Fourth-order Nonlinear Differential Equation Research Articles

Exploring the Bio-Mechanisms of Rod-Shaped Bacteria Moving on Rigid Substrate Coated with Non-Newtonian Williamson Slime

Bacteria are single-celled organisms that come in various shapes and sizes, often classified based on characteristics like shape, DNA, and movement mechanisms. Among them, gliding bacteria have a unique way of moving: they slide along surfaces without external structures like flagella or cilia. Instead, they create wave-like motions along their surface to glide across slimy areas. How fast can bacteria move, and does their speed change on different surfaces? Do they glide faster on smooth surfaces or slow down on rough or sticky ones? And when they move through thicker, more complex fluids, how do they adapt, and does this impact their speed? Researchers have investigated these questions by studying how surface properties and fluid dynamics affect bacterial movement. This study also aims to analyze the locomotion of gliding bacteria over a layer of Williamson fluid with a rigid substrate using the dynamics of an undulating sheet. The methodology involves converting the fundamental partial differential equations into a fourth-order nonlinear ordinary differential equation through the Stokes flow approximation and solving it by the regular perturbation technique. The impacts of different physical parameters on gliding speed, flow rate, energy loss, and slime velocity are visually illustrated and discussed. The results show that the glider's speed decreases with a lower viscosity ratio, while higher Weissenberg numbers and occlusion ratios improve the glider's motility. Additionally, the flow rate of the Williamson fluid decreases as the viscosity increases, and power loss experienced by the glider rises with an increase in viscosity. These findings highlight how fluid properties influence bacterial movement and energy efficiency, providing insights for developing microfluidic devices and studying microbial motility.

On the Oscillation of Fourth-Order Delay Differential Equations via Riccati Transformation

This paper deals with the oscillatory behavior of solutions of a general class of fourth-order non-linear delay differential equations. New oscillation criteria are established using Riccati transformation and a Philos-type technique. The obtained results not only improve and extend some published results in the literature, but also relax some traditional conditions on the function ψ(χ(ι)). Three examples are provided to illustrate the main results.

Multiple soliton and singular wave solutions with bifurcation analysis for a strain wave equation arising in microcrystalline materials

A microcrystalline material refers to a crystallized substance or rock comprised of tiny crystals that can only be observed under a microscope. The strain wave equation is a fourth-order nonlinear partial differential equation encountered in the examination of non-dissipative strain wave propagation within microstructured solids. In this paper, the transmission of waves in microcrystalline materials is dictated by the non-dissipative case of strain wave equation’s structure, accounting for multiple dimensions within microcrystalline structures. The simplest equation method is employed to extract multi-soliton solutions, while the modified Sardar subequation method is applied to identify additional soliton solutions, including bright, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Furthermore, the dynamical system bifurcation theory approach is utilized to investigate the phase diagrams of the governing equation. Further elaboration on the physical dynamical representation of the presented solutions is provided through profile illustrations. A comparison with the existing literature is also provided, highlighting the efficacy of our work. The significance of the acquired outcomes lies in their capacity to portray a wide array of intricate and diverse phenomena observed in both mathematical and physical systems.

Existence of Solutions for Generalized Nonlinear Fourth-Order Differential Equations

In this article, we studied the existence of solutions for a more general form of nonlinear fourth-order differential equations by using a new generalization of the Arzelá–Ascoli theorem and Schauder fixed theorem under easier and general conditions. Moreover, we provided some sufficient conditions on the nonlinear function that allowed us to deduce the nonexistence results. Finally, we outlined an example to illustrate our main results.

The Improved tan (\(\frac{\Phi(\xi)}{2}\)) -Expansion Method for The Non-dissipative Double Dispersive Equation in Microstructured Solids

In microstructured solids, the non-dissipative double dispersive equation is a fourth-order non-linear partial differential equation that arises in the study of non-dissipative strain wave propagation. Seeking the exact solution of a nonlinear partial differential equations with a physical background is helpful to understand the motion law of matter and to explain the corresponding physical phenomena scientifically. This equation has been studied widely, and there have been a lot of research methods to find exact solutions to this equation. In this manuscript, we try to study the non-dissipative double dispersive equation by the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. As far as our known, no one has used this method to study the non-dissipative double dispersive equation, partly because of the complicated calculation. We obtain a lot of exact traveling wave solutions, including hyperbolic function solutions, trigonometric function solutions, exponential function solutions, and rational function solutions. Compared with the other methods, some solutions obtained in this manuscript are consistent with existing solutions, which shows the effectiveness of the improved tan (\(\frac{\Phi(\xi)}{2}\)) -expansion method. Through this method, Our manuscript obtains more general and new exact solutions. These solutions may play an important role in engineering and physics. What's more, we plot the 3D graphs of some of the solutions obtained in this manuscript, which helps us understand the physical phenomenon of the non-dissipative double dispersive equation. In the future, we will continue to explore its physical significance from the new analytical solution we have obtained.

Fluid flow between two parallel active plates

This paper investigates the fluid flow phenomenon arising from the combined action of two parallel plates, which can expand/squeeze, absorb/inject, and stretch/shrink at different rates. These physical mechanisms are incorporated into the governing unsteady Navier–Stokes equations, which are then reduced to a fourth-order nonlinear differential equation with boundary conditions reflecting the imposed wall constraints. By letting the permeable Reynolds number (controlling the nonlinear convective terms) limit to zero, we demonstrate the existence of exact solutions expressed in terms of advanced mathematical functions. Additionally, in the absence of wall expansion/contraction, elementary exponential solutions are obtained under particular relationships between the stretching/shrinking and permeability parameters. A shear-like exact solution with broader applicability across various physical parameters is also identified. For moderate values of the expansion/squeezing parameters and permeable Reynolds numbers, we propose an efficient double-expansion perturbation analysis to approximate the flow behavior. Otherwise, for general physical parameters, a comprehensive mathematical analysis is provided and numerical simulations are employed to extract insights into the complex fluid motion between the parallel plates.

Considerable traveling wave solutions of a generalized Hietarinta-type equation

Abstract This work effectively utilizes the modified extended tanh− function approach to scientifically deduce semi-analytic traveling wave solutions for the (2+1)-dimensional fourth-order non-linear generalized Hietarinta-type problem, leading to previously unidentified satisfactory solutions. The proposed model has been transformed into a fourth-order non-linear ordinary differential equation via a traveling wave transformation. Some periodic-solitary, original, and oscillating wave solutions to the model under experimentation are acquired in mixed complex trigonometric and logarithmic functions combined with hyperbolic trigonometric functions, and complex rational functions. Assorted solutions have been shown using two- and three-dimensional graphics and suitable arbitrary parameters to demonstrate their physical and dynamic results. Two-dimensional graphs have shown how changes in time formally impact the features and structures of the solution. The free parameters (unrestricted parameters) that keep going in the solutions have a big impact on the dynamic behavior of the solutions. Traveling wave, oscillating, periodic, and breather wave solutions have also been figured out with the help of the operation that gives values to the free parameters.

Complexiton and interaction solutions to a specific form of extended Calogero–Bogoyavlenskii–Schiff equation via its bilinear form

Abstract In this article, we are focusing on an extended Calogero–Bogoyavlenskii–Schiff equation which was altered originally from a new generalized fourth-order nonlinear differential equation that obtained from Calogero–Bogoyavlenskii–Schiff equation. We apply simplified Hirota method, which is an exclusive form of the direct Hirota bilinear method, to a specific form of a new generalized fourth-order nonlinear differential equation. The key point in applicability of referred method is attainability proper forms of dispersion relations and phase shifts. Through this procedure, we present different types of solutions for three different cases. We also give constrictions for each solution type in this work so that readers can distinguish differences among types of solutions. In addition, we introduce some graphical representations for obtained solutions, even for existence of complexiton and interaction solutions.

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.

Symmetry Analysis and Wave Solutions of Time Fractional Kupershmidt Equation

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem.

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-order Ordinary Differential Equation

The paper considers a boundary value problem with symmetric boundary conditions for one nonlinear fourth-order ordinary differential equation on the segment $[0,1]$, which describes the deformation of an elastic beam. Using special topological means in semi-ordered spaces with a cone, based on the fixed point principle, sufficient conditions for the existence and uniqueness of a positive solution to the problem under study are established. The proof of the existence of at least one positive solution to the boundary value problem was carried out using the index of the operator's fixed point. To prove the uniqueness of the solution, the theorem on fixed points of alpha-concave operators - was accordingly used.

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.