Fourth-order Partial Differential Equation Research Articles

A boundary value problem for the fourth-order degenerate equation of the mixed type

Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.

Denoising of poisson-corrupted microscopic biopsy images using fourth-order partial differential equation with ant colony optimization

Microscopic biopsies play a critical role in the identification of lung cancer. The readable features of lung such as tissue and cell are usually affected by Poisson noise. Moreover, the existence of noise reduces the quality of the image that limits the interpretation of the disease by medical specialists. To address this, an ant colony optimization based reformed fourth order partial differential equation i.e., ACO-RFPDE is proposed. In order to develop RFPDE, two major advancements have been incorporated. In first advancement, the restoration of edges and fine details preservation has been achieved by maximum likelihood estimation. In second advancement, instead of user based manual value, a mean absolute deviation of the image has been employed for automatic calculation of the edge threshold constraint of diffusion coefficient of FPDE. Furthermore, ACO is utilized for tuning of the controlling parameters of the proposed method where peak signal to noise ratio is used as a fitness function. The performance is analyzed comparatively with existing methods as well as qualitatively and quantitatively for the LC25000 dataset. The quantitative analysis includes mean square error, correlation, normalized absolute error, no reference, universal quality index, peak signal-to-noise ratio, and structural similarity index. The following values 0.175, 0.964, 0.111, 42.45, 0.946, 55.68 and 0.985 of these metrics have been obtained by the proposed method for the images of lung squamous carcinoma cell. The results of the proposed method demonstrate the successful elimination of Poisson noise while preserving the image features like edges, boundaries and enhancing the abnormal region.

Analysis of the chemical diffusion master equation for creation and mutual annihilation reactions

We propose an infinite dimensional generating function method for finding the analytical solution of the so-called chemical diffusion master equation (CDME) for creation and mutual annihilation chemical reactions. CDMEs model by means of an infinite system of coupled Fokker–Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Using our method we are able to link certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space. Our technique extends the one presented in Lanconelli [J. Math. Anal. Appl. 526, 127352 (2023)] and Lanconelli et al. [arXiv:2302.10700 [math.PR] (2023)] which allowed for an explicit representation for the solution of birth-death type CDMEs.

Analytical Studies of Fluid Conveying Pipes on Viscoelastic Foundation Using Differential Transforms Method

This study presents an analytical investigation of the vibration of fluid-conveying pipes on viscoelastic foundations using the differential transform method. The effects of a new time dependent viscosity parameter in the modified Winkler viscoelastic foundation is studied and analyzed. The governing equation is a fourth-order partial differential equation with pinned-pinned boundary conditions, which required a special analytical method for solution. The differential transform method was applied to obtain the solution of the governing partial differential equation for the fluid-conveying pipes on viscoelastic foundations. The time-dependent viscosity parameter in the modified Winkler viscoelastic model was implemented and simulated to determine the behavior of the viscoelastic foundation. The obtained analytical solution was validated with Runge-Kutta order four numerical method. The effects of foundation stiffness , coefficient of foundation damping and the frequency mass ratio on the governing model equation were investigated. In addition, the bending and deflection of the pipe on a viscoelastic foundation are compared with those on an elastic foundation. The analytical and the numerical solutions are in good agreement. From the study, it is observed that an increase in the foundation stiffness results in increase in the pipe inherent frequencies. Furthermore, the vibration of the pipe on a viscoelastic foundation shows better control and reduction compared with its vibration on an elastic foundation.

A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.

On the Dynamic Characteristics of Orthotropic Rectangular Plates under the Influence of Moving Distributed Masses and Resting on a Variable Elastic Pasternak Foundation

This work studies the dynamic characteristics of orthotropic rectangular plates under the influence of moving distributed masses and resting on a variable elastic Pasternak foundation. The governing equation is a fourth order partial differential equation with variable and singular coefficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al, then simplified using asymptotic method of Struble [11]. The closed form solution is analyzed, resonance conditions are obtained and the results are depicted graphically for both cases of moving distributed mass and moving distributed force

On a boundary problem for the fourth order equation with the third derivative with respect to time

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

Invariance Analysis and Closed-form Solutions for The Beam Equation in Timoshenko Model

Our research focuses on a fourth-order partial differential equation (PDE) that arises from the Timoshenko model for beams. This PDE pertains to situations where the elastic moduli remain constant and an external load, represented as F, is applied. We thoroughly analyze Lie symmetries and categorize the various types of applied forces. Initially, the principal Lie algebra is two-dimensional, but in certain noteworthy cases, it extends to three dimensions or even more. For each specific case, we derive the optimal system, which serves as a foundation for symmetry reductions, transforming the original PDE into ordinary differential equations. In certain instances, we successfully identify exact solutions using this reduction process. Additionally, we delve into the conservation laws using a direct method proposed by Anco, with a particular focus on specific classes within the equation. The findings we have presented in our study are indeed original and innovative. This study serves as compelling evidence for the robustness and efficacy of the Lie symmetry method, showcasing its ability to provide valuable insights and solutions in the realm of mathematical analysis.

Retracted: Construction of Computer Microscope Image Segmentation Model Based on Fourth-Order Partial Differential Equation Smoothing

Retracted: Construction of Computer Microscope Image Segmentation Model Based on Fourth-Order Partial Differential Equation Smoothing

УТОЧНЁННАЯ ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧЁТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

A variant of the linear theory is proposed, based on taking into account in Taylor expansions for stresses and in power series and the expansion of the radical into a binomial series for deformations of not only linear, but also quadratic terms. In this case, static equilibrium equations in stresses are written in the form of second-order partial differential equations. The law of pairing of tangential stresses is observed. The relationship between deformations and displacements is described by relations similar to Cauchy relations, but taking into account the second derivatives of displacements along spatial coordinates. The equations of continuity of deformations for this version of the theory of elasticity are written in the form of two groups of relations. The first group of relations establishes the continuity equations for the linear terms of the deformation components; The second group of relations establishes the continuity equations for the quadratic terms of the deformation components. The physical equations for this variant of the linear theory of elasticity are written in the same way as for the classical linear theory of elasticity. Equilibrium equations in stresses are represented by second-order partial differential equations. The equilibrium equations in displacements are written in the form of fourth-order partial differential equations. The equations of equilibrium, as well as the equations of continuity of deformations, along with other parameters – the physical constants of the medium – contain small parameters dx, dy, dz, the magnitude of which, as shown by numerical modeling, has little effect on the nature of the stress-strain state. To determine them, it is necessary, apparently, to use experimental data. The construction of resolving equations in both stresses and displacements for special cases of the stress-strain state of an elastic continuous medium is considered: uniaxial dressed-up state; uniaxial deformed state; plane strain; generalized plane stress state. The problem of the uniaxial deformed state of a continuous medium (solution in stresses) is numerically solved. The proposed version of the linear theory of elasticity extends the classical linear theory of elasticity and, with appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

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.

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.

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.

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.

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.

Retracted: Construction of Biologic Microscopic Image Segmentation Model Based on Smoothing of Fourth-Order Partial Differential Equation.

[This retracts the article DOI: 10.1155/2022/1908644.].