Third-order Differential Equation Research Articles

ON THE ASYMPTOTICS OF SOLUTIONS OF NON-AUTONOMOUS DIFFERENTIAL EQUATIONS ASYMPTOTICALLY CLOSE TO LINEAR

The necessary and sufficient conditions are obtained for the existence of all possible types of Pω(Y0, λ0)-solutions of non-autonomous second and third-order ordinary differential equations that are asymptotically close to linear equations. The asymptotic representation of each possible type of Pω(Y0, λ0)-solutions has been determined, and the question of their quantity has been clarified. Consequences have also been derived from the obtained theorems for cases when the equation is a linear differential equation.

More Effective Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of a Class of Third-Order Functional Differential Equations

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations.

Turing-Hopf bifurcation and bi-stable spatiotemporal periodic orbits in a delayed predator-prey model with predator-taxis

When Turing-Hopf bifurcation occurs, it is naturally worth discussing the types of spatiotemporal patterns and their stabilities. In this paper, for partial functional differential equations with nonlinear diffusion, we establish the explicit formulae for the coefficients of normal form associated with Turing-Hopf bifurcation. This provides an explicit connection between the original system and the third-order ordinary differential equations (i.e. normal form) restricted on center manifold. Subsequently, with the aid of extended formulae, we discuss a delayed predator-prey model with predator-taxis and find that Turing-Hopf bifurcation leads to the emergence of a pair of stable single-peak spatially inhomogeneous steady states, a pair of stable multi-peak spatially inhomogeneous periodic orbits and so on. Also, the parameter regions in which these phenomena arise are given quantitatively. It is further shown that the pair of bistable spatially inhomogeneous periodic orbits arise due to the interaction of predator-taxis and time delay, which won't emerge in the same system without predator-taxis.

A class of single-step hybrid block methods with equally spaced points for general third-order ordinary differential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary differential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the effectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.

On Determining the Equation of a Salkowski Curve Satisfying tau/kappa=1/s

In this paper, we determine the equation of a Salkowski curve whose ratio of torsion to curvature is given by 1/s, where s is the arc length of the curve. The Frenet-Serret equations provide the third-order vector differential equation for the unit tangent vector T(s) and the general (series) solution was obtained. In the end, the series solution is entirely determined by the given initial conditions.

Positive periodic solutions of third-order neutral differential equations with delayed derivative terms

By applying fixed point index theory on cones, the existence results of positive 2π-periodic solutions are obtained for the third-order neutral differential equation with delayed derivative terms in nonlinearity(x(t)−cx(t−δ))‴+a(t)x(t)=f(t,x(t),x(t−τ0),x′(t−τ1),x″(t−τ2)),t∈R under the condition that the nonlinear term f satisfies superlinear or sublinear growth, where constants δ>0,|c|<1, a:R→(0,+∞) and f:R×[0,+∞)2×R2→[0,+∞) are continuous functions which are 2π periodic in t and constants τ0,τ1,τ2>0.

Ulam Stability for Third-Order Linear Differential Equations with Variable Coefficients

Ulam Stability for Third-Order Linear Differential Equations with Variable Coefficients

On the Global Asymptotic Stability and 4-Period Oscillation of the Third-Order Difference Equation

The main objective of this paper is to study the global behavior and oscillation of the following third-order rational difference equation x n + 1 = α x n x n − 1 x n − 2 / β x n − 1 2 + γ x n − 2 2 , where the initial conditions x − 2 , x − 1 , x 0 are nonzero real numbers and α , β , γ are positive constants such that α ≤ β + γ . Visual examples supporting solutions are given at the end of the study. The figures are found with the help of MATLAB.

On the existence of $6$-cycles for some families of difference equations of third order

On the existence of $6$-cycles for some families of difference equations of third order

Kneser-Type Oscillation Criteria for Half-Linear Delay Differential Equations of Third Order

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings.

Spectral analysis for some third-order differential equations: a semigroup approach

Spectral analysis for some third-order differential equations: a semigroup approach

SOME QUALITATIVE PROPERTIES OF SOLUTIONS OF CERTAIN NONLINEAR THIRD-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY

This study considered certain nonlinear third-order stochastic differential equations with delay. The third-order equation is reduced to an equivalent system of first-order differential equations and used to construct the desired complete Lyapunov-Krasovski\v{\i} functional. Standard conditions guaranteeing stability when the forcing term is zero, boundedness of solutions when the forcing term is non-zero, and lastly the existence and uniqueness of solutions are derived. The obtained results indicated that the adopted technique is effective in studying the qualitative behaviour of solutions. The obtained results are not only new but extend the frontier of knowledge of the qualitative behaviour of solutions of nonlinear stochastic differential with delay. Finally, two special cases are given to illustrate the derived theoretical results.

New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation.

Existence of Infinite, Everywhere Discontinuous Spectra of Upper Exponents of Oscillation of Signs, Zeros, and Roots of Third-Order Differential Equations

Existence of Infinite, Everywhere Discontinuous Spectra of Upper Exponents of Oscillation of Signs, Zeros, and Roots of Third-Order Differential Equations

Scaled boundary perfectly matched layer for wave propagation in a three-dimensional poroelastic medium

In this paper, we introduce a novel direct time-domain artificial boundary condition, called the scaled boundary perfectly matched layer, for wave problems in 3D unbounded fluid-saturated poroelastic medium. The poroelastic medium is conceptualized as a two-phase medium composed of a solid skeleton and an interstitial fluid based on the u-p formulation of Biot's theory. The scaled boundary perfectly matched layer proposed in this work originates from an extension of a recently developed perfectly matched layer for wave problems in a single-phase solid medium. The dependence of the conventional perfectly matched layer on the global coordinate system is overcome in the present method. As a result, this method permits the utilization of an artificial boundary with general geometry (not necessarily convex) and can consider planar physical surfaces and interfaces extending to infinity. The proposed method is formulated by a mixed unsplit-field form, which includes both the mixed displacement-stress field and pore pressure-relative displacement field. The resulting scaled boundary perfectly matched layer formulation can be directly described by a system of third-order ordinary differential equations in time, which can be easily integrated into the standard finite-element framework of poroelastic theory. Finally, numerical benchmark examples are presented to demonstrate the accuracy and robustness of the proposed scaled boundary perfectly matched layer, as well as the versatility in practical engineering problems.

Method of Variation of Parameters for the Third-Order Linear Proportional Dynamic Equations

The method of variation of parameters for the third-order linear nonhomogeneous differential equations is given in this article, utilizing the proportional derivative and its characteristics on the time scale. Then, an example is provided to illustrate how to apply the provided approach.

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems

In this study, we developed a new modification of the well-known differential transform method (DTM) that differs from the classical one by the algorithm for calculating the coefficients of an approximate solution given as a series. The proposed new algorithm we will call α-parameterized DTM (α-p DTM,for short). By using the proposed α-p DTM, we solve the boundary value transmission problem for the third-order differential equation. For the same problem, an approximate solution is also found using also the classical DTM. The solutions obtained were compared graphically.

On the Size of the Safety Area around the Launch Trajectory of a Rocket

The safety zone around the flight path of a rocket is determined by the fall of debris in the case of an accidental explosion or commanded termination. The trajectory of a tumbling body in a vertical plane is determined by specifying the velocity, flight path angle and angle of attack as functions of time. This involves the lift, drag and pitching moment coefficients as functions of the angle of attack over a full circle—0 to 360 degrees—to account for the tumbling motion. The problem is reduced to a third-order non-linear differential equation for the angle of attack by using the approximation of free fall coordinates. The analytical and numerical solutions show that two types of tumbling fall are possible, one with rotation and the other with oscillation. The tumbling trajectories are plotted and discussed for a variety of initial conditions, mass and aerodynamic properties of the tumbling body.

Solvability of an infinite system of three point BVP in c0 and ℓ1 spaces using measure of noncompactness

Various existing works in the literature have addressed the solvability of different infinite systems of boundary value problems in different types of Banach spaces. In most of these works, the authors have restricted themselves in systems of differential equations with boundary conditions defined at two points. In this work, we have addressed the solvability of an infinite system of third-order differential equations with boundary conditions defined at three points in the sequence spaces [Formula: see text] and [Formula: see text]. We have also illustrated the results for both sequence spaces with suitable examples.