Higher Order Nonlinear Differential Equations Research Articles

Lyapunov-type inequalities for a class of nonlinear higher order differential equations

In this work, we establish several Lyapunov-type inequalities for a class of nonlinear higher order differential equations having a form , with anti-periodic boundary conditions, where m > n ≥ 0 are integers, ψ or fi(i = 0,1,2,…,n) satisfies Lieberman-type structural condition. The obtained inequalities are extensions and complements of the existing results in the literature.

A new technique for obtaining approximate solution of higher order nonlinear differential equation

In this study, a new technique is presented combining perturbation method and harmonic balance (HB) method and adopted to investigate a fourth order nonlinear Duffing equation with strong nonlinearity. The advantage of the proposed method is that the method can provide good results even the nonlinearity is strong. The result obtained by the proposed method is compared with the corresponding numerical results to verify the accuracy of the method. The results are also compared with the previous perturbation method results found in literature. Besides, the damped and undamped cases are studied, and the internal resonance is found for undamped case.

Legendre wavelet solution of high order nonlinear ordinary delay differential equations

The purpose of this paper is to illustrate the use of the Legendre wavelet method in the solution of high-order nonlinear ordinary differential equations with variable and proportional delays. The main advantage of using Legendre polynomials lies in the orthonormality property, which enables a decrease in the computational cost and runtime. The method is applied to five differential equations up to sixth order, and the results are compared with the exact solutions and other numerical solutions when available. The accuracy of the method is presented in terms of absolute errors. The numerical results demonstrate that the method is accurate, effectual and simple to apply.

Riemann\u2013Hilbert approach and N-soliton solution for an eighth-order nonlinear Schr\xf6dinger equation in an optical fiber

This paper aims to present an application of the Riemann–Hilbert approach to treat higher-order nonlinear differential equation that is an eighth-order nonlinear Schrödinger equation arising in an optical fiber. Starting from the spectral analysis of the Lax pair, a matrix Riemann–Hilbert problem is formulated strictly. Then, by solving the obtained Riemann–Hilbert problem under the reflectionless case, N-soliton solution is generated for the eighth-order nonlinear Schrödinger equation. Finally, the localized structures and dynamic behaviors of one- and two-soliton solutions are illustrated by some figures.

Nonlinear diffusion based image segmentation using two fast algorithms

In this paper, a new variational model is proposed for image segmentation based on active contours, nonlinear diffusion and level sets. It includes a Chan-Vese model-based data fitting term and a regularized term that uses the potential functions (PF) of nonlinear diffusion. The former term can segment the image by region partition instead of having to rely on the edge information. The latter term is capable of automatically preserving image edges as well as smoothing noisy regions. To improve computational efficiency, the implementation of the proposed model does not directly solve the high order nonlinear partial differential equations and instead exploit the efficient alternating direction method of multipliers (ADMM), which allows the use of fast Fourier transform (FFT), analytical generalized soft thresholding equation, and projection formula. In particular, we creatively propose a new fast algorithm, normal vector projection method (NVPM), based on alternating optimization method and normal vector projection. Its stability can be the same as ADMM and it has faster convergence ability. Extensive numerical experiments on grey and colour images validate the effectiveness of the proposed model and the efficiency of the algorithms.

Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions

Global Behavior of a Higher-Order Nonlinear Difference Equation with Many Arbitrary Multivariate Functions

Stability analysis of higher order nonlinear differential equations in β–normed spaces

In this paper, we interrogate different Ulam type stabilities, ie, β–Ulam–Hyers stability, generalized β–Ulam–Hyers stability, β–Ulam–Hyers–Rassias stability, and generalized β–Ulam–Hyers–Rassias stability, for nth order nonlinear differential equations with integrable impulses of fractional type. The existence and uniqueness of solutions are investigated by using the Banach contraction principle. In the end, we give an example to support our main result.

Existence of periodic solutions for a higher-order neutral difference equation

Based on a continuation theorem of Mawhin, the existence of a periodic solution for a higher-order nonlinear neutral difference equation is studied. Our conclusion is new and interesting.

Coupled soliton solutions of modeled equations in a Noguchi electrical line with crosslink capacitor

An electrical line is one of the transmission media used worldwide for the transmission of signals. This is for the fact that it is less expensive and easy to manufacture than the other transmission media. In linear domain, the electrical lines transmit sinusoidal signals whose amplitude decreases exponentially and loss a lot of energy. It is therefore necessary to valorize electrical lines by enabling them to transmit solitary waves which are more stable and do not dissipate energy. This will be possible if the media of transmission line are dispersive and nonlinear. We define the analytical expressions that the charges of capacitors must obey so that the Noguchi electrical line with crosslink capacitor accepts to propagate a coupled solitary wave of the need type. The application of those definitions and Kirchhoff laws to the networks of a modified Noguchi electrical line with crosslink capacitor have enable to obtain new set of higher-order nonlinear partial differential equations which govern the dynamics of a coupled solitary wave in the said line. The construction of coupled solitary waves solutions of those set of equations by direct and effective mathematical methods notably that of Bogning-Djeumen Tchaho-Kofane [16–22] has permitted to discover that, solitary waves of type (Kink; Kink) and type (Pulse; Pulse) are easily propagated in that line when certain conditions we have elaborated are respected. The Noguchi electrical line with crosslink capacitor that we have studied is advantageous for the fact that it permits simultaneously the propagation of a set of two solitary waves contrary to a non-coupled Noguchi electrical line which only enables the propagation of one solitary wave when the signal considered is the voltage; the more we will multiply the crosslink in the line, the more we will multiply the simultaneous propagation of solitary wave in the line.

Dispersive optical soliton solutions of the higher-order nonlinear Schrödinger dynamical equation via two different methods and its applications

In this paper, we apply two methods which are the arbitrary nonlinear parameters and the exponential rational function method to construct many new exact solutions of the higher-order nonlinear partial differential equations, namely, the higher-order nonlinear Schrodinger (HNLS) equation. The solutions obtained by the current methods are generalized periodic solutions. The shape of the solutions can be well controlled by adjusting the parameters of the system. Optical soliton solutions obtained can be used to transport information in the telecommunication domain. It also comes from this work that the behavior of this HNLS equation may be easily studied by means of the phase plane plot which is the best tool to predict some solutions.

The Uniqueness of Positive Solution for Higher-Order Nonlinear Fractional Differential Equation With Fractional Multi-Point Boundary Conditions

In this paper, we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-orderfractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions andthe error estimations are also given. The result is illustrated with an example.

Existence and Multiple Solutions for Higher Order Difference Dirichlet Boundary Value Problems

Abstract In this paper, a higher order nonlinear difference equation is considered. By using the critical point theory, we obtain the existence and multiplicity for solutions of difference Dirichlet boundary value problems and give some new results. The proof is based on the variational methods and linking theorem.

Variation of Parameters Method for Thermal Analysis of Straight Convective- Radiative Fins with Temperature Dependent Thermal Conductivity

In this study, thermal performance across straight convecting- radiating fin with temperature dependent thermal conductivity is considered. The variation of parameters (VPM) is adopted to analyze the nonlinear higher order differential equations arising due to thermal conductivity and heat transfer coefficient on temperature distribution. Pertinent parameters such as thermo geometric and radiation parameters effect on temperature profile are investigated. Result obtained illustrates that quantitative increase of thermo geometric parameter causes a significant increase in temperature distribution due to increase in ratio of convective to conduction heat transfer which influence is significant toward fin base while increasing radiation parameter leads to decrease in temperature distribution due to increasing heat transfer from fins surface to ambient environment . Comparative analysis of result obtained in study against literature proves to be in satisfactory agreement. Therefore study provides useful insight to fins operational performance in applications such as radiators, boilers, refrigeration devices, oil pipelines amongst others.

Sum operator methods for the existence and uniqueness of solution to infinite-point boundary value problems for fractional differential equations

In this paper, we study infinite-point boundary value problems for a class of higher-order nonlinear factional differential equations involving the Riemann-Liouville derivative. By using sum operator methods, the existence and uniqueness of solution to this kind of problems is obtained and iterative sequence of the positive solution is structured. Two examples are provided for our new results.

Sum operator methods for the existence and uniqueness of solution to infinite-point boundary value problems for fractional differential equations

In this paper, we study infinite-point boundary value problems for a class of higher-order nonlinear factional differential equations involving the Riemann-Liouville derivative. By using sum operator methods, the existence and uniqueness of solution to this kind of problems is obtained and iterative sequence of the positive solution is structured. Two examples are provided for our new results.

Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method

We apply the generalized projective Riccati equations method with the aid of Maple software to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the nonlinear quantum Zakharov-Kuznetsov (QZK) equation. The obtained exact solutions include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. The given nonlinear PDEs have been derived and can be reduced to nonlinear ordinary differential equations (ODEs) using a simple transformation. A comparison of our new results with the well-known results is made. Also, we drew some graphs of the exact solutions using Maple. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

A convergence criterion for the solutions of nonlinear difference equations and dynamical systems

We show that the solutions of nonlinear higher order difference equations may have convergent subsequences, even when the solution as a whole does not converge, if the defining function sequence is suitably bounded near the origin. The delay size and pattern in the higher order equation play an essential role in determining which subsequences of its solutions converge. We then show that this method can be extended to planar systems and discuss applications to discrete dynamical systems that have been used in biological population models.

A numerical scheme for solutions of a class of nonlinear differential equations

In this paper, a collocation method based on Bessel functions of the first kind is presented to compute the approximate solutions of a class of high-order nonlinear differential equations under the initial and boundary conditions. First, the matrix forms of the Bessel functions of the first kind and their derivatives are constructed. Second, by using these matrix forms, collocation points and the matrix operations, a nonlinear differential equation problem is converted to a system of nonlinear algebraic equations. The solutions of this system give the coefficients of the assumed approximate solution. To demonstrate the validity and applicability of the technique, numerical examples are included and comparisons are made with existing results. The results show the efficiency and accuracy of the present work.

Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

高阶非线性微分方程非局部边值问题的解法

本文讨论高阶非线性微分方程非局部边值问题的数值方法。通过建立满足非局部边值条件的再生核空间，获得简单易行的再生核数值解法。证明近似解及其导数的收敛性。 This paper discusses the numerical method for the higher order nonlinear differential equation with nonlocal boundary value problem. By constructing the reproducing kernel space which satis-fies the nonlocal boundary value conditions, the simple reproducing kernel numerical approximate method is established. Convergence of approximate solution and its derivatives is proved, respectively.