Third-order Differential Equation Research Articles

Asymptotic Behavior and Oscillation of Third-Order Nonlinear Neutral Differential Equations with Mixed Nonlinearities

In this paper, we investigate the asymptotic properties of third-order nonlinear neutral differential equations with mixed nonlinearities using the comparison principle. Our results not only vastly improve upon but also broadly generalize many previously known ones. Examples demonstrating the applicability and efficacy of our results are provided.

Existence theory for a third-order ordinary differential equation with non-separated multi-point and nonlocal Stieltjes boundary conditions

<abstract><p>This study develops the existence of solutions for a nonlinear third-order ordinary differential equation with non-separated multi-point and nonlocal Riemann-Stieltjes boundary conditions. Standard tools of fixed point theorems are applied to prove the existence and uniqueness of results for the problem at hand. Further, we made use of the fixed point theorem due to Bohnenblust-Karlin to discuss the existence of solutions for the multi-valued case. Lastly, we clarify the reported results by means of examples.</p></abstract>

Asymptotic behavior of solutions of the third-order nonlinear advanced differential equations

<abstract><p>The aim of this work is to study some asymptotic properties of a class of third-order advanced differential equations. We present new oscillation criteria that complete, simplify and improve some previous results. We also provide many different examples to clarify the significance of our results.</p></abstract>

Оцінка ергономічної стійкості транспортного потоку на дільницях дорожньої мережі. Ідентифікація математичної моделі

The paper discusses the structure of the mathematical model featuring assessment of the ergonomic stability of the traffic flow in various sections of the road network under the influence of external disturbances, which is the result of structural identification. The mathematical model differs from the known ones in that it takes into account the dynamics of the process development. In addition to the gradients of speed and density of traffic flows, the research takes into account the dynamic properties of vehicles and the multi-lane road network, as well as the time of delays at pedestrian crossings and traffic lights. The dynamic properties of the traffic flow are described with a third-order differential equation. The mathematical model is parametrically identified; expressions for determining the gains and time constants included in the differential equation are obtained. The input parameters impacting and affecting the stability of the traffic flow are substantiated – these are the gradients of the density and speed of the flow. The parameters characterizing the response of the traffic flow to disturbances are substantiated – these are time constants, the physical meaning of which is the inertia of all links included in the model. The study presents expressions for determining the gains and time constants included in the differential equation. The gain coefficient K1 characterizes the degree of influence of the density of the traffic flow on the reaction time of the driver. The gain coefficient K2 characterizes the influence of the degree of dynamism of the traffic flow on the time of delays during movement and loss of stability. The gain coefficient K3 characterizes the degree of influence of a change in the traffic situation on the delay time when moving in the stream and loss of stability. The value of the time constant T1 characterizes the inertia of the driver depending on the density and intensity of the traffic flow. The value of the time constant T2 characterizes the inertia of the car and is expressed in the ability to maneuver. The value of the time constant T3 characterizes the inertia of changing the traffic situation.

Property $ \bar{A} $ of third-order noncanonical functional differential equations with positive and negative terms

<abstract><p>In this article, we have derived a new method to study the oscillatory and asymptotic properties for third-order noncanonical functional differential equations with both positive and negative terms of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} (p_2 (t)(p_1 (t) x'(t) )')'+a(t)g(x(\tau(t)))-b(t)h(x(\sigma(t)) = 0 \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>Firstly, we have converted the above equation of noncanonical type into the canonical type using the strongly noncanonical operator and obtained some new conditions for Property $ \bar{A} $. We furnished illustrative examples to validate our main result.</p></abstract>

Observer-Based Fuzzy Quantized Control for a Stochastic Third-Order Parabolic PDE System

The work addresses an observer-based fuzzy quantized control for stochastic third-order parabolic partial differential equations (PDEs) using discrete point measurements. For the first time, we contribute in introducing three types of quantizer—logarithmic quantizer, uniform quantizer, and hysteresis quantizer into the controller designs for the stochastic PDE system. The main advantage of the quantized control law lies in reducing the energy that the system spends. The stability analysis is nontrivial due to the complex stochastic PDE model. Different stability results are presented for each case. Sufficient LMI-based conditions are investigated to ensure the internal mean-square exponential stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance of the perturbed actuated system. Consistent simulation results that support the proposed theoretical statements are provided.

Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation

Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation

On the properties of solutions for nonautonomous third-order stochastic differential equation with a constant delay

In this work, complete Lyapunov functionals (LFs) are constructed and used for the established conditions on the nonlinear functions appearing in the main equation, to guarantee stochastically asymptotically stable (SAS), uniformly stochastically bounded (USB) and uniformly exponentially asymptotically stable (UEAS) in probability of solutions to the nonautonomous third-order stochastic differential equation (SDE) with a constant delay as \begin{align*} \begin{split} \dddot{x}(t)&+a(t)f(x(t),\dot{x}(t))\ddot{x}(t)+b(t)\phi(x(t))\dot{x}(t) +c(t)\psi(x(t-r))\\&+g(t,x)\dot{\omega}(t)=p(t,x(t),\dot{x}(t),\ddot{x}(t)). \end{split} \end{align*} In Section 4, we give two numerical examples as an application to illustrate the results.

Двумерное течение Пуазейля ньютоновской жидкости с вязкостью, зависящей от температуры

The problem of a two-dimensional Poiseuille flow of a Newtonian fluid with a viscosity exponen-tially dependent on temperature under the influence of longitudinal pressure and temperature gra-dients is reduced to a two-parameter boundary value problem for a third-order ordinary differential equation with respect to a dimensionless unknown characterizing the change in temperature across the channel. In the range of negative values of the dimensionless longitudinal temperature gradient, the solution has two branches. By closing, the branches form a boundary on the parameter plane beyond which Poiseuille-type flows do not exist. The solutions belonging to different branches dif-fer in the flow rate and intensity of heat transfer by the liquid through solid boundaries. The latter circumstance entails deceleration of the flow in the near-wall areas. In the region of positive values of the dimensionless longitudinal temperature gradient, the solution is unique. With an increase in the mentioned parameter, the heat flow to the liquid from the walls increases, which entails the formation of a more bulged velocity profile compared to the isothermal case. At sufficiently large values of the longitudinal temperature gradient, there appears a ‘stick’ flow with a uniform velocity distribution everywhere except for narrow layers near solid boundaries. It is shown that the rate is proportional to the ratio of the densities of the transverse and longitudinal heat flows. Explicit ex-pressions for the pressure and shear stress fields are given. The properties of monotonicity of the velocity and temperature profiles as functions of the transverse coordinate have been studied. It is shown that, regardless of the values of the dimensionless parameters of the problem, the velocity has a positive maximum on the central axis of the channel and monotonically decreases to zero at the solid boundary.

On Novel Spline Function Method for Solving Third-Order Boundary Value Problems

In this paper, a numerical method is suggested for solving general a nonlinear third order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.

Исследование в окрестности подвижной особой точки некоторого класса нелинейного дифференциального уравнения третьего порядка для комплексной области

В настоящей работе проводится исследование некоторого класса дифференциальных уравнений с подвижными особенностями. Приводится обобщение полученных ранее результатов исследования, а именно, теоремы существования и единственности решения рассматриваемого класса уравнений на комплексную область. Получена структура аналитического приближенного решения и априорные оценки погрешности. Данные оценки были оптимизированы с помощью апостериорных. Теоретические положения были протестированы с помощью численного эксперимента. In this paper, we study a certain class of differential equations with moving singularities. A generalization of the previously obtained results of the study, namely, the existence and uniqueness theorem for the solution of the considered class of equations, to the complex domain is given. The structure of the analytical approximate solution and a priori error estimates are obtained. These estimates were optimized using post hoc estimates. Theoretical provisions were tested using a numerical experiment.

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

Symmetries and solutions for the inviscid oceanic Rossby wave equation

The -and -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method.

Existence of positive solutions for third-order semipositone boundary value problems on time scales

In this paper, we consider the existence of positive solutions for a semipositone third-order nonlinear ordinary differential equation on time scales. In suitable growth conditions, by considering the properties on time scales and establishing a special cone, some new results on the existence of positive solutions are established when the nonlinearity is semipositone.

Asymptotic representation of some classes of solutions third-order differential equation

The conditions for the existence of one class of solutions of a binomial non-autonomous differential equation of the third order with a nonlinearity close in some sense to a linear one are established. Using the a priori properties of the so-called Pω(λ0)-solutions, asymptotic at t↑ω(ω≤+∞) images were obtained for such solutions connections and their derivatives of the first and second order in the case λ0 = 0. The propositions proved for the nonlinear equation are transferred to linear differential equations of the third order with asymptotically small coefficients. This made it possible, to some extent, to supplement the known results regarding the asymptotic properties of solutions of linear differential equations of the third order.

A Simplification and Generalization of Elsayed and Ibrahim’s Two-Dimensional System of Third-Order Difference Equations

A full Lie analysis of a system of third-order difference equations is performed. Explicit solutions, expressed in terms of the initial values, are derived. Furthermore, we give sufficient conditions for the existence of two-periodic and four-periodic solutions in certain cases. Our results generalize and simplify some work in the literature.

Solution of initial-value problem for linear third-order fuzzy differential equations

Solution of initial-value problem for linear third-order fuzzy differential equations

New Conditions for Testing the Oscillation of Third-Order Differential Equations with Distributed Arguments

In this paper, we consider a certain class of third-order nonlinear delay differential equations with distributed arguments. By the principle of comparison, we obtain the conditions for the nonexistence of positive decreasing solutions as well as, and by using the Riccati transformation technique, we obtain the conditions for the nonexistence of increasing solutions. Therefore, we get new sufficient criteria that ensure that every solution of the studied equation oscillates. Asymmetry plays an important role in describing the properties of solutions of differential equations. An example is given to illustrate the importance of our results.

Charged black holes in Einsteinian quartic gravity

In this paper, we studied Einsteinian quartic gravity minimally coupled to electrodynamics in four dimensions. First, by varying the action, we obtain the field equations, and by integration, we obtain a nonlinear third-order differential equation for the metric function and as well as the electromagnetic potential. Then, in the context of the Maxwell field, we discussed the conditions under which the charged black hole exists. Then, we have demonstrated the thermodynamics and stability of the solution for the case of positive coupling of quartic theory. Finally, we showed that the charged black hole solutions of EQG (unlike GR and like ECG) have no inner horizon and do not conform to the extremal bound of GR. Also, the uniqueness of BH solutions in this theory does not work anymore.

Study of periodic solutions for third-order iterative differential equations via Krasnoselskii-Burton’s theorem

Abstract This paper studies the existence of periodic solutions of a third order iterative differential equation. The main tool used here is Krasnoselskii-Burton’s fixed point theorem dealing with a sum of two mappings, one is a large contraction and the other is compact.