Fourth-order Differential Equation Research Articles

Symmetries and Conservation Laws for a Class of Fourth-Order Reaction–Diffusion–Advection Equations

We have studied a class of (1+1)-dimensional equations that models phenomena with heterogeneous diffusion, advection, and reaction. We have analyzed these fourth-order partial differential equations within the framework of group methods. In this class, the diffusion coefficient is constant, while the coefficients of advection and the reaction term are assumed to depend on the unknown density u(t,x). We have identified the Lie symmetries extending the Principal Algebra along with all the conservation laws corresponding to the different forms of the coefficients, and have derived several brief applications.

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

Simultaneous Approximation via Laplacians on the Unit Ball

We study the orthogonal structure on the unit ball textbf{B}^d of mathbb {R}^d with respect to the Sobolev inner products f,gΔ=λL(f,g)+∫BdΔ[(1-‖x‖2)f(x)]Δ[(1-‖x‖2)g(x)]dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\langle f,g\\right\\rangle _{\\Delta } =\\lambda \\, \\mathscr {L}(f,g) + \\int _{\ extbf{B}^d}{\\Delta [(1-\\Vert x\\Vert ^2) f(x)] \\, \\Delta [(1-\\Vert x\\Vert ^2) g(x)]\\,\ extrm{d}x}, \\end{aligned}$$\\end{document}where mathscr {L}(f,g) = int _{textbf{S}^{d-1}}f(xi ),g(xi ),textrm{d}sigma (xi ) or mathscr {L}(f,g) = f(0) g(0), lambda >0, sigma denotes the surface measure on the unit sphere textbf{S}^{d-1}, and Delta is the usual Laplacian operator. Our main contribution consists in the study of orthogonal polynomials associated with langle cdot , cdot rangle _{Delta }, giving their explicit expression in terms of the classical orthogonal polynomials on the unit ball, and proving that they satisfy a fourth-order partial differential equation, extending the well-known property for ball polynomials, since they satisfy a second-order PDE. We also study the approximation properties of the Fourier sums with respect to these orthogonal polynomials and, in particular, we estimate the error of simultaneous approximation of a function, its partial derivatives, and its Laplacian in the L^2(textbf{B}^d) space.

Optimal boundary control of a model thin-film fiber coating model

This paper considers the control of fluid on a solid vertical fiber, where the fiber radius is larger than the film thickness. The fluid dynamics is governed by a fourth-order partial differential equation (PDE) that models this flow regime. Fiber coating is affected by the Rayleigh–Plateau instability that leads to breakup into moving droplets. In this work, we show that control of the film profile can be achieved by dynamically altering the input flux to the fluid system that appears as a boundary condition of the PDE. We use the optimal control methodology to compute the control function. This method entails solving a minimization of a given cost function over a time horizon. We formally derive the optimal control conditions, and numerically verify that subject to the domain length constraint, the thin film equation can be controlled to generate a desired film profile with a single point of actuation. Specifically, we show that the system can be driven to both constant film profiles and traveling waves of certain speeds.

What makes a shear-free spherical perfect fluid be inhomogeneous with tidal effects?

This is an important and natural question as the spacetime shear, inhomogeneity and tidal effects are all intertwined via the Einstein field equations. Though many solutions with these properties exist in the literature, in this paper we identify, via a geometrical analysis, the important physical reason behind these solutions. We show that such scenarios are possible for limited classes of equations of state that are solutions to a highly nonlinear and fourth order differential equation. To show this, we use a covariant semitetrad spacetime decomposition and present a novel geometrical classification of shear-free locally rotationally symmetric perfect fluid self-gravitating systems, in terms of the covariantly defined fluid acceleration and the fluid expansion. Noteworthily, we deduce the governing differential equation that gives the possible limited equations of state of matter.

TRAVELING WAVES IN HEMODYNAMIC EQUATIONS WITH THE KOITER SHELL

We consider a one-dimensional model of hemodynamics for a tube defined by a Koiter shell, and investigate traveling wave solutions. The system of partial differential equations is reduced to a fourth-order ordinary differential equation for such solutions. We find a single equilibrium point and determine the condition for the changing type of the equilibria for the corresponding system of first-order differential equations. The conditions for changing the type of the equilibrium point are formulated. Then we introduce the classification of blood flow regimes relative to the type of the equilibrium point. Numerical experiments are carried out to confirm and analyze the obtained results, various regimes of blood flow are considered.

A computational study of the calendering processes using Oldroyd 8-constant fluid with slip effects

The analyses of the calendering processes using Oldroyd 8-constant fluid model with the non-linear slip condition effects are discussed in this paper. To model the flow equations for the study of calendering, we utilized the conservation of mass, momentum equations, and Oldroyd 8-constant model. The final equations are reduced into fourth order differential equations by utilizing lubrication approximation theory (LAT). MATLAB’s built-in function bvp4c is employed to calculate the stream function and the fluid velocity. Furthermore, pressure gradient, pressure, and mechanical quantities of calendering processes are obtained by fourth-order Runge-Kutta method. The effects of the slip and material parameters of Oldroyd 8-constant fluid on the physical quantities related to calendering are illustrated via graphical figures. The presence of slip is resistive to length of contract and pressure distribution. On the other hand the pressure showed the increasing trend with increase the value of material parameter ([Formula: see text]). The engineering quantity such as force function show decreasing with increase the values of the slip parameter K. As evident from the study, the material parameters play a crucial role in controlling the pressure, thereby influencing the final sheet thickness and separating force. With an increase in the material parameter [Formula: see text], both the sheet thickness and roll-separating force show a corresponding increase.

A new recovery based [formula omitted] element method for fourth-order equations

In this paper, we study a new Hessian recovery based method for fourth-order differential equations by combining finite element methods (FEMs) and finite difference methods (FDMs). The new C0 finite element method is established under the framework of Petrov–Galerkin methods. Meanwhile, some unsuccessful higher-order methods based on the Hessian recovery operator are also discussed. Finally, we extend our method to solve Cahn-Hilliard equations. Numerical results validate that our proposed C0 method works well for Cahn-Hilliard, with good convergence property and easy implementation.

Investigation of the Oscillatory Properties of Solutions of Differential Equations Using Kneser-Type Criteria

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study’s methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using a comparison technique to connect the oscillation of the studied equation with some second-order equations, the second aspect takes advantage of the significant progress made in the study of the oscillation of second-order equations. Numerous applications of functional differential equations of the neutral type served as the inspiration for the study of a subclass of these equations.

Optimizing the Monotonic Properties of Fourth-Order Neutral Differential Equations and Their Applications

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields.

Segmentation of Brain MRI Images using Multi-Kernel FCM EHO Method.

In image processing, image segmentation is a more challenging task due to different shapes, locations, image intensities, etc. Brain tumors are one of the most common diseases in the world. So, the detection and segmentation of brain tumors are important in the medical field. The primary goal of this work is to use the proposed methodology to segment brain MRI images into tumor and non-tumor segments or pixels. In this work, we first selected the MRI medical images from the BraTS2020 database and transferred them to the contrast enhancement phase. Then, we applied thresholding for contrast enhancement to enhance the visibility of structures like blood arteries, tumors, or abnormalities. After the contrast enhancement process, the images were transformed into the image denoising phase. In this phase, a fourth-order partial differential equation was used for image denoising. After the image denoising process, these images were passed on to the segmentation phase. In this segmentation phase, we used an elephant herding algorithm for centroid optimization and then applied the multi-kernel fuzzy c-means clustering for image segmentation. Peak signal-to-noise ratio, mean square error, sensitivity, specificity, and accuracy were used to assess the performance of the proposed methods. According to the findings, the proposed strategy produced better outcomes than the conventional methods. Our proposed methodology was reported to be a more effective technique than existing techniques.

A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach

A Note on the Spectral Analysis of Some Fourth-Order Differential Equations with a Semigroup Approach

Consensus tracking and vibration control for multiple single-link flexible manipulators with input signal quantization

This study examines consensus tracking and vibration control of a multiagent system consisting of multiple single-link flexible manipulators with input signal quantization. The dynamics of each flexible manipulator are described by fourth-order partial differential equations (PDEs). A logarithmic quantizer with changeable parameters is adopted to quantize control inputs, and adaptive laws are designed to compensate for the effect caused by quantization. A distributed adaptive boundary control protocol is proposed based on the PDE model to ensure all manipulators track the desired angular position and eliminate vibrations without knowledge of quantizer parameters. The proposed control protocol is proven to ensure the uniformly bounded stability of the closed-loop system. Simulations are carried out under three cases to illustrate the effectiveness of the designed strategy.

Extending Density Phase-Field Simulations to Dynamic Regimes

Density-based phase-field (DPF) methods have emerged as a technique for simulating grain boundary thermodynamics and kinetics. Compared to the classical phase-field, DPF gives a more physical description of the grain boundary structure and chemistry, bridging CALPHAD databases and atomistic simulations, with broad applications to grain boundary and segregation engineering. Notwithstanding their notable progress, further advancements are still warranted in DPF methods. Chief among these are the requirements to resolve its performance constraints associated with solving fourth-order partial differential equations (PDEs) and to enable the DPF methods for simulating moving grain boundaries. Presented in this work is a means by which the aforementioned problems are addressed by expressing the density field of a DPF simulation in terms of a traditional order parameter field. A generic DPF free energy functional is derived and used to carry out a series of equilibrium and dynamic simulations of grain boundaries in order to generate trends such as grain boundary width vs. gradient energy coefficient, grain boundary velocity vs. applied driving force, and spherical grain radius vs. time. These trends are compared with analytical solutions and the behavior of physical grain boundaries in order to ascertain the validity of the coupled DPF model. All tested quantities were found to agree with established theories of grain boundary behavior. In addition, the resulting simulations allow for DPF simulations to be carried out by existing phase-field solvers.

Alcubierre warp drive in Bohmian Quantum Gravity

In the realm of theoretical physics, the Alcubierre warp drive metric has long been a topic of fascination. However, the feasibility of this seemingly miraculous method of faster-than-light travel has been limited by the requirement of exotic matter, which has proven elusive to obtain. This research has revealed a potential breakthrough in the quest for exotic matter, through the coupling of the Alcubierre warp drive metric to a quantum mechanical scalar matter field. By mapping the requirement of exotic matter into a conformal wave equation, it becomes possible to express the problem as a fourth-order partial differential equation in terms of the quantum mechanical density. This research reveals that the key to solving the exotic matter problem lies in finding a proper quantum mechanical density that satisfies the proposed partial differential equation.

Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial.

Fourth-Order Emden–Fowler Neutral Differential Equations: Investigating Some Qualitative Properties of Solutions

In this article, we investigate some of the qualitative properties of a class of fourth-order neutral differential equations. We start by obtaining new inequalities and relations between the solution and its corresponding function, as well as with its derivatives. The new relations allow us to improve the monotonic and asymptotic properties of the positive solutions of the studied equation. Then, using an improved approach, we establish new criteria that test the oscillation of all solutions. We also rely on the principle of symmetry between positive and negative solutions to obtain the new criteria. The paper provides illustrative examples that highlight the significance of our findings.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.

Approximate Solution to Fractional Order Models Using a New Fractional Analytical Scheme

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions.