Quasilinear Differential Equations Research Articles

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.

Unpredictable solutions of quasilinear differential equations with generalized piecewise constant arguments of mixed type

"An unpredictable solution is found for a quasilinear differential equation with generalized piecewise constant argument (EPCAG). Sufficient conditions are provided for the existence, uniqueness and exponential stability of the unpredictable solution. The theoretical results are confirmed by examples and illustrated by simulations."

The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations

Abstract The article uses mathematical theory to derive a quasi-linear differential equation describing the law of memory forgetting in psychology. The nonlinear automatic control machine model is used to preliminary model the indiscriminate memory problem in psychology. The research obtains the recommended evaluation value with the forgetting characteristics in memory psychology. The experimental results found that the model reasonably solves the problem of quantification of timeliness.

On the existence of mild solutions for a class of nonlocal quasi-linear differential equations with iterated deviating arguments

On the existence of mild solutions for a class of nonlocal quasi-linear differential equations with iterated deviating arguments

Hyperbolicity of First Order Quasi-Linear Equations

The theorem about equivalence of the strong hyperbolicity concept and the Friedrichs hyperbolicity concept for partial quasi-linear differential equations of the first order is proved. On the basis of this theorem, the necessary and sufficient conditions of hyperbolicity are found in terms of the matrix of the corresponding linearized first order equations system.

Bifurcations of Invariant Tori in Second-Order Quasilinear Evolution Equations in Hilbert Spaces and Scenarios of Transition to Turbulence

In this paper, we consider second-order quasilinear differential equations in a separable Hilbert space for which the well-known Landau–Hopf scenario of transition to turbulence can be realized. We prove increasing of the control parameter leads to the consecutive appearance of invariant tori of increasing dimensions. In this case, the invariant torus of the largest possible dimension appears to be attractive. The results are obtained by using methods of the qualitative theory of dynamical systems with an infinite-dimensional space of initial conditions: the method of integral manifolds, the theory of normal forms, and also asymptotic methods of analysis of dynamical systems.

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

Periodic solutions of impulsive differential equations with piecewise alternately advanced and retarded argument of generalized type

We investigate the existence of the periodic solutions of a quasilinear impulsive differential equation with alternately advanced and retarded arguments of generalized type, in short IDEPCAG. By using some fixed point theorems and some new analysis techniques, sufficient conditions are obtained for the existence and uniqueness of periodic solutions of these IDEPCAG systems. A new IDEPCAG’s Gronwall-type lemma is proved. Some examples concerning impulsive biological models such as Lasota–Ważewska model, Nicholson’s blowflies and logistic models are treated.

On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

Oscillation Results for Third-Order Quasi-Linear Emden-Fowler Differential Equations with Unbounded Neutral Coefficients

Abstract Some new oscillation criteria are obtained for a class of thirdorder quasi-linear Emden-Fowler differential equations with unbounded neutral coefficients of the form ( a ( t ) ( z ″ ( t ) ) α ) ′ + f ( t ) y λ ( g ( t ) ) = 0 , \[(a(t){(z(t))^\alpha })' + f(t){y^\lambda }(g(t)) = 0,\] where z(t) = y(t) + p(t)y(σ(t)) and α, λ are ratios of odd positive integers. The established results generalize, improve and complement to known results.

Алгоритмизация решения динамических краевых задач теории гибких пластин с учетом сдвига и инерции вращения

Most of the problems on flexible plates are solved in the Föppl-von Karman formulation, which is Love's special case. The constructed algorithms are not economical in terms of implementation on a computer. Therefore, construction of algorithms for the complete calculation of flexible plates with a given degree of accuracy with allowance for the shear and inertia of rotation is becoming a topical issue. The problem of creating an automated inference system and solving the equations of the theory of elasticity and plasticity were first posed in the monograph by V.K. Kabulov. In this work, for the first time, the main problems of algorithmization are formulated and ways of their machine solution are outlined. The problem of algorithmization is solved as follows: depending on geometric characteristics of the object and physical properties of the material, a design scheme of this model is selected; derivation of the initial differential equations and the corresponding boundary and initial conditions; selection of a computational algorithm and numerical solution of the obtained equations; analysis of the obtained numerical results describing the stress-strain state of the structure under consideration. This work consists of an introduction, three sections and a conclusion. In the first paragraph, the equations of motion of rectangular plates are given. Substituting the expression for the force of moments and shearing forces and introducing a dimensionless value, a system of equations in displacements is obtained. In the second section, using the central difference formulas, a system of quasilinear ordinary differential equations is obtained. Taking into account the boundary and initial conditions, the system of equations is reduced to matrix form, which can be solved by the Runge-Kutta method. In the third paragraph, an analysis of the results obtained is presented.

ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION

<abstract> We investigate a singularly perturbed boundary value problem for a piecewise-smooth second-order quasilinear differential equation in the case when the discontinuous curve which separates the domain is monotone. Applying the boundary layer function method, the asymptotic expansion of a solution with internal layer appearing in the neighborhoods of some point on the monotone curve and the point itself is constructed. For sufficiently small parameter values, using the matching method, the existence of a smooth solution with an internal transition layer in the neighborhood of a point of the monotone curve is proved. A simple example is given to show the effectiveness of our method. </abstract>

Oscillatory Behavior of Third-Order Quasi-Linear Neutral Differential Equations

In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results.

Sobolev regularity solutions for a class of singular quasilinear ODEs

AbstractThis paper considers an initial-boundary value problem for a class of singular quasilinear second-order ordinary differential equations with the constraint condition stemming from fluid mechanics. We prove that the existence of positive Sobolev regular solutions for this kind of singular quasilinear ODEs by means of a suitable Nash-Moser iteration scheme Meanwhile, asymptotic expansion of those positive solutions is shown.

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations.

In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Разрешимость задачи Коши для одной системы квазилинейных дифференциальных уравнений первого порядка

Рассмотрена задача Коши для одной системы квазилинейных дифференциальных уравнений первого порядка. Для указанной задачи Коши в исходных координатах с помощью метода дополнительного аргумента исследуется ее разрешимость; определены достаточные условия существования и единственности локального решения, при которых решение имеет такую же гладкость по независимой переменной, как и начальные функции задачи Коши. Сформулирована и доказана теорема о существовании и единственности локального решения. Определены достаточные условия существования и единственности глобального решения. Сформулирована и доказана теорема о~существовании и единственности глобального решения. Доказательство глобальной разрешимости опирается на глобальные оценки.

Quasilinear elliptic equations with sub-natural growth terms in bounded domains

We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type $$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta _{p, w} u = \sigma u^{q} &{} \text {in }\Omega ,\\ u = 0 &{} \text {on }\partial \Omega \end{array}\right. } \end{aligned}$$ in the sub-natural growth case $$0< q < p - 1$$ , where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^{n}$$ , $$\Delta _{p, w}$$ is a weighted p-Laplacian, and $$\sigma $$ is a nonnegative (locally finite) Radon measure on $$\Omega $$ . We give criteria for the existence problem. For the proof, we investigate various properties of p-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.

Some new oscillation criteria of fourth-order quasi-linear differential equations with neutral term

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems.

Some New Oscillation Criteria of Even-Order Quasi-Linear Delay Differential Equations with Neutral Term

The neutral delay differential equations have many applications in the natural sciences, technology, and population dynamics. In this paper, we establish several new oscillation criteria for a kind of even-order quasi-linear neutral delay differential equations. Comparing our results with those in the literature, our criteria solve more general delay differential equations with neutral type, and our results expand the range of neutral term coefficient. Some examples are given to illustrate our conclusions.