Higher Order Nonlinear Differential Equations Research Articles

Forced oscillation of higher-order nonlinear neutral difference equations with positive and negative coefficients

AbstractIn this paper, we study the forced oscillation of the higher-order nonlinear difference equation of the formΔm[x(n)−p(n)x(n−τ)]+q1(n)Φα(n−σ1)+q2(n)Φβ(n−σ2)=f(n),wherem≥1,τ,σ1andσ2are integers,0<α<1<βare constants,Φ∗(u)=|u|∗−1u,p(n),q1(n),q2(n)andf(n)are real sequences withp(n)>0. By taking all possible values ofτ,σ1andσ2into consideration, we establish some new oscillation criteria for the above equation in two cases: (i)q1=q1(n)≤0,q2=q2(n)>0; (ii)q1≥0,q2<0.MSC:39A10.

Existence of Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Delay Difference Equations

This paper studies the existence of nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation \begin{align*} \Delta \big(a_{kn} \cdots & \Delta (a_{2n} \triangle (a_{1n} \Delta (x_n + b_n x_{n-d}) ) ) \big) \\ &{} + f(n,x_{n-r_{1n}}, x_{n-r_{2n}}, \ldots , x_{n-r_{sn}} ) = 0, \ \ n \ge n_0, \end{align*} where $n_0 \ge 0$, $n \ge 0$, $d > 0$, $k > 0$, $s > 0$ are integers, $\{ a_{in} \} _{n \ge n_0}$ ($i = 1, 2, \ldots , k)$) and $\{ b_n \} _{n \ge n_0}$ are real sequences, $f \colon \{ n : n \ge n_0 \} \times {\mathbb R}^n \to {\mathbb R}$ is a mapping and $\bigcup _{j=1}^s \{ r_{jn} \} _{n \ge n_0} \subseteq {\mathbb Z}$. By applying Krasnoselskii's Fixed Point Theorem, some sufficient conditions for the existence of nonoscillatory solutions of this equation are established and indicated through five theorems according to the range of value of the sequence $b_n$.

Invariant intervals and global behavior of a higher order difference equation

In this paper, we study the boundedness, the invariant intervals,the periodic character and the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation yn+1=p+qyn−k1+yn−k+ryn−k−1,n=0,1,…, where the parameters p,q,r∈(0,∞),k∈{0,1,2,…}, and the initial conditions y−k−1,y−k,…y0∈[0,∞). We show that the unique positive equilibrium of the above equation is globally asymptotically stable under certain conditions.

Study on derivation and optimization algorithm about thin plate bending large deformation higher-order nonlinear partial differential equations

For a thin plate bending large deformation problem, variational principle is applied, and higher-order nonlinear partial differential equations about thin plate bending large deformation is established. Based on difference method and dynamic design variable optimization method, making unknown deflection of discrete coordinate points as design variables, differential equations sets of the discrete coordinates points as building objective function, a dynamic design variable optimization algorithm for computing thin plate bending deflection is proposed. Universal computing program is designed. Practical example about rectangular thin plate with fixed boundary under uniform load is analyzed. Comparing the program computing result with finite element solution. Effectiveness and feasibility of the method are verified. This method can be used to solve engineering problem.

Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of thenth order delay differential equations(E)(r(t)[x(n−1)(t)]γ)′+q(t)xγ(τ(t))=0. The results obtained utilize also the comparison theorems.

APPLICATION OF ADOMIAN METHOD TO NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Adomian Decomposition Method (ADM) is of a great important to mathematicians nowadays. In this paper, ADM and Kutta-4-Method were applied to nonlinear systems of ordinary differential equations. From a comparison study, it was found that ADM offers a simple and more accurate approximate solution. Also, a higher-order nonlinear ordinary differential equation was transformed to the corresponding system of equations. Both equation and corresponding system were solved. A comparison study using ADM shows that, this transformation simplifies calculations.

Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [r(t)(x(t)+P(t)x(t-τ))(n-1)]′+∑i=1mQi(t)fi(x(t-σi))=0, t≥t0, where m≥1,n≥2 are integers, τ>0, σi≥0, r,P,Qi∈C([t0,∞),R), fi∈C(R,R) (i=1,2,…,m), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Qi(t) (i=1,2,…,m) which means that we allow oscillatory Qi(t) (i=1,2,…,m). In particular, our results improve essentially and extend some known results in the recent references.

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.

Positive Solutions for a Higher-Order m-Point Boundary Value Problem

We study the existence of positive solutions for a higher-order nonlinear differential system subject to some m-point boundary conditions. As applications of the main results, we present two existence theorems for the positive solutions of a higher-order nonlinear differential equation with boundary conditions of the same form as those for the studied system.

On a periodic problem for higher-order differential equations with a deviating argument

For higher-order nonautonomous linear and nonlinear differential equations with deviating arguments, new sufficient conditions of existence and uniqueness of a periodic solution are found.

A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. A high order nonlinear partial differential equation (PDE) needs to be solved, and the gradient descent method usually takes high computational cost. In this paper, we propose a fast and efficient numerical algorithm to solve minimization problems related to Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition

Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition

Dynamical properties of a class of higher-order nonlinear difference equations

In this note, we consider the dynamical properties of the following higher-order nonlinear difference equationxn=∏i=1vxn-iβi+1+∏i=1vxn-iβi-1∏i=1vxn-iβi+1-∏i=1vxn-iβi-1,n=0,1,⋯with x−v,x−v+1,…,x−1∈(0,∞), β1∈(0,1], βi∈(0,+∞)(i=2,3,…,v). Our result extends and includes corresponding results in the recent literatures.

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.

A new technique of using homotopy analysis method for solving high-order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high-order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth-order nonlinear differential equation to a system of n first-order equations. Second- and third- order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

This paper investigates the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying Krasnoselskii's fixed-point theorem in cones. The nonexistence of positive solutions is also studied.

Global stability of some classes of higher-order nonlinear difference equations

We give elegant and short proofs of some recent results on global stability of some classes of higher-order nonlinear difference equations.

Global attractivity of a higher-order nonlinear difference equation

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation x n + 1 = 1 - x n A + x n - k , n = 0 , 1 , … , where A ∈ ( - ∞ , - 1 ) , k is a positive integer and initial conditions x - k , … , x 0 ∈ ( - ∞ , 0 ] . It is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient.

Oscillations of a higher order nonlinear differential equations with impulses

AbstractImpulsive differential equations are important in simulation of processes with jump conditions. Although there are quite a few studies on the oscillation properties of low order equations, there are not too many studies of higher order equations. In this paper, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.