System Of Quasilinear Differential Equations Research Articles

The nonlocal solvability conditions for a system with constant terms and coefficients of the variable t

We consider the Cauchy problem for a system of quasilinear differential equations with constantterms and coefficients of the variablet.We investigate the solvability of the Cauchy problem fora system of quasilinear differential equations with constant terms and coefficients of the variabletusing the additional argument method. A theorem on the existence and uniqueness of the localsolution of the Cauchy problem for a system of quasilinear differential equations with constantterms and coefficients of the variabletis formulated. We obtain sufficient conditions for theexistence and uniqueness of a nonlocal solution of the Cauchy problem in original coordinates fora system of quasilinear differential equations with constant terms and coefficients of the variablet.A theorem on the existence and uniqueness of the nonlocal solution of the Cauchy problem fora system of quasilinear differential equations with constant terms and coefficients of the variabletis formulated. A theorem on the existence and uniqueness of the nonlocal solution of the Cauchyproblem for a system of quasilinear differential equations with constant terms and coefficients ofthe variabletis proved. The proof of the nonlocal solvability of the Cauchy problem for a systemof quasilinear differential equations with constant terms and coefficients of the variabletrelies onglobal estimates.

ASYMPTOTIC BEHAVIOR OF Pω(Y0, Y1,±∞)-SOLUTIONS OF THE SECOND ORDER DIFFERENTIAL EQUATIONS WITH THE PRODUCT OF DIFFERENT TYPES OF NONLINEARITIES FROM AN UNKNOWN FUNCTION AND ITS FIRST DERIVATIVE

The task of establishing the conditions of existence, as well as finding asymptotic images of solutions of differential equations, which contain nonlinearities of various types in the righthand side, is one of the most important tasks of the qualitative theory of differential equations. In this work, second-order differential equations, which contain in the right part the product of a regularly varying nonlinearity from an unknown function and a rapidly varying nonlinearity from the derivative of an unknown function when the corresponding arguments are directed to zero or infinity, are considered. Necessary and sufficient conditions for the existence of slowly varying P!(Y0; Y1;±∞) solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives have also been obtained. When additional conditions are imposed on the coefficients of the characteristic equation of the corresponding equivalent system of quasi-linear differential equations, it is established that there is a one-parameter family of such P!(Y0; Y1;±∞)-solutions to the equation. Similar results were obtained earlier when considering second-order equations, which contain on the right-hand side the product of a rapidly varying function from an unknown function and a properly varying function from the derivative of an unknown function when the arguments go to zero or infinity. Results for the equation, considered in this paper, are new.

Asymptotic behaviour of solutions of one class of the fourth-order nonlinear differential equations

The asymptotic behaviour as t↑ω of one of the possible types of Pω(Y0, λ0)-solutions of a binomial non-autonomous fourth-order ordinary differential equation with exponential nonlinearity and a continuous and non-zero in some left neighbourhood ω(ω≤+∞) coefficient p(t) is investigated. First, using a priori asymptotic properties of the considered Pω(Y0, λ0)-solutions, necessary conditions for their existence are established, as well as asymptotic representations of these solutions and their derivatives up to the third order. The question of the actual existence of solutions with the obtained asymptotic representations is solved by reducing it to the question of the existence of solutions that vanish at a specific point of a system of quasilinear differential equations. This system is obtained as a result of some transformations of the original equation, taking into account the kind of established asymptotic representations. In addition, the question of the number of solutions with found asymptotic representations is also resolved.

Asymptotic representations of the solutions of some type $n$-th order differential equations

In this work there is considered the n order differential equation (r(t)u(m))(n-m) = ∑mk=0 pku(k), for which the existence conditions and asymptotic representations of solutions under certain conditions on the function pk and the function r are found. Solutions of this type of equation for m = 0 were considered in the work of Hinton, and for s ≡ 1 and m = n − 1 were considered in the work of I. T. Kiguradze. The results obtained in this work for the indicated equation in some sense generalize the results obtained in the works of Hinton and I. T. Kiguradze. For obtaining asymptotic images using substitutions, the equation is transformed into an equivalent system of quasi-linear differential equations for which the well-known Levinson results are satisfied, the equation is in some sense asymptotically equivalent to the corresponding binomial differential equation of the nth order.

ПОСТРОЕНИЕ РЕШЕНИЯ СИСТЕМЫ КВАЗИЛИНЕЙНЫХ УРАВНЕНИЙ МЕТОДОМ ПРОСТОЙ ИТЕРАЦИИ

This article considers the problem of constructing a solution to a system of quasilinear algebraic equations by the method of simple iteration. The method of simple iteration refers to the type of successive approximations. This method is characterized by the fact that when finding the unknowns in the kth approximation, the values of the unknowns found in the (k-1)-approximation are substituted into the right parts of the initial equations. Attention is paid to the analysis of the iterative process from the point of view of convergence, the region of existence of solutions, as well as the estimation of the error between approximations and the exact solution. As an example, the problem of reducing a system of quasi-linear differential equations with a small parameter to a system of quasi-linear algebraic equations is considered.

Про гіперболічність і властивості розв’язків континуальних моделей агрегації та седиментації мікро наночастинок у концентрованих суспензіях

In continual mechanics sedimentation of aggregating particles in concentrated suspensions are determined by the mass and momentum conservation laws for each component of the suspension. The resulting quasilinear system of differential equations governing the flow could be hyperbolic, strongly strictly or weakly hyperbolic depending on the model accepted. The type and Eigenvalues of the matrix influence the characteristics of the pattern formation during the sedimentation that is essential for the model application in modern medical, microbiological and nanofluidic technologies. In this paper the hyperbolicity of the three-phase model of aggregation and sedimentation of micro/nanoparticles is studied.

On the Solutions of Oscillating-Type Countable Differential Systems with Slowly Varying Parameters

For a countable quasilinear system of differential equations whose coefficients have the form of absolutely and uniformly convergent Fourier series with slowly varying coefficients and frequency, we obtain conditions for the existence of a partial solution of a similar structure. As a result, we establish conditions for the possibility of complete decomposition of a countable linear homogeneous differential system with coefficients of the same structure.

Existence and behaviour of parameter classes of solutions of a system of quasilinear differential equations

The paper presents some results on the existence and behaviour of parameter classes of solutions for system of quasilinear differential equations. Behaviour of integral curves in neighborhood of an arbitrary or a certain curve is considered. The obtained results contain the answer to the question on stability as well as approximation of solutions whose existence is established. The errors of the approximation are defined by the functions that can be sufficiently small. To obtain our main results, the theory of qualitative analysis of differential equations and topological retraction method are used.

Discontinuous almost periodic type functions, almost automorphy of solutions of differential equations with discontinuous delay and applications

In this work, using discontinuous almost periodic type functions, exponential dichotomy and the notion of Bi-almost automorphicity we give sufficient conditions to obtain a unique almost automorphic solution of a quasilinear system of differential equations with piecewise constant arguments. Finally, an application to the Lasota– Wazewska model with piecewise constant delayed argument is given.

Dynamics of Cosmic Neutrinos in Galaxies

AbstractThe cosmic background of massive (about 1 eV rest-energy) neutrinos can be cooled to extremely low temperatures, reaching almost completely degenerated state. The Fermi velocity of the neutrinos becomes less than 100 km/s. The equations of dynamics for the cosmic background neutrinos are derived for the spherical and axisymmetrical thin circular disk galaxies. The equations comprise the gravitational potential and gravity of the uniform baryonic disk galaxies. Then the equations are integrated analytically over the disk radius. The constant radial neutrino flux in spherical galaxies favors formation of the wide unipotential wells in them. The neutrino flux in the axisymmetrical galaxies suggests to favor the evolution in the direction of a spherically symmetrical potential. The generated unipotential wells are observed as plateaux in the velocity curves of circular stellar orbits. The constant neutrino density at galactic centers gives the linear part of the curves. The derived system of quasilinear differential equations for neutrinos in the axisymmetrical galaxies have been reduced to the system of the Lagrange-Charpit equations: the coupled differential equations, specifying the local neutrino velocities and dynamics of motion along trajectories, and an additional interconnected equation of the neutrino mass conservation, which can be applied for the determination of density of the neutrino component in galaxies.

Application of Exponential-Based Methods in Integrating the Constitutive Equations with Multicomponent Nonlinear Kinematic Hardening

The von-Mises plasticity model, in the small strain regime, along with a class of multicomponent nonlinear kinematic hardening rules is considered. The material is assumed to be stabilized after several load cycles and therefore, isotropic hardening will not be accounted for. Application of exponential-based methods in integrating plasticity equations is provided, which is based on defining an augmented stress vector and using exponential maps to solve a system of quasi-linear differential equations. The solutions obtained by this new technique give very accurate updated stress values that are consistent with the yield surface. The classical forward Euler method is reformulated in details and applied to the multicomponent form of the nonlinear kinematic hardening in order to provide a comparison for the suggested technique. Moreover, a consistent tangent operator for the exponential-based integration strategy and also for the classical forward Euler algorithm is presented. In order to show the robustness and performance of the proposed formulation, an extensive numerical investigation is carried out.

Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.

Bifurcation results to some quasilinear differential equation systems

The main purpose of this paper is to present a new global bifurcation approach to the studies of certain quasilinear differential equation systems.

Necessary conditions of optimality in quasilinear systems with delays

AbstractThe optimal control problem for the processes described with quasilinear system of differential equations in presence of mixed restrictions on phase and control variables in case of delays on both variables is considered. Using the Lagrange principle of taking restrictions off necessary conditions of optimality for such kind of systems are given. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On periodic solutions of differential equations with piecewise constant argument

The periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA, 66 (2007) 367–383, M.U. Akhmet, On the reduction principle for differential equations with piecewise argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646–663] is addressed. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results of our paper is an analogue of Gronwall–Bellman Lemma for functions with piecewise constant and retarded–advanced type arguments. Dependence of solutions on the parameter is investigated. Appropriate examples are given to show our results.

Asymptotic behavior of solutions of differential equations with piecewise constant arguments

The main goal of the work is to obtain sufficient conditions for the asymptotic equivalence of a linear system of ordinary differential equations and a quasilinear system of differential equations with piecewise constant argument.

Mountain pass solutions for a system of differential equations with Dirichlet boundary condition

In this article, the Dirichlet problem for a class of quasilinear differential equation systems is studied. Existence results are given by combining the dual least action principle and mountain pass theorem.

On Stability of Solutions to Quasilinear Periodic Systems of Differential Equations

We consider a quasilinear system of differential equations with periodic coefficients in the linear terms. We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity. The results are stated in terms of the integrals of the norm of a periodic solution to the Lyapunov differential equation.

Modified projection-iterative method for systems of quasilinear differential equations with delay and restrictions

We consider a quasilinear system of differential equations with constant delay, initial condition, and restriction and justify the application of a modified projection-iterative method to this system.

Asymptotic Representations for Solutions of One Class of Systems of Quasilinear Differential Equations

We establish asymptotic representations for solutions of one class of systems of differential equations appearing in the investigation of the asymptotic behavior of nth-order quasilinear differential equations.