Third-order Differential Equation Research Articles

Boundedness properties of the generalized Todd's difference equation

ABSTRACTWe consider the third-order difference equation with and We show that if then there exist positive initial conditions such that the solutions are unbounded except for the case and

Solvability of infinite systems of third-order differential equations in $$c_{0}$$ by Meir–Keeler condensing operators

Throughout this work, using the technique of measure of noncompactness together with Meir–Keeler condensing operators, we study the solvability of the following infinite system of third-order differential equations in the Banach sequence space $$c_0 $$ as a closed subspace of $$ \ell ^{\infty } $$ : $$\begin{aligned} u_{i}^{\prime \prime \prime }+au_{i}^{\prime \prime }+bu_{i}^{\prime }+cu_{i} =f_{i}(t,u_{1}(t),u_{2}(t),\ldots ) \end{aligned}$$ where $$ f_{i}\in C(\mathbb {R}\times \mathbb {R}^{\infty }, \mathbb {R}) $$ is $$ \omega $$ -periodic with respect to the first coordinate and $$ a,b,c \in \mathbb {R} $$ are constant. Our approach depends on the Green’s function corresponding to the aforesaid system and deduce some conclusions relevant to the existence of $$ \omega $$ -periodic solutions in Banach sequence space $$c_{0} $$ . In addition, some examples are supplied to illustrate the usefulness of the outcome.

Solution of initial-boundary value problem for a system of partial differential equations of the third order

The author consider initial-boundary value problem for a system of differential equations in partial derivatives of third order. We study questions of existence and uniqueness of classical solution of the problem, and ways for its evaluation. The introduction of a new desired function enables us to reduce the problem to equivalent nonlocal problem with integral condition for a system of integro-differential euqtions of hyperbolic type with a functional relation. We establish conditions for unique solvability of the nonlocal problem by means of the method of functional parameters, propose algorithms for solving of the equivalent problem and prove their convergence. As a result, there are obtained conditions for existence of unique classical solution of initial-boundary value problem for the considered system of differential equations of third order in terms of initial data.

Exact Analytic Solution and Investigation of the Guyer-Krumhansl Heat Equation

The heat propagation in a Guyer-Krumhansl one-dimensional thermal conductivity model is studied. The exact analytic solutions of a one-dimensional equation of the Guyer-Krumhansl type are obtained and investigated by the operator method, taking into account the medium motion, and also a harmonic solution for a system of third-order inho-mogeneous partial differential equations is obtained. The evolution is investigated for several types of initial distributions with periodic, power-law, exponential-power-law coordinate dependence and for the distribution of the Dirac delta function. The exact solutions thus obtained can simulate, in particular, the propagation of heat pulses of diverse forms under real experimental conditions, such as an isolated spatial heat wave, a laser pulse, and a periodic heating. The results are applied to simulate the propagation of heat in ultrathin films. The problem of heat propagation in small-size systems like nanofibres and nanofilms is solved analytically, taking into account ballistic effects; the maximum principle and the positivity of solutions in the context of heat conduction are discussed.

Application of Laplace-Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations.

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

Exact results for the anisotropic honeycomb lattice Green function with applications to three-step Pearson random walks

The mathematical properties of the lattice Green function for the anisotropic honeycomb lattice are studied, where is a complex variable which lies in a plane, and is a real anisotropy parameter with . This double integral defines a single-valued analytic function provided that a cut is made along the real axis from to . In order to analyse the behaviour of along the edges of the cut it is convenient to define the limit function where . It is proved that the limit functions and can be sectionally evaluated exactly for all , in terms of various elliptic integrals of the first kind , where is a rational function of and u.Next, it is demonstrated that is a solution of a second order linear differential equation with eight ordinary regular singular points and two apparent singular points. It is shown that the apparent singularities can be removed by constructing a particular differential equation of third order. The series solution where and is investigated. In particular, we show that, in general, satisfies a four-term linear recurrence relation. This result is used to determine the asymptotic behaviour of as .Integral representations are established for and . It is found that where J0(z) and Y0(z) denote Bessel functions of the first and second kind, respectively, and .Finally, the results are applied to the lattice Green function for the anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane.

A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock–Paper–Scissors

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models.

Asymptotic properties of solutions to third order neutral differential equations with delay

This paper concerns the asymptotic properties of solutions of a class of third-order neutral differential equations with delay. We give sufficient conditions for every solution to be converges to zero, bounded and square integrable. An example is also given to illustrate the results.

Existence Theory for Nonlinear Third-Order Ordinary Differential Equations with Nonlocal Multi-Point and Multi-Strip Boundary Conditions

We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely on the modern tools of functional analysis and are well illustrated with the aid of examples. An analogue problem involving non-separated integro-multi-point boundary conditions is also discussed.

Holographic duality for 3d spin-3 gravity coupled to scalar field

The 3d spin-3 gravity theory is holographically dual to a 2d W3-extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to SU(1,2)×SU(1,2). On ground of symmetry the dual bulk space–time will be given by an 8d group manifold SU(1,2). Hence we need to introduce five extra coordinates in addition to three ordinary ones. The 3d space–time is a 3d hyper-surface Σ embedded at constant values of the extra variables. Operators in the CFT at the boundary of Σ are expressed in terms of W descendants of the operators at the boundary of Σ0, where the extra variables vanish. In this paper it is shown that AdS/CFT correspondence for a scalar field coupled to 3d spin-3 gravity is realized in this auxiliary 8d space. A bulk-to-boundary propagator of a scalar field is found and a generating functional of boundary two-point functions of scalar W-descendant operators is obtained by using the classical action for the scalar field. Classically, the scalar field must satisfy both Klein–Gordon equation and a third-order differential equation, which are related to the quadratic and cubic Casimir operators of su(1,2). It is found that the coefficient function of the derivatives of the scalar field in the latter equation is the spin-3 gauge field, when restricted to the hypersurface. An action integral in the 8d auxiliary space for the 3d spin-3 gravity coupled to a scalar field is presented. In general, this 8d auxiliary space is a deformation of the manifold SU(1,2). An 8d local frame is introduced and the equations of motion for the 8d connections Aμ, A‾μ are solved. By restricting those solutions onto Σ, flat connections in 3d SL(3,R)×SL(3,R) Chern–Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.

Study on variable coefficients singular differential equation via constant coefficients differential equation

In this paper, we consider the following third-order singular differential equation with variable coefficients: \t\t\tx‴+a(t)x=f(t,x)+e(t).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x'''+a(t)x=f(t,x)+e(t). $$\\end{document} By using the Green’s function of the linear differential equation with constant coefficients and some fixed point theorems, i.e., Leray–Schauder alternative principle and Schauder’s fixed point theorem, we prove the existence of positive periodic solutions of the above equation.

Third Boundary-Value Problem for a Third-Order Differential Equation with Multiple Characteristics

We prove the unique solvability of the third boundary-value problem for a third-order differential equation with multiple characteristics containing the second time derivative in a rectangular domain.

Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w ( x , t ) = ( x + t ) λ x α e − x with x ≥ 0 , t > 0 , α > 0 , α + λ + 1 > 0 . The Deift–Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling s = 4 n t , n → ∞ , t → 0 such that s is positive and finite, at the hard edge, the limiting kernel can be described by the φ -function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P V ) transcendent via a simple transformation. Moreover, this P V transcendent is equivalent to a general Painlevé III transcendent. For large s , the P V kernel reduces to the Bessel kernel J α . For small s , the P V kernel reduces to another Bessel kernel J α + λ . At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.

A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

In this paper, we propose a geometric Hamilton–Jacobi (HJ) theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ theory is that if a Hamiltonian vector field [Formula: see text] can be projected into a configuration manifold by means of a one-form [Formula: see text], then the integral curves of the projected vector field [Formula: see text] can be transformed into integral curves of the vector field [Formula: see text] provided that [Formula: see text] is a solution of the HJ equation. This procedure allows us to reduce the dynamics to a lower-dimensional manifold in which we integrate the motion. On the other hand, the interest of a NJ structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric HJ equation on a NJ manifold and apply it to the third-order Riccati differential equation as an example.

Linearization criteria for two-dimensional systems of third-order ordinary differential equations by complex approach

Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs.

Oscillatory and asymptotic properties of third-order quasilinear delay differential equations

In this paper, we consider a class of quasilinear third-order differential equations with a delayed argument. We establish new sufficient conditions for all solutions of such equations to be oscillatory or almost oscillatory. Those criteria improve, simplify and complement a number of existing results. The strength of the criteria obtained is tested on Euler type equations.

Gaps between zeros of solutions for a certain class of third-order differential equations

In this paper, we establish some lower bounds for the distance between zeros of nontrivial solutions or/and their derivatives and extend some results about distribution of zeros for a certain class of third-order differential equations of the formThe main results of this paper will be proved by employing some Hardy’s and Opial’s type inequalities.

PERIODICITY AND SOLUTIONS OF SOME RATIONAL DIFFERENCE EQUATIONS SYSTEMS

In this paper we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case. Moreover, we study the periodicity of solutions for such systems. Finally, some numerical examples are presented.

Existence of positive periodic solutions for third-order nonlinear delay differential equations with variable coefficients

In this paper, the following third-order nonlinear delay differential equation with periodic coefficients x'''(t) + p(t)x''(t) + q(t)x'(t) + r(t)x(t) = f(t,x(t), x(t - t (t))) + d/dt g (t, x(t - t (t))), is considered. By employing Green's function and Krasnoselskii's fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order nonlinear delay differential equation.

Boundedness and Asymptotic Behaviour of the Solutions for a Third- Order Fuzzy Difference Equation

Our aim in this paper is to investigate the dynamics of a third-order fuzzy difference equation. By using new iteration method for the more general nonlinear difference equations and inequality skills as well as a comparison theorem for the fuzzy difference equation, some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for the nonlinear fuzzy system are obtained. Moreover, some numerical solutions of the equation describing the system are given to verify our theoretical results.