Fourth-order Differential Equation Research Articles

IVOMFuse: An image fusion method based on infrared-to-visible object mapping

Due to the significant difference in the structure of infrared and visible images, most of the existing algorithms use globally consistent strategy, which may weaken the contrast of the target region. To solve this problem, we propose an infrared and visible image fusion method based on infrared-to-visible object mapping (IVOMFuse) First of all, we use the probability induced intuitionistic fuzzy C-means algorithm (PIIFCM) to process the infrared image, and extract the target region of the infrared image according to the image structure features. However, because of the poor contrast of the target of the visible image, it can not be segmented directly. For this reason, we map the infrared target region to the visible image, and the target region with the same coordinates of the two images can be obtained very quickly. Secondly, due to the significant differences in image structure, we set a hybrid fusion strategies for different regions. We use the fusion strategy based on the Expectation-Maximization (EM) algorithm to iterate and optimize the probability model for the target region. The fourth-order partial differential equation (FPDE) is used for the background region to decompose the source image into high-frequency texture and low-frequency approximation images respectively. Then, the principal component analysis (PCA) and the average fusion strategy are used to obtain the fused image. Finally, we evaluated our algorithm with fourteen other state-of-the-art image fusion methods on three commonly used datasets, namely TNO, CVC14, and RoadScene. Among the ten evaluation metrics, our algorithm achieved the best six times.

An Improved Relationship between the Solution and Its Corresponding Function in Fourth-Order Neutral Differential Equations and Its Applications

This work aims to derive new inequalities that improve the asymptotic and oscillatory properties of solutions to fourth-order neutral differential equations. The relationships between the solution and its corresponding function play an important role in the oscillation theory of neutral differential equations. Therefore, we improve these relationships based on the modified monotonic properties of positive solutions. Additionally, we set new conditions that confirm the absence of positive solutions and thus confirm the oscillation of all solutions of the considered equation. We finally explain the importance of the new inequalities by applying our results to some special cases of the studied equation, as well as comparing them with previous results in the literature.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Математическое моделирование воздействия слабых низкочастотных сейсмических волн на ледниковый массив

Introduction. The Kolka glacial massif repeatedly broke off from the freezing surface and thereby formed powerful glacial mudflows that caused great material damage to the Republic of the Republic of Alania with numerous human casualties. In the work of Muzaev I. D. and Sozanov V. G., the first step was taken in mathematical modeling of seismic vibrations of the glacial massif and the statement was proved that weak earthquakes of 3-3.5 MSK points can provoke bending and shear stresses several times higher than the destructive stresses of ice on the underlying surface of the glacial massif. The considered mathematical model proposed in the article has been amended, significantly clarifying the scientific results. The purpose of the work is to formulate and solve an initial boundary value problem that resembles the problems of mathematical physics and simulates the fluctuations of a glacial massif. The oscillation of the array is caused by the freezing of a weak traveling harmonic seismic wave with an intensity of 3-3.5 MSK points falling on it and on its underlying surface. Materials and methods. Due to the fact that the longitudinal dimensions of glacial massifs are an order of magnitude larger than their transverse dimensions, the article assumes that the mechanics of such massifs are most adequately modeled if their geometric configurations are considered as girder bodies with their physical and mechanical characteristics. The losses of mechanical energy due to internal resistance in the array material are taken into account in accordance with the hypothesis of S. A. Sorokin. The elastic rebound force from a seismically oscillating base is taken into account in accordance with Winkler’s hypothesis. Due to the fact that the coefficient of the differential equation of vibrations of the beam array is complex, the equation splits into a system of two fourth-order differential equations with real coefficients, where the desired functions are the real and imaginary parts of the originally sought complex function. The initial boundary value problem is solved analytically. Results. On the example of the Kolka glacier (North Caucasus Highlands), which collapsed in 2002 and claimed 126 human lives, computer experiments were performed to calculate induced stresses. Discussion. Comparative computational experiments have proved the claims that the seismic stresses caused by the Winkler elastic rebuff of the seismically oscillating surface of the base: a) are significantly greater than the stresses from the bending component of the vibrations of the array; b) can significantly exceed all the values of the destructive stresses of the ice. Conclusion. It is proved that the superiority of the Winkler stress over the destructive stress of ice can also occur in weak earthquakes with an intensity of 3-3.5 MSK points. Such superiority can play the role of a trigger for the breakaway of the glacial massif from the freezing surface and thereby the formation of a powerful glacial mudflow, as happened on the Kolka glacier. Suggestions for practical application and directions for future research. The paper presents a method for directly determining the cause of the breakaway of the glacial massif from the freezing surface. In the future, it is desirable to check the effect of a sharp induced maximum stress for steels intended for mechanical engineering purposes. It is desirable to study in detail the time dependence of the effect of inducing maximum stresses in arrays using low-intensity waves.

The existence and uniqueness theorem for linear flexoelectricity and application to the Galerkin approximation

Flexoelectricity represents the electric polarization response to strain gradient and is governed by a boundary value problem (BVP) of fourth-order partial differential equations (PDEs). Due to the complicities of the BVP, we are frequently prevented from searching for analytical formulas of solutions and have to make a detour to numerical approximations. The well-posedness of the BVP has not been rigorously confirmed yet, although it serves as the cornerstone of numerical methods for flexoelectricity. We address this issue by establishing a theorem indicating a unique solution to the electric Gibbs variational principle (EGVP) that is a weak form of the BVP. The theorem leads us to bear out the solvability and error-estimation formula for some Galerkin methods, e.g., the finite element method (FEM) and isogeometric analysis (IGA). The reliability of the numerical theory is further verified by numerical experiments of benchmark problem, where the convergence rate given by FEM results agrees well with that predicted by error-estimation formulas. This study confirms the well-posedness of the flexoelectric theory and further establishes computational flexoelectricity on rigorous mathematical foundations instead of numerical tests.

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.

RETRACTED ARTICLE: Biological interactions between micro swimmers and cross fluid with inclined MHD effects in a complex wavy canal

The significance of studying biological interactions of micro swimmers in a complex wavy canal with MHD effects lies in its potential to provide insight into the behavior and dynamics of microorganisms in natural environments that contain complex fluid flow and magnetic fields. Current investigation explores the biological interactions between micro swimmers and Cross fluid with inclined MHD effects in a complex wavy canal. There are several factors of medium for micro swimmers like nature, magnetic field, liquid rheology and position. These interactions are hereby witnessed by utilizing the Cross fluid along with Taylor swimming sheet model under the influence of porous and inclined magnetic effect. Furthermore, two-dimensional complex wavy cervical canal is being utilized at inclined angle. By help of lubrication assumption, reduction of momentum equation is made and hence fourth-order differential equation associated with boundary conditions is obtained. Bvp4c command in Matlab is utilized for this boundary value problem. Obtained bvp4c solution is verified with finite difference method and found smooth agreement. Magnetic force enhances the swimming speed and reduction is seen for the power dissipation and effective role is seen for swimming motion with cross fluid rheology. The channel walls (peristaltic nature) and porous medium can be utilized as alternative factors to control the speed of the propeller.

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.

Oscillatory Properties of Fourth-Order Advanced Differential Equations

This paper presents a study on the oscillatory behavior of solutions to fourth-order advanced differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first and second-order delay differential equations. The results of this work contribute to a deeper understanding of fourth-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area.

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.

A corrected finite-difference scheme for the flexure equation with abrupt changes in coefficient

Abstract. The fourth-order differential equation describing elastic flexure of the lithosphere is one of the cornerstones of geodynamics that is key to understanding topography, gravity, glacial isostatic rebound, foreland basin evolution, and a host of other phenomena. Despite being fully formulated in the 1940s, a number of significant issues concerning the basic equation have remained overlooked to this day. We first explain the different fundamental forms the equation can take and their difference in meaning and solution procedures. We then show how numerical solutions to flexure problems as they are currently formulated are in general potentially unreliable in an unpredictable manner for cases in which the coefficient of rigidity varies in space due to variations of the elastic thickness parameter. This is due to fundamental issues related to the numerical discretisation scheme employed. We demonstrate an alternative discretisation that is stable and accurate across the broadest conceivable range of conditions and variations of elastic thickness, and we show how such a scheme can simulate conditions up to and including a completely broken lithosphere more usually modelled as an end-loaded, single, continuous plate. Importantly, our scheme will allow breaks in plate interiors, allowing, for instance, the creation of separate blocks of lithosphere which can also share the support of loads. The scheme we use has been known for many years but remains rarely applied or discussed. We show that it is generally the most suitable finite-difference discretisation of fourth-order, elliptic equations of the kind describing many phenomena in elasticity, including the problem of bending of elastic beams. We compare the earlier discretisation scheme to the new one in one-dimensional form and also give the two-dimensional discretisation based on the new scheme. We also describe a general issue concerning the numerical stability of any second-order finite-difference discretisation of a fourth-order differential equation like that describing flexure wherein contrasting magnitudes of coefficients of different summed terms lead to round-off problems, which in turn destroy matrix positivity. We explain the use of 128 bit floating-point storage for variables to mitigate this issue.

New Conditions for Testing the Asymptotic and Oscillatory Behavior of Solutions of Neutral Differential Equations of the Fourth Order

In this work, in the noncanonical case, we find new properties for a class of positive solutions of fourth-order differential equations. These properties allow us to obtain iterative criteria that exclude positive decreasing solutions, and we then establish sufficient conditions to guarantee that all solutions to the examined equation oscillate. The importance of applying the results to a special case of the investigated equation is demonstrated.

Boundary Value Problem for an Inhomogeneous Fourth-Order Equation with Lower-Order Terms

We consider the first boundary value problem in a rectangular domain for an inhomogeneous fourth-order differential equation with lower-order terms. The uniqueness of a solution of the stated problem is proved. The solution is obtained explicitly using the Green’s function constructed.

Existence and uniqueness of solutions to the nonlinear boundary value problem for fourth-order differential equations with all derivatives

This article addresses the existence and uniqueness of solution for fully fourth-order differential equations modeling beams on elastic foundations with nonlinear boundary conditions. The proof will rely on Perov’s fixed point theorem in complete generalized metric spaces to overcome the problems due to the presence of all lower-order derivatives in the nonlinearity. Finally, some illustrating examples of the theory are presented.

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.

Gravitational Baryogenesis: Problems and Possible Resolution

The coupling of baryonic current to the derivative of the curvature scalar, R, inherent to gravitational baryogenesis (GBG), leads to a fourth-order differential equation of motion for R instead of the algebraic one of general relativity (GR). The fourth-order differential equation is generically unstable. We consider a possible mechanism of stabilization of GBG by the modification of gravity, introducing an R2 term into the canonical action of GR. It is shown that this mechanism allows for the stabilization of GBG with bosonic and fermionic baryon currents. We establish the region of the model parameters leading to the stabilization of R. Still, the standard cosmology would be noticeably modified.

Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids

Flexoelectricity is an electromechanical coupling between the electric field and the mechanical strain gradient, as well as between the mechanical strains and the electric field gradient, observed in all dielectric materials, including those with centrosymmetry. Flexoelectricity demands C1-continuity for straightforward numerical implementation as the governing equations in the gradient theory are fourth-order partial differential equations. In this work, an alternative collocation-based mixed finite element method for direct flexoelectricity is used, for which a newly developed quadratic element with a high capability of capturing gradients is introduced. In the collocation method, mechanical strains and electric field through independently assumed polynomials are collocated with the mechanical strains and electric field derived from the mechanical displacements and electric potential at collocation points inside a finite element. The mechanical strain gradient and electric field are obtained by taking the directional derivative of the independent mechanical strain and electric field gradients. However, an earlier proposed linear element is unable to capture all mechanical strain gradient components and, thus, simulate flexoelectricity correctly. This problem is solved in the present work by using quadratic shape functions for the mechanical displacements and electric potential with fewer degrees of freedom than the traditional mixed finite element method. A Fortran user-element code is developed by the authors: first, for the linear and, after that, for the quadratic element. After verifying the linear element with numerical results from the literature, both linear and quadratic elements’ behaviors are tested for different problems. It is shown that the proposed second-order collocation-based mixed FEM can capture the flexoelectric behavior better compared to the existing linear formulations.

Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE

This paper contemplates a numerical investigation of the convection-diffusion type's fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coefficient of the first-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic B-splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems.

The Exact WKB analysis for asymmetric scalar preheating

Using the exact WKB analysis of the higher-order differential equations, we analyze the asymmetry in dynamical particle production of a complex scalar field. The solution requires the Stokes phenomena of the fourth-order differential equation. We found that the interference of different types of the Stokes phenomena causes matter-antimatter asymmetry. We also showed a specific example where asymmetry is forbidden in the exact calculation, but a false asymmetry appears in the perturbative expansion.

Probabilistic analysis of a cantilever beam subjected to random loads via probability density functions

This paper addresses the probabilistic analysis of the deflection of a cantilever beam by means of a randomization of the classical governing fourth-order differential equation with null boundary conditions. The probabilistic study is based on the calculation of the first probability density function of the solution, which is a stochastic process, as well as the density function of further quantities of interest associated with this engineering problem such as the maximum slope and deflection at the free end of the cantilever beam, that are treated as random variables. In addition, the probability density function of the bending moment and the shear force will also be computed. The study takes extensive advantage of the so called Random Variable Transformation method, also known as Probability Transformation Method, that allows us to fully unify the probabilistic analysis in three relevant cases commonly studied in the deterministic setting. All the theoretical findings are illustrated via detailed numerical examples corresponding to each one of the three scenarios.