Third-order Linear Differential Equation Research Articles

Asymptotic and Oscillatory Properties of Third-Order Differential Equations with Multiple Delays in the Noncanonical Case

This paper investigates the asymptotic and oscillatory properties of a distinctive class of third-order linear differential equations characterized by multiple delays in a noncanonical case. Employing the comparative method and the Riccati method, we introduce the novel and rigorous criteria to discern whether the solutions of the examined equation exhibit oscillatory behavior or tend toward zero. Our study contributes to the existing literature by presenting theories that extend and refine the understanding of these properties in the specified context. To validate our findings and demonstrate their applicability in a general setting, we offer two illustrative examples, affirming the robustness and validity of our proposed criteria.

On special solutions to the Ermakov–Painlevé XXV equation

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

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

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

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.

On the Control of the Spectra of Upper Strong Oscillation Exponents of Signs, Zeros, and Roots of Third-Order Differential Equations

We construct an example of a third-order linear homogeneous differential equation with continuous coefficients on the half-line whose spectra of upper strong oscillation exponents of the signs, zeros, and roots coincide with a given Souslin set containing zero in the nonnegative half-line of the extended real line.

An exact finite difference scheme for a three-layer beam model

ABSTRACT In this work, we propose an efficient computational technique for a recognized three-layer beam model which is described by a third-order linear ordinary differential equation. By using the Mickens' methodology, an exact finite difference (EFD) scheme for this model is constructed. The EFD scheme provides an explicit recurrence formula to compute the solution of the beam model, but it is more efficient than the formula of the exact solution. Moreover, the EFD scheme is also more efficient than well-known standard finite difference schemes in both computational time and accuracy. Finally, two numerical experiments are performed to illustrate the theoretical results as well as to show advantages of the EFD scheme.

Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations

We extend the invariant subspace method (ISM) to a class of Hamilton–Jacobi equations (HJEs) and a family of third-order time-fractional dispersive PDEs with the Caputo fractional derivative in this letter. More precisely, the complete classification is presented for such HJEs that admit invariant subspaces governed by solutions of the second-order and third-order linear ordinary differential equations (ODEs). Meanwhile, some concrete equations are derived for the construction of new exact solutions u(x,t)=∑i=1nCi(t)fi(x). Then a set of invariant subspaces of the considered third-order time-fractional non-linear dispersive equations are obtained. Based on the Laplace transform method (LTM) and applying several properties of the well known Mitta-Leffer (ML) function, the different types of explicit solutions of a family of third-order time-fractional dispersive PDEs are finally derived.

On essential values of Sergeev's frequencies and exponents of oscillation for solutions of a third-order linear differential periodic equation

In this paper, we study various types of Sergeev's frequencies and exponents of oscillation for solutions of linear homogeneous differential equations with continuous bounded coefficients. For any preassigned natural number $N$, a periodic third-order linear differential equation is constructively built in this paper, which has the property that its upper and lower Sergeev frequency spectra of strict signs, zeros and roots, as well as the spectra of all upper and lower strong and weak oscillation indices of strict and non-strict signs, zeros, roots and hyperroots contain the same set, consisting of $N$ different essential values, both metrically and topologically. Moreover, all these values are implemented on the same set of solutions of the constructed equation, that is, for each solution from this set, all the frequencies listed above and the oscillation exponents coincide with each other. When constructing the indicated equation and proving the required results, analytical methods of the qualitative theory of differential equations were used, in particular, methods of the theory of perturbations of solutions of linear differential equations, as well as the author's technique for controlling the fundamental system of solutions of such equations in one particular case.

Positing the problem of stationary distributions of active particles as third-order differential equation.

In this work, we obtain a third-order linear differential equationfor stationary distributions of run-and-tumble particles in two dimensions in a harmonic trap. The equationrepresents the condition j=0, where j is a flux. Since an analogous equationfor passive Brownian particles is first-order, a second- and third-order term are features of active motion. In all cases, the solution has a form of a convolution of two distributions: the Gaussian distribution representing the Boltzmann distribution of passive particles, and the beta distribution representing active motion at zero temperature.

Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations

We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of this linear equation and the maximal regularity estimate for their solution. Using these results, we prove the solvability of a nonlinear differential equation and estimate the norms of its terms.

ASYMPTOTIC PROPERTIES OF KNESER SOLUTIONS TO THIRD-ORDER DELAY DIFFERENTIAL EQUATIONS

<abstract> The aim of this paper is to extend and complete the recent work by Graef et al. (J. Appl. Anal. Comput., 2021) analyzing the asymptotic properties of solutions to third-order linear delay differential equations. Most importantly, the authors tackle a particularly challenging problem of obtaining lower estimates for Kneser-type solutions. This allows improvement of existing conditions for the nonexistence of such solutions. As a result, a new criterion for oscillation of all solutions of the equation studied is established. </abstract>

Dynamical invariants for time-dependent real and complex Hamiltonian systems

The Struckmeier and Riedel (SR) approach is extended in real space to isolate dynamical invariants for one- and two-dimensional time-dependent Hamiltonian systems. We further develop the SR-formalism in zz̄ complex phase space characterized by z = x + iy and z̄=x−iy and construct invariants for some physical systems. The obtained quadratic invariants contain a function f2(t), which is a solution of a linear third-order differential equation. We further explore this approach into extended complex phase space defined by x = x1 + ip2 and p = p1 + ix2 to construct a quadratic invariant for a time-dependent quadratic potential. The derived invariants may be of interest in the realm of numerical simulations of explicitly time-dependent Hamiltonian systems.

Computer Calculation of Green Functions for Third-Order Ordinary Differential Equations

In this paper, we present a method of computer calculation of Green functions in the form of generalized power series for third-order linear differential equations admitting regular singularities. For specific boundary-value problems, we construct Green functions by using the software proposed.

Asymptotic solutions of (q(qy′)′)′+py′+ry=0

We are concerned with the third-order linear differential equation (q(qy′)′)′+py′+ry=0 defined on an interval [t0,∞). We derive asymptotic solutions for cases characterized by p being “small” and “large”, respectively. Our main tool is a result by Eastham on the asymptotic solution of linear differential systems.

Self-associated three-dimensional cones

For every proper convex cone \(K \subset {\mathbb {R}}^3\) there exists a unique complete hyperbolic affine 2-sphere with mean curvature \(-1\) which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This equivalence relation is generated by the groups \(SL(3,{\mathbb {R}})\) and \(S^1\), where the former acts by linear transformations of the ambient space, and the latter by multiplication of the cubic holomorphic differential of the affine sphere by unimodular complex constants. The action of \(S^1\) generalizes conic duality, which acts by multiplication of the cubic differential by \(-1\). We call a cone self-associated if it is linearly isomorphic to all its associated cones, in which case the action of \(S^1\) induces (nonlinear) isometries of the corresponding affine sphere. We give a complete classification of the self-associated cones and compute isothermal parameterizations of the corresponding affine spheres. Their metrics can be expressed in terms of degenerate Painlevé III transcendents. The boundaries of generic self-associated cones can be represented as conic hulls of vector-valued solutions of a certain third-order linear ordinary differential equation with periodic coefficients, but there exist also self-associated cones with polyhedral boundary parts. The self-associated cones are the second family of non-trivial 3-dimensional cones for which the affine spheres can be computed explicitly, the first being the semi-homogeneous cones.

INCEPTION OF GREEN FUNCTION FOR THE THIRD-ORDER LINEAR DIFFERENTIAL EQUATION THAT IS INCONSISTENT WITH THE BOUNDARY PROBLEM CONDITIONS

Regarding the importance of teaching linear differential equations, it should be noted that every physical and technical phenomenon, when expressed in mathematical sciences, is a differential equation. Differential equations are an essential part of contemporary comparative mathematics that covers all disciplines of physics (heat, mechanics, atoms, electricity, magnetism, light and wave), many economic topics, engineering fields, natural issues, population growth and today’s technical issues. Used cases. In this paper, the theory of third-order heterogeneous linear differential equations with boundary problems and transforming coefficients into multiple functions p(x) we will consider. In mathematics, in the field of differential equations, a boundary problem is called a differential equation with a set of additional constraints called boundary problem conditions. A solution to a boundary problem is a solution to the differential equation that also satisfies the boundary conditions. Boundary problem problems are similar to initial value problems. A boundary problem with conditions defined at the boundaries is an independent variable in the equation, while a prime value problem has all the conditions specified in the same value of the independent variable (and that value is below the range, hence the term "initial value"). A limit value is a data value that corresponds to the minimum or maximum input, internal, or output value specified for a system or component. When the boundaries of boundary values in the solution of the equation to obtain constants D1, D2, D3 to lay down Failure to receive constants is called a boundary problem. We solve this problem by considering the conditions given for that true Green expression function. Every real function of the solution of a set of linear differential equations holds, and its boundary values depend on the distances. Key words: Green Function, Boundary Problem, Private Solution, Public Solution, Wronskian Determinant.

Conditions of Oscillation for a Linear Third-Order Differential Equation

We introduce some new oscillation criteria for a third-order linear differential equation with variable coefficients in this study. We found out the corollary as a result of the Storm comparison theory and used it to prove some theorems. Through it, we were able to achieve the necessary conditions for oscillation. We concluded that the solution to the differential equation is oscillating if it is bounded from below, and also if the discriminant of the equation is negative, its solution is oscillatory. We have given examples illustrating these results.

Similarity solution for unsteady magnetohydrodynamic flow between two parallel porous plates with an angular velocity

The objective of this paper is to analyse the unsteady magnetohydrodynamic flow of viscous fluid with an angular velocity between parallel porous plates when the fluid is being withdrawn through both the walls of the channel at the same rate. The problem is reduced to a third-order linear differential equation which depends on a Suction Reynolds number R and Hartmann number with angular velocity M1 for which an exact solution is obtained.

Moments of the ground state density for the d-dimensional Fermi gas in an harmonic trap

We consider properties of the ground state density for the [Formula: see text]-dimensional Fermi gas in an harmonic trap. Previous work has shown that the [Formula: see text]-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the [Formula: see text]th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re[Formula: see text]. The moments, and the smoothed density, permit expansions in [Formula: see text], where [Formula: see text], with [Formula: see text] denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the [Formula: see text] expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.

ON BOUNDEDNESS OF SOLUTIONS OF THIRD ORDER LINEAR DIFFERENTIAL EQUATIONS WITH FUNCTIONAL ON HALF-AXIS

Established the sufficient conditions of boundedness on half-axis all solutions and their first, second derivatives of third-order linear differential equations with functional. For this aim developed the non-standard method of reduce of equations to system. The illustrative example is given.