New System Of Differential Equations Research Articles

Nonlocal effects in a functionally graded thermoelastic layer due to volumetric absorption laser

The primary focus of the present contribution is to develop a new system of differential equations describing the nonlocal thermoelasticity theory. The problem deals with the thermoelastic interaction in a nonhomogeneous thermoelastic layer induced by absorbing penetrating laser radiation throughout its volume. The heat conduction equation has been formulated in the context of memory dependent three-phase lag model. The surfaces of the layer are thermally insulated and free of traction. Employing the integral Laplace transform technique, the solutions of the field quantities have been achieved in the transformed domain. The corresponding solutions in the space–time domain have been derived using the numerical inversion of the Laplace transform by Riemann-sum approximation technique. The numerical estimates of the field quantities have been depicted graphically to address the effects of nonlocality and memory effects on the temperature profile, displacement and stress components. The effect of the delay time, nonhomogeneity and various choices of linear and nonlinear kernel functions in the heat transport law have also been reported.

Existence and uniqueness of periodic solutions for a system of differential equations via operator methods

This article investigates the existence and uniqueness of periodic solutions for a new system of differential equations. By employing fixed point theorems for increasing φ-(h,tau )-concave operators, we establish the existence of unique periodic solution for our differential system and then give a monotone iterative scheme to approximate the unique periodic solution. Some examples are presented in the end to illustrate the validity of our main results.

A novel model of nonlocal thermoelasticity with time derivatives of higher order

The objective of this work is to introduce a new system of differential equations describing the nonlocal thermoelasticity theory with higher time derivatives and two‐phase lags. In order to obtain this model, we used the nonlocal continuum theory proposed by Eringen and the methodology of the Taylor series expansion of higher‐order time derivatives. Some generalized thermoelasticity theories follow as limited cases. This model is used to study the thermoelastic interaction in a nonlocal medium. The medium is exposed to an applied magnetic field and a periodic time heat source with a constant strength. Some comparisons have been displayed in figures to estimate the influences of the nonlocal parameter and magnetic field as well as the parameters of higher‐order on all the field quantities.

Method for the study of dynamic characteristics in the mechanisms of motion transmission

The process of motion transmission in mechanisms consisting of two links connected by straight stems is considered. The Zener phenomelogic model was used, where the phenomena of relaxation and retardation are taken into account. It is shown that the elastic properties of a substance should be considered as a complex function. A new system of differential equations is presented, which explicitly describes the oscillations of the speeds of movement and stresses in the body. It is shown that the new method allows to take into account the degree of distribution of the parameters of force lines, and for short lines it makes it possible to describe the dynamics with simple expressions.

On the existence of new integrable cases for Euler-Poisson equations in Newtonian fields

In this article, the rotational motion of a non-symmetric rigid body about a fixed point under the action of multi-Newtonian centers located on one fixed axis is considered. The aim of the work is to derive a technique to obtain such new (particular) integrable cases for the rotational motion of a rigid body in newtonian field of forces whose centers located on the vertical fixed axis. This technique reduces the governing equations and their known first integrals describing the motion to the unknown fourth integral and new system of differential equations. The well-known fourth integral cases are tested to verify the validity of the derived system and some new (particular) integrable cases are obtained with their geometrical interpretations of motion. The particular scope from this work is to extend Arkangel’skii’s theorem and the other related theorems to obtain such new integrable cases of single valuedness of solutions for this problem and its related special cases.

Self-consistent description of the tangential-discontinuity-type current sheet, using the particle trajectory method and angular variables

The description of the dynamics of charged-particles in an inhomogeneous magnetic field is a fundamental problem in space plasma physics. Since, this dynamics has a character of a nonlinear oscillator, the traditionally used approaches involve certain limiting conditions regarding the scales of magnetic field, particle motion, and the assumptions about conservation of specific invariants (e.g., the magnetic momentum, integrals of action, etc.). Such approaches naturally restrict the detailization the considered particle dynamics which is described in terms of the integral characteristics and averaged parameters of motion. However, in some regions the precise account of the particle trajectory details and the motion features (e.g., the phase of gyration) are of crucial importance. In this paper, we present a method for the description of particle dynamics, based on a new system of differential equations for the particle pitch-angle θ and phase of rotation ϕ, which are derived from the analysis of the particle trajectory in a given magnetic field. It enables an easy and comprehensive description of a number of elementary problems, which form the basis for more complex natural cases in space physics. The developed method admits generalization to the case of the particle ensemble, which makes it possible to find a set of the self-consistent solutions for tangential current sheets within the frame of the kinetic approach.

Theory of the interaction of flat sensing organ with the head of the sugar beet root

Sugar beet leaves now are very widely used for livestock feeding, as an organic fertiliser, and also as a raw material for the production of biogas. Therefore the harvest of the sugar beet tops (including leaves) can be considered as current task for the sugar beet growing system. Modern technologies involve harvest of the tops of sugar beet in two stages: flat basic cut and collecting of the entire green mass at higher altitude and the subsequent cutting of the heads of root crops from the residues. Therefore, topical issues of the sensing of the heads of sugar beet roots arranged in rows, are related to the majority of the sugar beet toppers, cleaners of the sugar beet heads, leaves cutters and, digging up working bodies of some designs. The aim of this study is theoretical determination the optimum design and kinematic parameters of a new sensing mechanism of the sugar beet heads located in the soil on the basis of the theory of interaction of flat passive swath board sensing organ with the sugar beet heads during their topping when located in the soil. In the study there are used methods of creation of mathematical models of functioning of the agricultural machines and their working bodies with the using of main provisions of mathematics, theoretical mechanics, programming and numerical calculations on the PC. In this paper, there is presented a theoretical study of the interaction of passive sensing organ with the head of the sugar beet root when there are located residues of the leaves on a root head spherical surface in the form of short elastic rods. Thus, for such an interaction of the sensing organ and the head of sugar beet root head there is taken into account elastic-damping properties of the sugar beet leaves residues. In the study there was first of all developed a new design of the topper for sugar beet heads with the use of a flat swath board sensing organ, there was developed the equivalent scheme of the interaction of the sensing organ with a fixed sugar beet root head. There was also selected and arranged the axis of coordinates, and determined the active forces and angle parameters. Using the basic law of dynamics there was made up a new system of differential equations and the integration of which has made it possible to find the laws of variation of the speed of movement and of the movement of the flat passive sensing organ on the head of sugar beet root head at the beginning of the contact. After determining the initial values of force and design parameters it is possible to solve them on PC to optimise kinematic and structural parameters of the sensing mechanism.

Symplectic geometry of the moduli space of projective structures in homological coordinates

We study the symplectic geometry of the space of linear differential equations with holomorphic coefficients of the form $$\varphi ''-u\varphi =0$$ on Riemann surfaces of genus g. This space coincides with the moduli space of projective connections which is an affine bundle modelled on the cotangent bundle $$T^*{\mathcal {M}}_g$$ . We show that for several choices of the origin, or base, holomorphic projective connection (such as Bergman, Wirtinger or Schottky) the canonical Poisson structure on $$T^*{\mathcal {M}}_g$$ induces the Goldman bracket on the monodromy character variety. These different choices give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations; we find the corresponding generating functions. Combined with a prior theorem of Kawai, our results show the symplectic equivalence between the embeddings of $$T^*{\mathcal {M}}_g$$ induced by the Bers and Bergman projective connections into the space of projective structures. The main technical tools are variational formulas with respect to homological Darboux coordinates on the space of holomorphic quadratic differentials. In particular, we get a new system of differential equations for the Prym matrix of the canonical two-sheeted covering of a Riemann surface defined by a quadratic differential.

Сопоставление координат больших планет, Луны и Солнца, полученных на основе нового принципа взаимодействия и банка данных DE405

В данной статье проведено сравнение координат и элементов орбит больших планет, Луны и Солнца, полученных на основе нового принципа взаимодействия и банка данных DE405. Под окружающим пространством можно понимать физический вакуум. Гравитация рассматривается как результат взаимодействия окружающего пространства с движущимися материальными телами. Тяготение объясняется свойством сжатия пространства относительно движущихся материальных тел. Получены дифференциальные уравнения движения больших планет, Луны и Солнца. Следует отметить, что система дифференциальных уравнений не содержит явно масс тел и силовых взаимодействий, кроме того, Земля рассматривается как сфероид. Путем численного интегрирования уравнений движения вычислены координаты Луны, Солнца больших планет и оскулирующие элементы орбит внутренних планет на интервале времени 1602-2193 гг. Результаты вычислений сопоставлены с координатами и элементами орбит, определенными по данным координат и скоростей DE405. Показано, что в отличие от механики Ньютона и релятивистских уравнений движения, координаты больших планет Луны и Солнца, основанные на решении новой системы дифференциальных уравнений, удовлетворительно согласуются с координатами этих объектов, вычисленных с помощью банка данных DE405. Полученные уравнения не содержат членов, учитывающих несферичность Земли и Луны, являясь при этом нерелятивистскими уравнениями. На основе исследований сделаны следующие выводы: полученные дифференциальные уравнения движения удовлетворительно описывают движение больших планет, Луны и Солнца на интервале времени 600 лет; они значительно проще и точнее дифференциальных уравнений, учитывающих релятивистские эффекты.

Reduction Methods and Chaos for Quadratic Systems of Differential Equations

We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension n. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of m differential equations, where . Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large n that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.

A class of exact strange quark star model

Static spherically symmetric space-time is studied to describe dense compact star with quark matter within the framework of MIT Bag Model. The system of Einstein’s field equations for anisotropic matter is expressed as a new system of differential equations using transformations and it is solved for a particular general form of gravitational potential with parameters. For a particular parameter, as an example, it is shown that the model satisfies all major physical features expected in a realistic star. The generated model also smoothly matches with the Schwarzschild exterior metric at the boundary of the star. It is shown that the generated solutions are useful to model strange quark stars.

Exact anisotropic sphere with polytropic equation of state

We study static spherically symmetric spacetime to describe compact objects with anisotropic matter distribution. We express the system of Einstein field equations as a new system of differential equations using a coordinate transformation, and then write the system in another form with polytropic equation of state and obtain two classes of exact models. The models satisfy all major physical features expected in a realistic star. For polytropic index n = 2, we obtain expressions for mass and density which are comparable with the reported experimental observations.

Application of the Hori Method in the Theory of Nonlinear Oscillations

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near‐identity transformations. These simplified algorithms are applied in determining second‐order asymptotic solutions of two well‐known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

Interaction of Point Sources and Vortices for Incompressible Planar Fluids

We consider a new system of differential equations which is at the same time gradient and locally Hamiltonian. It is obtained by just replacing a factor in the equations of interaction for N point vortices, and it is interpreted as an interaction of N point sources. Because of the local Hamiltonian structure and the symmetries it obeys, it does possess some of the first integrals that appear in the N vortex problem. We will show that binary collisions are easily blown up in this case since the equations of motion are of first order. This method may be easily generalized to the blow up of higher order collisions. We then generalize the model further to interactions of sources and vortices.

Analytic solutions for gas-phase chemical mechanism compression

A mathematical technique for combining unoxygenated hydrocarbons in gas-phase chemical reaction mechanisms is presented. Unlike earlier methods of combining hydrocarbons, the new method directly alters the differential equations for the species of the chemical mechanism. The new system of differential equations is mathematically equivalent to the original system, but contains fewer variables, representing a considerable decrease in computer memory. Processing time requirements were greatly decreased for scalar compiler tests, and relatively unchanged for vector processor tests. Numerical integrations were performed using an uncompressed reaction set, a reaction set condensed using the new method, and a reaction set parameterized using an older method (the “integrated reactivity weighting” of the RADM mechanism). The new method produces results nearly identical to the original system of equations, while integrated reactivity weighting can make large errors. These include large underpredictions of ozone concentrations in low NO x environments. Two examples of the new method in which eight hydrocarbons are replaced by four and then by a single condensed variable are given. The maximum error between the uncompressed and compressed systems over a 2 d simulation (for species not approaching the low concentration limit) was 1.8 × 10 −3%. The tests imply that current methods of lumping hydrocarbons may underestimate ozone production by as much as a factor of two, depending on the initial conditions. In contrast, the new method allows large numbers of hydrocarbons to be combined into a single variable, with negligible losses in accuracy.

Mathematical models for fine grinding of powders

In this paper we consider a mathematical model of the grinding process based upon the use of the one-dimensional rheological equation connecting tensions, deformations, variable mass and specific surface. New differential equations which describe specific surface dependence upon time are suggested. Specific surface enters the rheological equation, certifying its connection with the kinetic curves. A new system of differential equations of the continuum mechanics has been created in which the variability of the mass (size) of the demolished particles is allowed for. The author proposes a new way for solving this non-linear system. The models suggested are true for all types of mincing machines. However, some recommendations for the vibration grinding process are specified. They include the choice of zones of frequency and amplitude changes, as well as the experimental verification of these recommendations by the example of fine grinding of metallic powders used in powder metallurgy. Some conclusions for vibromills construction are made on the basis of the theoretical research.

An approach to spherically symmetric perfect fluid solution in general relativity using thin layers

The system of Einstein's field equations for a spherically symmetric perfect fluid is integrable in the event of constant mass density. Using this result we consider thin layers with the same symmetric properties. From the fitting conditions and by reducing the thickness of the layers to zero we obtain a new system of differential equations of the former constants of integration.

Localized RNA sequences in insect eggs: [formula omitted] and [formula omitted] Center for Developmental Biology, Department of Zoology, The University of Texas, Austin TX 78712-1064

The problem of variable aggregation in simulation model is considered. By aggregation is understood the unification of some variables of the system which are related to one another, into some sort of blocks each of which is a new variable with the properties defined by the aggregation laws. Let us take such a model, described by a system of differential equations with the set of variables {Xi} (i = 1,…,N). The problem is tofind a set of variables {Yj} (j=1,…,M; M⩽N) that is related to the set {Xi} by equations Y=φj(X) such that the new system of differential equations can be described by variables {Yi} only. it brings us to the problem of the representability of a function of many variables {Xi} in the aggregated variables {Yj}. Two criteria of representability are given. The withthe help of these criteria, the criterion of aggregation for some general model of a two-level ecological system is deduced.

Aggregation of variables in simulation models of water ecosystems

The problem of variable aggregation in simulation model is considered. By aggregation is understood the unification of some variables of the system which are related to one another, into some sort of blocks each of which is a new variable with the properties defined by the aggregation laws. Let us take such a model, described by a system of differential equations with the set of variables { X i } ( i = 1,…, N). The problem is tofind a set of variables { Y j } ( j=1,…, M; M⩽ N) that is related to the set { X i } by equations Y= φ j ( X) such that the new system of differential equations can be described by variables { Y i } only. it brings us to the problem of the representability of a function of many variables { X i } in the aggregated variables { Y j }. Two criteria of representability are given. The withthe help of these criteria, the criterion of aggregation for some general model of a two-level ecological system is deduced.

Zonal Spherical Functions on Some Symmetric Spaces

Let G be a real semisimple Lie group with finite center, and K a maximal compact subgroup of G. A zonal spherical function on the symmetric space X=G/K is an simulatneous eigeiifunction (p(x) of all the invariant differential operators on X satisfying (p(kx) = cp(x) for any x^X, k^K, and (p(eK) =1, where e is the identity element in G. By the Cartan decomposition G = KAK, (p(x) is considered as a function on A. And by the separation of variables, we obtain differential operators on A from the invariant differential operators, which are called their radial components. In this paper, we investigate the radial components of the invariant differential operators and the zonal spherical functions when G is a real, complex or quanternion unimodular group. The eigenvalues of the zonal spherical functions is parametrized by the element in a*. Therefore, the system of differential equations on A satisfied by the zonal spherical function has as many parameters as dim a. However, we can construct a new system of differential equations which admits the other parameter V. It is shown that the zonal spherical function on the real, complex or quaternion unimodular group corresponds to the case in which V = -o-, 1, 2, respectively.