Third-order Nonlinear Differential Equation Research Articles

Series Solution for Unsteady Boundary-Layer Flows Due toImpulsively Stretching Plate

The third-order nonlinear partial differential equation modelling theunsteady boundary-layer flows caused by an impulsively stretchingflat plate is solved by using the Adomian decomposition method (ADM).The ADM yields analytic solution in the form of a rapidly convergentinfinite series with easily computable terms. The series solutionusing the ADM for the unsteady flows is presented for the first time.

Further results on global stability of third-order nonlinear differential equations

Sufficient conditions are established for the global stability of certain third-order nonlinear differential equations. Our result improves on Qian’s [C. Qian, On global stability of third-order nonlinear differential equations, Nonlinear Anal. 42 (2000) 651–661].

Continuation of Periodic Orbits of a Third Order Oscillator under Autonomous Perturbation

We consider systems of nonlinear third-order differential equations depending on a small parameter such that the unperturbed system has a three dimension manifold of periodic solutions. We impose some conditions on the system in order to reduce the third order perturbed system to the second one. By using Poincare-Andronov-Hopf theorem we prove the existence of many periodic solutions the perturbed system.

Self-similar solutions of the magnetohydrodynamic boundary layer system for a dilatable fluid

A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting dilatable fluid along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness, existence, and nonexistence results for generalized normal solutions are established.

On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.

Harmonic balance approach to limit cycles for nonlinear jerk equations

The method of harmonic balance (HB) is employed to estimate the attributes of limit cycles of some nonlinear third-order (jerk) differential equations which are parity-invariant but not time-reversal-invariant. Two examples with cubic nonlinearities show that the HB method can give good values for the frequency and both the velocity and displacement amplitudes of a period one limit cycle.

Existence and uniqueness of solutions for third-order nonlinear boundary value problems

In this paper, we present some general results of existence and uniqueness of solutions of nonlinear two-point boundary value problems for third-order nonlinear differential equations by using the Shooting method. As applications we give certain concrete sufficient conditions for the existence and uniqueness.

On the complete integrability and linearization of nonlinear ordinary differential equations. II. Third-order equations

We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle–Singer method. We describe a procedure to deduce all the integrals of motion associated with the given equation, so that the general solution follows straightforwardly from these integrals. The method is illustrated with several examples. Further, we propose a powerful method of identifying linearizing transformations. The proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation. In addition to the above, we provide an algorithm to invert the non-local linearizing transformation. Through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation.

Remarks on Barbashin–Ezeilo Problem on third-order nonlinear differential equations

In this paper, we use the frequency domain criteria to initiate a general result on the Barbashin–Ezeilo Problem on third-order nonlinear differential equation. Earlier ideas of Burkin [I.M. Burkin, Orbital stability of second-kind limiting cycles for dynamical systems with cylindrical phase space, Differential Equations 29 (1993) 1262–1264], Leonov [G.A. Leonov, A frequency criterion for the existence of limit cycles of dynamical systems with cylindrical phase spaces, Differential Equations 23 (1987) 1375–1378] on the problem are being improved upon.

A Lie Symmetry Connection between Jacobi’s Modular Differential Equation and Schwarzian Differential Equation

In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi’s modular differential equation. A six-dimensional Lie symmetry algebra is obtained and its symmetry generators are found to be given in terms of elliptic integrals. As a consequence the transformation between Jacobi’s modular differential equation and the well-known Schwarzian differential equation is derived.

A boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls

A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation, which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the result above-mentioned is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one and a generalized normal solution if the power-law index is greater than one. Also the asymptotic behavior of the normal solution at the infinity is displayed.

Solution space of axisymmetric capsules enclosed by elastic membranes

Two principal methodologies for computing equilibrium shapes of axisymmetric capsules enclosed by elastic membranes are discussed. First, a variational formulation based on a bending energy functional defined in terms of the principal curvatures is considered. Constrained minimization results in a nonlinear, second-order differential equation for the meridional curvature with respect to the slope angle or arc length along the membrane contour in a meridional plane. Numerical solutions of the relevant boundary-value problem are presented over a broad range of conditions to illustrate the rich structure of the solution space. Second, the problem is formulated from the viewpoint of shell theory in terms of stress resultants and bending moments using force and torque equilibrium balances. Assuming isotropic tensions and linear constitutive equations relating the bending moments to the principal curvatures, the problem is reduced to solving a nonlinear third-order differential equation, which is distinct from that obtained using the variational method. Numerical solutions uncover a variety of physically realizable capsule shapes, including those resembling the biconcave, disk-like shape of healthy red blood cells.

Global asymptotic properties of third-order difference equations

The third-order nonlinear difference equations Δ ( p n Δ ( r n Δ x n ) ) + q n f ( x n + p ) = 0 , p ∈ { 0 , 1 , 2 } , ( E p ) where ( P n ), ( r n ), and ( q n ) are sequences of positive real numbers for n ε ℕ, ƒ : ℝ → ℝ is a continuous function such that ƒ (u)u > 0 for u ≠ 0, are investigated. All nonoscillatory solutions of these equations are classified according to the sign of their quasidifferences to classes N i, i = 0, 1, 2, 3, and sufficient conditions ensuring N i = , i ε {1,2,3} are given. Special attention is paid to equation ( E 1) for which the generalized zeros of solutions are studied and an energy function F is introduced. The relation between the class N 0 and solutions for which F n < 0 for n ε ℕ is established.

On the existence of periodic solutions for certain differential equations

Here we are concerned with the problem of the existence of periodic solution for certain second and third-order nonlinear differential equations. Our method here is to consider the problem as an eigenvalue problem and treat it by the topological degree theory. In particular we establish the conditions of the existence of periodic solution first for a simpler system which is homotopic to the original system and then generalize the obtained results for the focal system. The method employed here is applicable also for a system of nonlinear differential equations just with simple modifications. Finally, we present some specific examples numerically to show that the results are valid and realizable.

On the existence of periodic solutions for certain differential equations

Here we are concerned with the problem of the existence of periodic solution for certain second and third-order nonlinear differential equations. Our method here is to consider the problem as an eigenvalue problem and treat it by the topological degree theory. In particular we establish the conditions of the existence of periodic solution first for a simpler system which is homotopic to the original system and then generalize the obtained results for the focal system. The method employed here is applicable also for a system of nonlinear differential equations just with simple modifications. Finally, we present some specific examples numerically to show that the results are valid and realizable.

Sundman Symmetries of Nonlinear Second-Order and Third-Order Ordinary Differential Equations

We investigate the Sundman symmetries of second-order and third-order nonlinear ordinary differential equations. These symmetries, which are in general nonlocal transformations, arise from generalised Sundman transformations of autonomous equations. We show that these transformations and symmetries can be calculated systematically and can be used to find first integrals of the equations.

Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory

By a new approach, we prove in this paper that there exists λ 0 ϵ (− 1 2 , 0) such that the following third-order nonlinear boundary value problem for f(η): f‴+ff″+λ(1−f′2)=0, 0<η<∞ , f(0)=0, f′(0)=0, f′(+∞)=1, which arises in boundary layer theory in fluid mechanics, has a solution at least for any fixed λ ϵ ( λ 0, 0).

Asymptotic behavior of a third-order nonlinear differential equation

Consider the third-order nonlinear differential equation x‴+ψ(x,x′)x″+f(x,x′)=p(t), where ψ,f,fx∈C(R×R,R) and p∈C([0,∞),R). We obtain sufficient conditions for every solution of the equation to be bounded; we also establish criteria for every solution of the equation to converge to zero.

Beyond-all-orders effects in multiple-scales asymptotics: travelling-wave solutions to the Kuramoto-Sivashinsky equation

This paper concerns the possible `shock' patterns that can exist in the solution to a singularly perturbed, third-order nonlinear ordinary differential equation arising as the travelling-wave reduction of the Kuramoto-Sivashinsky equation. In particular, the existence (or otherwise) of oscillatory shocks and multiple shocks made up of combinations of oscillatory and monotonic shocks is examined, using an optimal truncation strategy to track crucial exponentially small terms lying beyond all orders of the (divergent) algebraic expansion. The results provide further understanding of numerical solutions previously obtained by others, as well as giving a methodology which is applicable to much broader classes of differential equations exhibiting multiscale phenomena; they also afford same new insight into the multi-scales technique.

MHD flow of a third-grade fluid due to eccentric rotations of a porous disk and a fluid at infinity

The problem of magnetohydrodynamics (MHD) flow of a conducting, incompressible third-grade fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is considered. An exact analysis is carried out to model the governing non-linear partial differential equation. A numerical solution of the third-order non-linear partial differential equation has been obtained. Several graphs and tables have been drawn to show the influence of porosity ε, magnetic parameter N, material parameters α and β on the velocity distribution.