Fourth-order Nonlinear Differential Equation Research Articles

An analytical investigation on nonlinear vibrations and stability of Timoshenko pipes conveying two-phase flow

The primary objective of this study is to conduct a comprehensive investigation on the nonlinear free vibration and stability analysis of the multilayer graphene reinforced pipe conveying liquid–gas two–phase flow based on the Timoshenko beam model. The Halpin–Tsai micromechanics model is employed which provides a means to evaluate and analyze the mechanical behavior and performance of the multilayer FG reinforced pipe. According to Von–Kármán strain relations and applying the Hamilton's principle, the governing equations of motion are derived. The Galerkin technique is operated to convert the partial differential equation (PDE) to the ordinary differential equation (ODE). The homotopy analysis method is employed to solve the fourth order nonlinear differential equation. Closed–form expressions for the nth nonlinear frequency and time history as well as the critical flow velocity based on the Timoshenko theory are achieved. The results demonstrate that neglecting the effects of rotary inertia and shear deformation may cause some errors by the increment of the initial amplitude. Furthermore, the critical flow velocity, depends not only on the mechanical and geometrical properties of the pipe but also on the initial mode shape excitation. The results indicate that by the increment of flow velocity, liquid density and slenderness ratio, the nonlinear frequency decreases, on the other hand, by the increment of the initial amplitude, the nonlinear frequency increases.

Asymptotic and Oscillatory Analysis of Fourth-Order Nonlinear Differential Equations with p-Laplacian-like Operators and Neutral Delay Arguments

This paper delves into the asymptotic and oscillatory behavior of all classes of solutions of fourth-order nonlinear neutral delay differential equations in the noncanonical form with damping terms. This research aims to improve the relationships between the solutions of these equations and their corresponding functions and derivatives. By refining these relationships, we unveil new insights into the asymptotic properties governing these solutions. These insights lead to the establishment of improved conditions that ensure the nonexistence of any positive solutions to the studied equation, thus obtaining improved oscillation criteria. In light of the broader context, our findings extend and build upon the existing literature in the field of neutral differential equations. To emphasize the importance of the results and their applicability, this paper concludes with some examples.

Oscillation of Fourth-Order Nonlinear Semi-Canonical Neutral Difference Equations via Canonical Transformations

The authors present a new technique for transforming fourth-order semi-canonical nonlinear neutral difference equations into canonical form. This greatly simplifies the examination of the oscillation of solutions. Some new oscillation criteria are established by comparison with first-order delay difference equations. Examples are provided to illustrate the significance and novelty of the main results. The results are new even for the case of nonneutral difference equations.

EXISTENCE RESULTS AND NUMERICAL SOLUTION OF FULLY FOURTH ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a fully fourth-order nonlinear functional differential equation which contains all lower derivatives of proportional delay arguments.By the reduction of the problem to operator equation for the right-hand side nonlinear function, we establish the existence and uniqueness of the solution and construct iterative methods on both continuous and discrete levels for solving it. We obtain the total error estimate for the discrete iterative solution. Many examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.

On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order

A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results.

A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions

This article encounters the use of two wavelet methods, namely the collocation method based on Haar wavelets (CMHW) and the higher-order collocation method based on Haar wavelets (HCMHW), to solve linear and nonlinear fourth-order differential equations with different forms of given data such as two-point boundary conditions and two-point integral boundary conditions. Managing these types of boundary conditions can be challenging in numerical methods. However, in this study, these types of equations are handled in a simple manner using the Haar wavelet expressions, as provided in the given information. In the case of nonlinear problems, the quasi-linearization technique is introduced to linearize the equation. Nonlinear fourth-order differential equations are transformed into a simple linear system of algebraic equations using the quasi-linearization technique and Haar wavelets. These equations are then solved very easily to find the solution of the differential equations. The convergence rate and stability of both the methods are studied in details. The convergence rate of the proposed HCMHW is faster than the CMHW (2+2s>2,s=1,2…). Some of the examples are given to indicate the better performance and accuracy of the proposed HCMHW.

Oscillation criteria of fourth-order nonlinear semi-noncanonical neutral differential equations via a canonical transform

In this work first we transform the semi-noncanonical fourth order neutral delay differential equations into canonical type. This simplifies the investigations of finding the relationships between the solution and its companion function which plays an important role in the oscillation theory of neutral differential equations. Moreover, we improve these relationships based on the monotonic properties of positive solutions. We present new conditions for the oscillation of all solutions of the corresponding equation which improve the oscillation results already reported in the literature. Examples are provided to illustrate the importance of our main results. For moreinformation see https://ejde.math.txstate.edu/Volumes/2023/70/abstr.html

Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersion.

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

BOUNDEDNESS SOLUTIONS OF A CERTAIN FOURTH-ORDER NONLINEAR DIFFERENTIAL EQUATION

We consider the boundedness solutions of a certain fourth-order nonlinear and nonhomogeneous differential equation. We prove that the solutions of this equation are bounded byCauchy formula

A meshless generalized finite difference scheme for the stream function formulation of the Naiver-Stokes equations

In this study, a meshless numerical scheme, which is the combination of the generalized finite difference method, the fictitious-nodes technique, and the two-step Newton-Raphson method, was proposed to solve the stream function formulation of the Navier-Stokes equations. To reduce the numbers of unknowns and partial differential equations for the two-dimensional flow field of a viscous incompressible fluid, the stream function formulation, which is a fourth-order nonlinear partial differential equation with only one unknown variable, can be derived by introducing the definition of vorticity and stream function. By applying the generalized finite difference method, the derivatives in the stream function formulation can be simply expressed as a linear combination of functional data and weighting values at several nearest nodes. To deal with the two boundary conditions for the fourth-order partial differential equations, the fictitious-nodes technique was utilized to set up the fictitious nodes outside the computational domain, and cooperate with the generalized finite difference method to form a system of nonlinear algebraic equations. The aforementioned numerical procedure can yield a square Jacobian matrix, so the Newton-Raphson method can be adopted to efficiently acquire the numerical solutions. Several numerical examples were tested to verify the practicability of the proposed meshless numerical scheme.

Axisymmetric necking of a circular electrodes-coated dielectric membrane

We investigate the stability of a circular electrodes-coated dielectric membrane under the combined action of an electric field and all-round in-plane tension. It is known that such a membrane is susceptible to the limiting point instability (also known as pull-in instability) which is widely believed to be a precursor to electric breakdown. However, there is experimental evidence showing that the limiting point instability may not necessarily be responsible for rapid thinning and electric breakdown. We explore the possibility that the latter is due to a new instability mechanism, namely localised axisymmetric necking. The bifurcation condition for axisymmetric necking is first derived and used to show that this instability may occur before the Treloar-Kearsley instability or the limiting point instability for a class of free energy functions. A weakly nonlinear analysis is then conducted and it is shown that the near-critical behaviour is described by a fourth order nonlinear ordinary differential equation with variable coefficients. This amplitude equation is solved using the finite difference method and it is demonstrated that a localised solution does indeed bifurcate from the homogeneous solution. Based on this analysis and what is already known for the purely mechanical case, we may deduce that the necking evolution follows the same three stages of initiation, growth and propagation as other similar localisation problems. The insight provided by the current study is expected to be relevant in assessing the integrity of dielectric elastomer actuators.

Similarity reduction and multiple novel travelling and solitary wave solutions for the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

Oscillation of Nonlinear Neutral Delay Difference Equations of Fourth Order

This paper focuses on the study of the oscillatory behavior of fourth-order nonlinear neutral delay difference equations. The authors use mathematical techniques, such as the Riccati substitution and comparison technique, to explore the regularity and existence properties of the solutions to these equations. The authors present a new form of the equation: Δ(a(m)(Δ3z(m))p1−1)+p(m)wp2−1(σ(m))=0, where z(m)=w(m)+q(m)w(m−τ) with the following conditions: ∑s=m0∞1a(1p1−1(s))=∞. The equation represents a system where the state of the system at any given time depends on its current time and past values. The authors demonstrate new insights into the oscillatory behavior of these equations and the conditions required for the solutions to be well-behaved. They also provide a numerical example to support their findings.

Monotonically Iterative Method for the Cantilever Beam Equations

In this paper, we consider the existence of extremal solutions for the nonlinear fourth-order differential equation. By use of a new comparison result, some sufficient conditions for the existence of extremal solutions are established by combining the monotone iterative technique and the methods of lower and upper solutions. Finally, an example is given to illustrate the validity of our main results.

A Study of a Mathematical Model with a Movable Singular Point in a Fourth-Order Nonlinear Differential Equation

This article introduces a mathematical model that utilizes a nonlinear differential equation to study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this equation is challenging due to the presence of moving singular points. The article addresses two main problems: first, it establishes the existence and uniqueness of the solution of the equation and, second, it provides precise criteria for determining the existence of a moving singular point. Additionally, the article presents estimates of the error in the analytical approximate solution and validates the results through a numerical experiment.

Asymptotic behaviour of solutions of one class of the fourth-order nonlinear differential equations

The asymptotic behaviour as t↑ω of one of the possible types of Pω(Y0, λ0)-solutions of a binomial non-autonomous fourth-order ordinary differential equation with exponential nonlinearity and a continuous and non-zero in some left neighbourhood ω(ω≤+∞) coefficient p(t) is investigated. First, using a priori asymptotic properties of the considered Pω(Y0, λ0)-solutions, necessary conditions for their existence are established, as well as asymptotic representations of these solutions and their derivatives up to the third order. The question of the actual existence of solutions with the obtained asymptotic representations is solved by reducing it to the question of the existence of solutions that vanish at a specific point of a system of quasilinear differential equations. This system is obtained as a result of some transformations of the original equation, taking into account the kind of established asymptotic representations. In addition, the question of the number of solutions with found asymptotic representations is also resolved.

Transforming Navier‐stokes equation to a fourth order nonlinear differential equation for the study of electrolyte flow in a circular microchannel

AbstractIn this paper, analytical solutions are derived for the study of steady‐state electrolyte flow in a circular microchannel under the simultaneous effects of an external electric field and uniform wall injection. The Navier‐Stokes equations containing an axial linear electric body force are transformed to a fourth‐order nonlinear differential equation and solved analytically using a series solution method. The most important results are presented and discussed in terms of radial Reynolds number and a fundamental non‐dimensional electric field parameter. The Electrical Double Layer theory is employed to model the effects of external electric field. Analytical relations are given for velocity profiles, exit flow rate, and total pressure drop in the microchannel. The results show that a high‐intensity electric field can completely reverse the velocity profile or make it more flat in the presence of wall injection. This electric effect can be used in practical applications to control the shear stress, heat, or mass transfer in microchannels or microelectromechanical systems. In addition, various graphs are presented showing simultaneous effects of electric field and wall injection of the same fluid on the flow physics.

Stability, Boundedness, and Square Integrability of Solutions of Neutral Fourth-Order Differential Equations

The purpose of this paper is to establish a new result, which guarantees the asymptotic stability and boundedness of the zero solution and the square integrability of solutions and their derivatives to neutral type nonlinear differential equations of fourth-order. We illustrate our results by an example at the end of the paper.

Existence of Periodic Solutions for a Class of Fourth-Order Difference Equation

We apply the continuation theorem of Mawhin to ensure that a fourth-order nonlinear difference equation of the form Δ 4 u k − 2 − a k u α k + b k u β k = 0 with periodic boundary conditions possesses at least one nontrivial positive solution, where Δ u k = u k + 1 − u k is the forward difference operator, α > 0 , β > 0 and α ≠ β , a k , b k are T -periodic functions and a k b k > 0 . As applications, we give some examples to illustrate the application of these theorems.

New symmetric bidirectional progressive surface-wave solutions to a generalized fourth-order nonlinear partial differential equation involving second-order time-derivative

New nonlinear fourth-order integrable equation of second-order in time has been established by Wazwaz via collaborating some recursion operators and the sense of Boussinesq equation. The proposed new equation has several real-life applications in the field of water-waves, such as gravity-capillary waves, waves in shallow water, and ocean engineering. The aim of the current study is threefold. First, we show that the propagation of Wazwaz equation is simultaneously moving symmetric bidirectional waves. Second, we extract new novel solutions such as bidirectional lumps and bidirectional kinks, singular-kinks, and periodic-kinks. Finally, by using graphical analysis, we show the physical structures of the recovery solutions and study the significance of the model’s parameters.