feasible extrusion dies corresponding to these solutions. The corresponding tools could be used to curve rods or slabs, or to shape a ring in an ideal plastic material by an extrusion process.

Symmetry group analysis of an ideal plastic flow

In this paper, we study a finite-dimensional Lie point symmetry group of a system describing an ideal plastic plane flow in two dimensions in order to find analytical solutions. The infinitesimal generators that span this Lie algebra are given. We completely classify the subalgebras of codimension up to two into conjugacy classes under the action of the symmetry group. Based on invariant forms, we use Ansätze to compute symmetry reductions in such a way that the obtained solutions simultaneously cover many invariant and partially invariant solutions. We calculate solutions of algebraic, trigonometric, inverse trigonometric and elliptic type. Some solutions depending on one or two arbitrary functions of one variable have also been found. In some cases, the shape of a potentially feasible extrusion die corresponding to the solution is deduced. These tools could be used to thin, curve, undulate or shape a ring in an ideal plastic material.

Hybrid classical integrable structure of squashed sigma models - a short summary

We give a short summary of our recent works on the classical integrable structure of two-dimensional non-linear sigma models defined on squashed three-dimensional spheres. There are two descriptions to describe the classical dynamics, 1) the rational description and 2) the trigonometric description. It is possible to construct two different types of Lax pairs depending on the descriptions, and the classical integrability is shown by computing classical r/s-matrices satisfying the extended Yang-Baxter equation in both descriptions. In the former the system is described as an integrable system of rational type. On the other hand, in the latter it is described as trigonometric type. There exists a non-local map between the two descriptions and those are equivalent. This is a non-local generalization of the left-right duality in principal chiral models.

STABILITY OF TRIGONOMETRIC TYPE FUNCTIONAL EQUATIONS IN RESTRICTED DOMAINS

We prove the Hyers-Ulam stability for trigonometric type functional inequalities in restricted domains with time variables. As consequences of the result we obtain asymptotic behaviors of the inequalities and stability of related functional inequalities in almost everywhere sense.

Bäcklund transformations and abundant exact explicit solutions of the Sharma–Tasso–Olver equation

By using an extension of the homogeneous balance method and Maple, the Bäcklund transformations for the Sharma–Tasso–Olver equation are derived. The connections between the Sharma–Tasso–Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma–Tasso–Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Bäcklund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma–Tasso–Olver equation is illustrated.