Transcendental Solutions Research Articles

WHOLE SOLUTIONS OF ONE CLASS OF ALGEBRAIC DIFFERENTIAL EQUATIONS GENERALIZING BRIOT–BOUQUET TYPE EQUATIONS

The paper examines entire solutions of differential equations of Brio–Bouquet type. It is shown that (under some conditions that the polynomials 𝑃, 𝑄 satisfy) all integer transcendental solutions of such equations are quasi-polynomials.

Results on solutions of several systems of the product type complex partial differential difference equations

Abstract This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u ( z + c ) [ α 1 u ( z ) + β 1 u z 1 + γ 1 u z 2 + α 2 v ( z ) + β 2 v z 1 + γ 2 v z 2 ] = 1 , v ( z + c ) [ α 1 v ( z ) + β 1 v z 1 + γ 1 v z 2 + α 2 u ( z ) + β 2 u z 1 + γ 2 u z 2 ] = 1 , \left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c = ( c 1 , c 2 ) ∈ C 2 c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2} , α j , β j , γ j ∈ C , j = 1 , 2 {\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2 . Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

Properties of meromorphic solutions of first-order differential-difference equations

Abstract For the first-order differential-difference equations of the form A ( z ) f ( z + 1 ) + B ( z ) f ′ ( z ) + C ( z ) f ( z ) = F ( z ) , A\left(z)f\left(z+1)+B\left(z)f^{\prime} \left(z)+C\left(z)f\left(z)=F\left(z), where A ( z ) , B ( z ) , C ( z ) A\left(z),B\left(z),C\left(z) , and F ( z ) F\left(z) are polynomials, the existence, growth, zeros, poles, and fixed points of their nonconstant meromorphic solutions are investigated. It is shown that all nonconstant meromorphic solutions are transcendental when deg B ( z ) < deg { A ( z ) + C ( z ) } + 1 {\rm{\deg }}B\left(z)\lt {\rm{\deg }}\left\{A\left(z)+C\left(z)\right\}+1 and all transcendental solutions are of order at least 1. For the finite-order transcendental solution f ( z ) f\left(z) , the relationship between ρ ( f ) \rho (f) and max { λ ( f ) , λ ( 1 ∕ f ) } \max \left\{\lambda (f),\lambda \left(1/f)\right\} is discussed. Some examples for sharpness of our results are provided.

On symmetric solutions of the fourth q-Painlevé equation

The Painlevé equations possess transcendental solutions y(t) with special initial values that are symmetric under rotation or reflection in the complex t-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a q-difference Painlevé equation. We focus on symmetric solutions of a q-difference equation known as or and provide their symmetry properties and solve the corresponding monodromy problem.

Differential Transcendence of Solutions of the Difference Equation Delta y=ay^2+by+c

By a purely algebraic way, we investigate a necessary and sufficient condition for the existence of differentially algebraic and transcendental solutions to certain difference equations whose form is related to Riccati equations. Our result roughly implies that all the solutions do not satisfy any algebraic differential equations if there is no algebraic solution.

Dirac Particle in the Coulomb Field on the Background of Hyperbolic Lobachevsky Model

The known systems of radial equations describing the relativistic hydrogen atom on the base of the Dirac equation in Lobachevsky hyperbolic space is solved. The relevant 2-nd order differential equation has six regular singular points, its solutions of Frobenius type are constructed explicitly. To produce the quantization rule for energy values we have used the known condition for determination of the transcendental Frobenius solutions. This defines the energy spectrum which is physically interpretable and similar to the spectrum arising for the scalar Klein-Fock-Gordon equation in Lobachevsky space. In the present paper, exact analytical solutions referring to this spectrum are constructed. Convergence of the series involved is proved analytically and numerically. Squared integrability of the solutions is demonstrated numerically. It is shown that the spectrum coincides precisely with that previously found within the semi-classical approximation.

Spectral Output of Homogeneously Broadened Semiconductor Lasers

Gain, spontaneous emission, and reflectance play important roles in setting the spectral output of homogeneously broadened lasers, such as semiconductor diode lasers. This paper provides a restricted-in-scope review of the steady-state spectral properties of semiconductor diode lasers. Analytic but transcendental solutions for a simplified set of equations for propagation of modes through a homogeneously broadened gain section are used to create a Fabry–Pérot model of a diode laser. This homogeneously broadened Fabry–Pérot model is used to explain the spectral output of diode lasers without the need for guiding-enhanced capture of spontaneous emission, population beating, or non-linear interactions. It is shown that the amount of spontaneous emission and resonant enhancement of the reflectance-gain (RG) product as embodied in the presented model explains the observed spectral output. The resonant enhancement is caused by intentional and unintentional internal scattering and external feedback.

Notes on Solutions for Some Systems of Complex Functional Equations in ℂ 2

The purpose of this article is to give the details of finding the transcendental entire solutions with finite order for the systems of nonlinear partial differential-difference equations ∂ f 1 z 1 , z 2 / ∂ z 1 + ∂ f 1 z 1 , z 2 / ∂ z 2 n 1 + P 1 z f 2 z 1 + c 1 , z 2 + c 2 m 1 = Q 1 z , ∂ f 1 z 1 , z 2 / ∂ z 1 + ∂ f 1 z 1 , z 2 / ∂ z 2 n 2 + P 2 z f 1 z 1 + c 1 , z 2 + c 2 m 2 = Q 2 z , where P 1 z , P 2 z , Q 1 z , and Q 2 z are polynomials in ℂ 2 ; n 1 , n 2 , m 1 , and m 2 are positive integers, and c = c 1 , c 2 ∈ ℂ 2 . We obtain that there exist some pairs of the transcendental entire solutions of finite order for the above system, which is a very powerful supplement to the previous theorems given by Xu and Cao and Xu and Yang.

On the transcendental solutions of Fermat type delay-differential and $c$-shift equations with some analogous results

In this paper, we mainly investigate on the finite order transcendental entire solutions of two Fermat types delay-differential and one Fermat type c-shift equations, as these types were not considered earlier. Our results improve those of [13] in some sense. In addition, we also extend some recent results obtained in [18]. A handful number of examples have been provided by us to justify our certain assertion as and when required.

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

Finite-Order Transcendental Entire Solutions of Generalized Non-linear Shift Equations

We have found out the form of the most generalized non-linear shift equations possessing finite-order transcendental entire solutions. We have extended two main results of Qi et al. (Adv. Differ. Equ. 256, 1–10, 2014). We have also pointed out some gaps in the proofs of two theorems in Latreuch (Mediterr. J. Math. 14(3), Art. 115, 2017) and showed that the results are partially true. In an attempt to rectify those results, we have presented their extended corrected forms in a compact manner. Finally, we have presented one more theorem relevant with our other results. A number of examples have been provided by us as and when required.

Dirac particle in the external coulomb field on the background of the Lobachevsky–Riemann space models

The known systems of the radial equations describing the hydrogen atom on the basis of the Dirac equation in the Lobachevsky–Riemann spaces of constant curvature are investigated. In the both geometrical models, the differential equations of second order with six regular singular points are found, and their exact solutions of Frobenius type are constructed. To produce the quantization rule for energy values we use the known condition which separates the transcendental Frobenius solutions. This provides us with the energy spectra that are physically interpretable and are similar to those for the Klein–Fock–Gordon particle in these space models. These spectra are similar to those that previously have appeared in studying the same systems of the equations with the use of the semi-classical approximation.

On the Riemann-Hilbert Problem for a q-Difference Painlevé Equation

A Riemann-Hilbert problem for a q-difference Painlevé equation, known as $$q{\text {P}}_{{\text {IV}}}$$ , is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $$q{\text {P}}_{{\text {IV}}}$$ and corresponding data on an associated q-monodromy surface. We also construct the moduli space of $$q{\text {P}}_{{\text {IV}}}$$ explicitly.

Solutions for systems of complex Fermat type partial differential-difference equations with two complex variables

<abstract><p>By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate some properties of the transcendental entire solutions for several systems of partial differential difference equations of Fermat type, and obtain some results about the existence and the forms of transcendental entire solutions of the above systems, which improve and generalize the previous results given by Cao, Gao, Liu <sup>[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b24">24</xref>,<xref ref-type="bibr" rid="b39">39</xref>]</sup>. Some examples are given show that there exist some significant differences in the forms of transcendental entire solutions with finite order of the systems of equations with between several complex variables and a single complex variable.</p></abstract>

Remarks on Painlevé’s differential equation P34

This paper is engaged with Painleve’s differential equation P34:2ww″ = w″2 + 2w2(2w − z) − α, also known as Ince’s equation XXXIV and closely related to Painleve’s second differential equation $${{\rm{P}}_\Pi}:\varpi\prime\prime= \alpha + z\varpi + 2{\varpi^3}$$ . We will show that the transcendental solutions belong to the Yosida class $${\mathfrak{Y}_{{\rm{1,}}{1 \over 2}}}$$ and have no deficient rational targets. We will also identify the sub-normal solutions and prove that they are characterised by the fact that their first integrals belong to the class $${\mathfrak{Y}_{{1 \over 2}{\rm{,}}{1 \over 2}}}$$ .

Restrictions on meromorphic solutions of Fermat type equations

AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

Closed-form approximate solution for linear buckling of Mindlin plates with SRSR-boundary conditions

The present work deals with the buckling analysis of rectangular Mindlin plates consisting of laminated composites with symmetrical, balanced lay-up or isotropic materials. Along the longitudinal edges the plate is rotationally restrained by springs. The transversal edges are simply supported. In agreement with common notation, the boundary conditions are abbreviated as follows: simply supported (S), rotationally restrained (R), and fully clamped (C). As loading situation axial compression is considered. Aiming at high computational efficiency the problem is solved by the Rayleigh-Ritz-method. Since well suited shape functions for deflection and rotations with very few variables are used, closed-form approximate solutions for the buckling load can be obtained. For verification exact transcendental solutions and/or high fidelity finite element analyses are employed. Additionally, results are compared to those of existing closed-form approximate solutions.Apart from the special case of simply supported longitudinal edges where all methods yield exact or nearly exact results, the present method shows clear advantages: 1.Due to the type of shape functions it is able deal with unsymmetrical boundary conditions. 2.For the case of both longitudinal edges fully clamped where all closed-form approximate solutions show the greatest deviations the present method is significantly more accurate.

Solutions of mixed Painlevé PIII—V model

We review the construction of the mixed Painlevé P system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of a hybrid differential equation that for special limits of its parameters reduces to either Painlevé PIII or PV.The aim of this paper is to describe solutions of the P model. In particular, we determine and classify rational, power series and transcendental solutions of the P model. A class of power series solutions is shown to be convergent in accordance with the Briot–Bouquet theorem. Moreover, the P equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of a P equation is quantized by integer .

First-order algebraic differential equations of genus zero

We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus zero.

Education for Peace in the Mexican context

This paper presents a case study of a peace-based education program piloted in Mexico. An analysis of the interactions among the micro, meso, and macro levels sheds light on how the benefits, challenges, and impact of synergic action contribute towards building a culture of peace. The paper concludes that the best strategy for crime prevention is education for peace. This requires multi-stakeholder collaboration to create transcendental solutions that go to the roots of the problem of violence in Mexico rather than simply fighting its symptoms.