Third-order Nonlinear Differential Equation Research Articles

On the Global Stability Properties and Boundedness Results of Solutions of Third-Order Nonlinear Differential Equations

We studied the global stability and boundedness results of third-order nonlinear differential equations of the form . Particular cases of this equation have been studied by many authors over years. However, this particular form is a generalization of the earlier ones. A Lyapunov function was used for the proofs of the two main theorems: one with and the other with . The results in this paper generalize those of other authors who have studied particular cases of the differential equations. Finally, a concrete example is given to check our results.

On the Existence of Positive Solutions of Resonant and Nonresonant Multipoint Boundary Value Problems for Third-Order Nonlinear Differential Equations

Positive solutions for a kind of third-order multipoint boundary value problem under the non-resonant conditions and the resonant conditions are considered. In the nonresonant case, by using Leggett-Williams fixed-point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using Leggett-Williams norm-type theorem due to O’Regan and Zima, existence result of at least one positive solution is established. The results obtained are valid and new for the problem discussed. Two examples are given to illustrate the main results.

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on ℝ and is convergent for any x ∊ℝ. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

Uniformly boundedness of a class of non-linear differential equations of third order with multiple deviating arguments

This paper deals with a certain third-order non-linear differential equation with multiple deviating arguments. Some sufficient conditions are set up for all solutions and their derivatives to be uniformly bounded.

Oscillation Criterion of Third-Order Nonlinear Neutral Damped Functional Differential Equations

In this paper, A class of third-order nonlinear neutral damped functional differential equations with distributed deviating arguments are studied. By using a generalized Riccati transformation and Kamenev-type or Philos-type integral averaging technique,we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero.

Solution of Blasius Equation by Variational Iteration

TheBlasius equation is a well known third-order nonlinear ordinary differential equation, which arises in cer- tain boundary layer problems in the fluid dynamics. This paper presents a way of applying Hes variational iteration method to solve the Blasius equation. Approximate analytical solution is derived and compared to the results obtained from Ado- mian decomposition method. Comparisons show that the present method is accurate and the using of Hes method does accelerate the convergence of the power series. A robust and efficient algorithm is also programmed using Matlab based on the present approach, which can be easily employed to solve Blasius equation problems

Conservation laws for a class of quasi self-adjoint third order equations

In this work we consider a class of third-order nonlinear partial differential equation containing two un-specified coefficient functions of the dependent variable which include various integrable and nonintegrable equations. We determine the subclasses of these equations which are self-adjoint and quasi self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov we find conservation laws for some of these partial differential equations without classical Lagrangians.

Properties of third-order nonlinear differential equations

We establish new comparison theorems, to offer criteria for all nonoscillatory solutions of the third-order functional differential equation [ r ( t ) [ x ′ ( t ) ] γ ] ″ +p(t) x β ( τ ( t ) ) =0 tend to zero. We consider both delay and advanced case of studied equation. The results obtained essentially improve and complement earlier ones.MSC:34K11, 34C10.

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (uxx − u)t + u xxx + uux = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.

Solving a laminar boundary layer equation with the rational Gegenbauer functions

In this paper, a collocation method using a new weighted orthogonal system on the half-line, namely the rational Gegenbauer functions, is introduced to solve numerically the third-order nonlinear differential equation, af‴+ff″=0, where a is a constant parameter. This method solves the problems on semi-infinite domain without truncating it to a finite domain and transforming the domain of the problems to a finite domain. For a=2, the equation is the well-known Blasius equation, which is a laminar viscous flow over a semi-infinite flat plate. We solve this equation by considering 1⩽a⩽2 and compare the new results with the established results to show the efficiency and accuracy of the new method.

Multiple Positive Solutions of Two-Point Boundary Value Problems for Systems of Nonlinear Third-Order Differential Equations

In this paper, we study the two-point boundary value problems for systems of nonlinear third-order dierential equations .Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point theorems in cones.

Oscillation of third-order nonlinear delay difference equations

Third-order nonlinear difference equations of the form \Delta (c_n\Delta (d_n\Delta x_n))+p_n\Delta x_{n+1}+q_nf(x_{n-\sigma})=0, n\geq n_{0} are considered. Here, {c_n}, {d_n}, {p_n}, and {q_n} are sequences of positive real numbers for n_0 \in N, f is a continuous function such that f(u) /u\geq K > 0 for u \neq 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results.

Heat Transfer in MHD Dusty Boundary Layer Flow over an Inclined Stretching Sheet with Non-Uniform Heat Source/Sink

This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of dusty fluid over an inclined stretching sheet with non-uniform heat source/sink, where the flow is generated due to a linear stretching of the sheet. 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 by Runge-Kutta-Fehlberg fourth-fifth-order method using symbolic software Maple. Our numerical solutions are shown to agree with the available results in the literature and then employ the numerical results to bring out the effects of the fluid-particle interaction parameter, local Grashof number, angle of inclination, heat source/sink parameter, Chandrasekhar number, and the Prandtl number on the flow and heat transfer characteristics. The results have possible technological applications in liquid-based systems involving stretchable materials.

On nonoscillatory solutions tending to zero of third-order nonlinear differential equations

Abstract The aim of this paper is to present some results concerning with the asymptotic behavior of solutions of nonlinear differential equations of the third-order with quasiderivatives. In particular, we state the necessary and sufficient conditions ensuring the existence of nonoscillatory solutions tending to zero as t → ∞.

Response control for the externally excited van der Pol oscillator with non-local feedback

A non-local control force is introduced in such a way to obtain a third-order nonlinear differential equation (jerk dynamics) and to control nonlinear vibrations in an externally excited van der Pol oscillator. Two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of solutions are derived and subsequently the performance of the control strategy is investigated. Excitation amplitude–response and frequency–response curves are shown. In certain cases when the excitation amplitude is very low an approximate analytic solution corresponding to a modulated two-period quasi-periodic motion can be obtained for the uncontrolled system. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the response and to exclude the possibility of quasi-periodic motion. Numerical simulation confirms the validity of the new method.

Boundedness of Certain Forms of Jerky Dynamics

In this paper, we find regions in the bifurcation parameter space of certain forms of jerky dynamic systems where they are bounded. The method of analysis is based on a recent result about the boundedness of solutions of a certain type of third-order nonlinear differential equation with bounded delay. In particular, the boundedness of some chaotic attractors displayed by these systems is confirmed analytically.

On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.

Oscillation of third-order nonlinear functional differential equations with mixed arguments

The objective of this paper is to offer sufficient conditions for the oscillation of all solutions and other asymptotic properties of the third-order nonlinear functional differential equation � a(t) � x �� (t) � γ � � = q(t)f (x (τ (t))) + p(t)h (x (σ(t))) with mixed arguments, where both cases � ∞ a −1/γ (s) ds = ∞ and � ∞ a −1/γ (s) ds < ∞ are dealt with. We deduce properties of the studied equations via new com- parison theorems. Our results essentially improve and complement earlier ones. We also repair one interesting result of Grace et al.

Vibration control for the parametrically excited van der Pol oscillator by nonlocal feedback

A nonlocal feedback is used for the control of nonlinear vibrations in a parametrically excited van der Pol oscillator. A nonlocal control force is introduced in order to obtain a third-order nonlinear differential equation (jerk dynamics). Using the asymptotic perturbation method, two slow flow equations on the amplitude and phase of the response are obtained, and subsequently the performance of the control strategy is investigated. Parametric excitation–response and frequency–response curves are shown. Uncontrolled and controlled systems are compared, and the appropriate choices of the feedback gains for reducing the amplitude peak of the response are found. Energy considerations are used in order to study the existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a two-period modulated motion for the van der Pol oscillator. To exclude the possibility of quasi-periodic motion and to reduce the amplitude peak of the parametric resonance, appropriate choices of the feedback gains are found. Numerical simulation confirms the validity of the new method.

On the qualitative behaviors of solutions of third-order nonlinear functional differential equations

The third-order nonlinear functional differential equations of the form ( r 2 ( t ) ( r 1 ( t ) y ′ ) ′ ) ′ + p ( t ) y ′ + q ( t ) f ( y ( g ( t ) ) ) = 0 are considered. We present some new oscillatory and asymptotic behavior of solutions of this equation by modifying a method given for second-order differential equations. Our results are applicable to nonlinear functional differential equations of the above form. Several examples are also given to illustrate the importance of our results.