Third-order Nonlinear Differential Equation Research Articles

Treatment for third-order nonlinear differential equations based on the Adomian decomposition method

The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

On Oscillatory Behavior of Solutions of Third-Order Nonlinear Differential Equations

On Oscillatory Behavior of Solutions of Third-Order Nonlinear Differential Equations

Simple Chaotic Flow with Circle and Square Equilibrium

Simple systems of third-order autonomous nonlinear differential equations can exhibit chaotic behavior. In this paper, we present a new class of chaotic flow with a square-shaped equilibrium. This unique property has apparently not yet been described. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are interesting and important in engineering applications. The mathematical model is accompanied by an electrical circuit implementation, demonstrating structural stability of the strange attractor. The circuit is simulated with PSpice, constructed, and analyzed (measured).

Oscillatory behavior of third-order nonlinear neutral delay differential equations

We provide an oscillation criterion for a class of third-order nonlinear neutral delay differential equations by using the double generalized Riccati substitutions. Our theorem complements and improves previous results. Two illustrative examples are included.

Stability and ultimate boundedness of solutions of some third order differential equations with delay

This paper is devoted to study the boundedness, ultimate boundedness, and the asymptotic stability of solutions for a certain class of third-order nonlinear differential equations using Lyapunov’s second method. Our results improve and form a complement to some earlier results in the literature.

Asymptotic Representations for the Solutions of Third-Order Nonlinear Differential Equations

We study the asymptotic behavior of solutions for a certain class of third-order nonlinear nonautonomous differential equations.

Oscillation of third-order nonlinear damped delay differential equations

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout.

On the Asymptotic Properties of Nonlinear Third-Order Neutral Delay Differential Equations with Distributed Deviating Arguments

This paper is concerned with the asymptotic properties of solutions to a third-order nonlinear neutral delay differential equation with distributed deviating arguments. Several new theorems are obtained which ensure that every solution to this equation either is oscillatory or tends to zero. Two illustrative examples are included.

Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations

Convergence behaviors of solutions arising from certain system of third-order nonlinear differential equations are studied. Such convergence of solutions corresponding to extreme stability of solutions when relates a pair of solutions of the system considered. Using suitable Lyapunov functionals, we prove that the solutions of the nonlinear differential equation are convergent. Result obtained generalizes and improves some known results in the literature. Example is included to illustrate the result.

Theory of Weightless Sagging Elasto-Flexible Cables

IN THIS PAPER, A THEORY OF SINGLE SAGGING PLANER WEIGHTLESS ELASTO-FLEXIBLE CABLES IS PROPOSED. IN REFERENCE TO THE INSTANTANEOUS NATURAL STATE, HYPOELASTIC RATE-TYPE CONSTITUTIVE EQUATIONS AND THIRD-ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF PLANER MOTION FOR MASSLESS CABLES WITH NODAL MASSES ARE DERIVED. NEW CONCEPTS LIKE CONFIGURATIONAL COMPLEMENTARY POTENTIAL AND CONTRAGRADIENT PRINCIPLES ARE PROPOSED. TYPICAL DYNAMIC RESPONSE OF THESE ELASTIC CABLES WITH DIFFERENT SUSTAINED LOADS, AXIAL RIGIDITY AND SAG/SPAN RATIOS IS DETERMINED. MODAL, SUBHARMONIC AND INTERNAL RESONANCES, JUMP AND BEAT PHENOMENA, ETC., ARE PREDICTED. A NEW MODE OF DAMPING OF €˜FREE€™ VIBRATION OF THE SYSTEM PERTURBED FROM ITS EQUILIBRIUM STATE IS IDENTIFIED. SIGNIFICANCE OF THE PROPOSED THEORY OF WEIGHTLESS SAGGING ELASTO-FLEXIBLE CABLES IS DISCUSSED.

Oscillation of third-order nonlinear neutral differential equations

We establish some new criteria for the oscillation and asymptotic behaviour of solutions of the third-order nonlinear neutral differential equation ( 1 p ( t ) ( 1 r ( t ) [ x ( t ) + a ( t ) x ( γ ( t ) ) ] ′ ) ′ ) ′ + q ( t ) f ( x ( δ ( t ) ) ) = 0 . Applications illustrating the results are given.

Oscillatory Solutions of Third-Order Nonlinear Differential Equations

Oscillatory Solutions of Third-Order Nonlinear Differential Equations

On The Existence of Multiple Solutions of a Class of Third-Order Nonlinear Two-Point Boundary Value Problems

A general approach is presented for proving existence of multiple solutions of the third-order nonlinear differential equation $$Au^{\prime\prime\prime}(x) + u^{\prime\prime}(x)u^\prime(x) + u^\prime(x)f(u(x))=0,\quad x \in [0,1] ,$$ subject to given proper boundary conditions. The proof is constructive in nature, and could be used for numerical generation of the solution or closed-form analytical solution by introducing some special functions. The only restriction is about f(u), where it is supposed to be differentiable function with continuous derivative. It is proved the problem may admit no solution, may admit unique solution or may admit multiple solutions.

Interconnections between various analytic approaches applicable to third-order nonlinear differential equations.

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle-Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

Application of the Generalized Decomposition Method for Solving the Nonlinear Problem of Jeffery–Hamel Flow

In this paper the nonlinear problem of the two-dimensional flow of an incompressible viscous fluid between nonparallel plane walls is investigated analytically. The third-order nonlinear differential equation governing the dynamic field of the considered flow has been treated by a new analytical method called the generalized decomposition method (GDM). Indeed, this method introduces a new strategy of decomposition that can use all necessary information concerning the terms of series solutions and the nonlinear term Nu. This paper investigates, on the one hand, the velocity distribution in convergent and divergent channels for various Reynolds numbers, Re and various channel half-angles, ?, and, on the other hand, the compares our results with the results obtained by the fourth-order Runge Kutta method and other applied analytical methods, thus showing the higher accuracy of the adopted generalized decomposition method.

Boundary layer flow past an inclined stationary/moving flat plate with convective boundary condition

In this study, the mathematical modeling for boundary layer flow and heat transfer past an inclined stationary/moving flat plate with a convective boundary condition is considered. Using a similarity transformation, the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations and are solved numerically using the shooting method. The obtained numerical solutions are compared with the available results in the literature and are found to be in excellent agreement. The features of the flow and heat transfer characteristics for various values of the angle of inclination, Prandtl number, local Grashof number and the Biot number are analyzed and discussed. It is found that the temperature of the stationary flat plate is higher than the temperature of the moving flat plate.

Logistic function as solution of many nonlinear differential equations

The logistic function is shown to be solution of the Riccati equation, some second-order nonlinear ordinary differential equations and many third-order nonlinear ordinary differential equations. The list of the differential equations having solution in the form of the logistic function is presented. The simple method of finding exact solutions of nonlinear partial differential equations (PDEs) is introduced. The essence of the method is based on comparison of nonlinear differential equations obtained from PDEs with standard differential equations having solution in the form of the logistic function. The wide application of the logistic function for finding exact solutions of nonlinear differential equations is demonstrated.

Stability and boundedness of solutions of a certain n-dimensional nonlinear delay differential system of third-order

AbstractIn this paper, by defining Lyapunov functionals, we investigate proper sufficient conditions for the uniform stability of the zero solution, and also for the uniform boundedness and uniform ultimate boundedness of all solutions of a certain third-order nonlinear vector delay differential equation of the type

Stability, Boundedness, and Existence of Periodic Solutions to Certain Third-Order Delay Differential Equations with Multiple Deviating Arguments

The behaviour of solutions for certain third-order nonlinear differential equations with multiple deviating arguments is considered. By employing Lyapunov’s second method, a complete Lyapunov functional is constructed and used to establish sufficient conditions that guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results not only are new but also include many outstanding results in the literature. Finally, the correctness and effectiveness of the results are justified with examples.

Multiplicity of Periodic Solutions for Third-Order Nonlinear Differential Equations

We study the existence of periodic solutions for third-order nonlinear differential equations. The method of proof relies on Schauder’s fixed point theorem applied in a novel way, where the original equation is transformed into second-order integrodifferential equation through a linear integral operator. Finally, examples are presented to illustrate applications of the main results.