Fourth-order Partial Differential Equation Research Articles

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.

Framework of fourth-order partial differential equation–based remote sensing image restoration

Framework of fourth-order partial differential equation–based remote sensing image restoration

Displacement-potential analysis of ply-stresses in a partially guided non-uniform short-column of hybrid laminated composites

Reliable and accurate analysis of local stresses in structural problems of fiber-reinforced laminated composites involving nonuniform geometry is of utmost practical importance, especially when they are subjected to local loads and supports. In this research, a new study of ply stresses in a partially guided, nonuniform, short composite column of hybrid laminates, subjected to an eccentrically distributed loading is conducted using an efficient computational scheme. The analytical modeling of the present mixed-boundary-value problem of laminated structure is realized in terms of a single scalar function of space variables (called as displacement-potential function), which is defined by the displacement components of plane elasticity, and is governed by a single fourth-order partial-differential equation of equilibrium. In this modeling approach, the original three-dimensional symmetric laminated column is reduced to a plane-stress scalar-field problem of an equivalent imaginary lamina in an attempt to obtain the corresponding strain field (global laminate strain), which is then used to determine the stress fields of individual plies through the respective transformed reduced stiffness matrices. A finite-difference based computational scheme is developed to solve the representative nonuniform equivalent lamina in terms of the displacement-potential function, which is however well capable of dealing with different ply materials as well as fiber orientations, together with the associated mixed and changeable physical conditions along the boundaries of the column. A 28-ply balanced hybrid laminate composed of boron-epoxy and glass-epoxy plies with various fiber orientation is considered as the structural material for the present column. The stress fields obtained for individual plies as well as the overall laminate are analyzed in the perspective of a number of practical issues of structural design, which include material hybridization, eccentricity of loading and attachment of partial guides to the column. The design stresses of the nonuniform laminated column are identified to be affected significantly by the material and orientation of fibers as well as the laminate hybridization. An attempt is also made to verify the soundness and accuracy of the present computational scheme through the comparison of the present potential-function solutions with those obtained by the conventional computational method.

FusionNGFPE: An Image Fusion Approach Driven by Non-global Fuzzy Pre-enhancement Framework

The majority of prevailing image fusion methods employ a global strategy, often resulting in a reduction of contrast. This study addresses this issue by proposing a novel image fusion approach called FusionNGFPE, specifically designed for the structural characteristics of infrared (IR) imagery. The approach introduces a contrast equalization algorithm based on the Fourth-order Partial Differential Equation (FPDE) to enhance background regions effectively. Considering the inherent differences between IR and visible (VIS) images, we developed a hybrid fusion strategy that combines the Expectation Maximization (EM) algorithm and Principal Component Analysis (PCA). Comparative analysis with state-of-the-art fusion methods shows that our proposed algorithm achieves superior performance in both qualitative and quantitative evaluations. To further demonstrate the practical significance of FusionNGFPE, we integrated this fusion framework into the RGBT target tracking task using the VOT-RGBT and OTCBVS datasets. Extensive comparative experiments confirm that the FusionNGFPE framework integrates seamlessly with the tracking task, significantly improving tracking accuracy across diverse scenarios.

Implicit-Explicit Schemes for Compressible Cahn–Hilliard–Navier–Stokes Equations

The isentropic compressible Cahn–Hilliard–Navier–Stokes equations are a system of fourth-order partial differential equations that model the evolution of some binary fluids under convection. The purpose of this paper is the design of efficient numerical schemes to approximate the solution of initial-boundary value problems with these equations. The efficiency stems from the implicit treatment of the high-order terms in the equations. Our proposal is a second-order linearly implicit-explicit time stepping scheme applied in a method of lines approach, in which the convective terms are treated explicitly and only linear systems have to be solved. Some experiments are performed to assess the validity and efficiency of this proposal.

A new spline method on graded mesh for fourth-order time-dependent PDEs: application to Kuramoto-Sivashinsky and extended Fisher-Kolmogorov equations

In this research, we introduce a two-tier non-polynomial spline approach with graded mesh discretization for addressing fourth-order time-dependent partial differential equations, which find applications in various physical scenarios like the nonlinear Kuramoto-Sivashinsky equation and extended Fisher-Kolmogorov equation. Our method involves considering three spatial points at each time step for scheme development, achieving spatial accuracy of three and temporal accuracy of two. Notably, our approach offers the advantage of directly tackling singular biharmonic problems without the need for discretizing boundary conditions or incorporating fictitious points. It demonstrates unconditional stability when tested on fourth-order problems and outperforms previous methods, as evidenced by comparative analyses. Additionally, we present 2D and 3D plots illustrating numerical solutions for different benchmark scenarios, including the depiction of the chaotic nature of the Kuramoto-Sivashinsky equation under Gaussian initial conditions at various time levels.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

Flexural Rigidity Influence on Dynamic Response of Orthoropic Rectangular Plate Resting on Constant Elastic Foundation

This research article considers the flexural rigidity influence along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions. The orthotropic rectangular plate model is a coupled fourth order partial differential equation having variables and singular coefficients. The solutions to this model are arrived at by reconstructing the fourth order partial differential. This partial differential equation model is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [11]. This set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to depict the influence of flexural rigidities along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions for both cases of moving distributed mass and moving distributed force.

Amplitude Variations of Moving Distributed Masses of Orthotropic Rectangular Plate with Elastically Supported Ends under Moving Loads

The deflection of thin orthotropic rectangular plate under moving loads is a classic problem in solid mechanics. However, the equations are challenging to solve due to their non linearity and complexity. At the same time, this equation is a coupled fourth order partial differential equation having variables and singular coefficients. In this research article, the partial differential equation is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [19]. This transformed set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble and Laplce transformation. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to solve the variations in amplitudes for some varying orthotropic plate parameters with elastically supported ends under moving loads for both cases of moving distributed force and moving distributed mass.

A Modified of Fourth-Order Partial Differential Equations Model Based on Isophote Direction to Noise Image Removal

Image denoising is one of the initial stages of image processing. Many models based on the diffusion method have been used to smooth the image. One of the problems, we face in the model based on the diffusion method is its possible loss of edges. The diffusion force is known to be more effective in areas of high frequency. So This paper suggests combining the direction of isophote and fourth-order partial differential equations to reduce the problem of loss edges and preserve important details of the image. The direction of the isophote can regulate the direction of diffusion. Thus, we have a proposed model that can remove the noise in the area while preserving the important edges and details of the image. We have proven the efficiency and superiority of the proposed model by applying it to a set of images and solving it numerically using the finite difference method (FDM).

Adaptive methods with [formula omitted] splines for multi-patch surfaces and shells

We introduce an adaptive isogeometric method for multi-patch surfaces and Kirchhoff–Love shell structures with hierarchical splines characterized by C1 continuity across patches. We extend the construction of smooth hierarchical splines from the multi-patch planar setting to analysis suitable G1 surfaces. The adaptive scheme to solve fourth order partial differential equations is presented in a general framework before showing its application for the numerical solution of the bilaplacian and the Kirchhoff–Love model problems. A selection of numerical examples illustrates the performance of hierarchical adaptivity on different multi-patch surface configurations.

Bending analysis of thin plates with variable stiffness resting on elastic foundation via a two-network strategy physics-informed neural network method

A physics-informed neural network method based on a two-network strategy is introduced to address the bending problem of thin plates with variable stiffness resting on an elastic foundation. This problem is governed by a fourth-order partial differential equation (PDE), and the use of a one-network strategy to solve the PDE directly may lead to convergence issues due to singular points. Following the principles of Kirchhoff plate theory, the governing PDE is equivalently transformed into four second-order PDEs. A two-network strategy is employed for solution. We present numerical examples under various load conditions, plate geometries, foundation types, thicknesses, and material properties. The obtained results are validated against finite element method (FEM) solutions and literature. Discussions on alternative network strategies were conducted, and this study reveals that the presented two-network strategy have proven to be effective and robust. It also facilitates easier imposition of boundary conditions.

A note on the telegraph type differential equation with time involution

In the present paper, the initial value problem for the telegraph type involutory in t second order linear partial differential equation with damping term is investigated. The equivalent initial value problem for the fourth order partial differential equations to the initial value problem for this second order linear partial differential equations with involution and damping term is obtained. Applying the operator tools, the stability estimates for the solution and its first and second order derivatives of this problem are established.

Mode-III fracture of four cracks emanating from an elliptical hole in one-dimensional orthorhombic quasicrystals with piezoelectric effect

In this study, a new generalized conformal mapping is introduced using the generalized complex variable method, focusing on the investigation of the anti-plane problem of an elliptical hole with four cracks in one-dimensional (1D) orthogonal piezoelectric quasicrystals. Under the consideration of electrically impermeable and electrically permeable boundary conditions, the governing equations of the anti-plane problem in 1D orthogonal piezoelectric quasicrystals are successfully simplified into an eighth-order partial differential equation and a fourth-order partial differential equation by introducing a potential function. Based on this, analytical solutions for the stress field and electric field are obtained, and the exact solution for the stress intensity factor (SIF) is derived. In addition, formulas for the electric displacement intensity factor (EDIF) and energy release rate are provided. Subsequently, the influence of geometric parameters and external loads on the field intensity factors and energy release rate is analyzed through numerical examples, providing a theoretical basis for engineering mechanics analysis.

Modeling and solution of eigenvalue problems of laminated cylindrical shells consisting of nanocomposite plies in thermal environments

This work is dedicated to the modeling and solution of eigenvalue problems within shear deformation theory (SDT) of laminated cylindrical shells containing nanocomposite plies subjected to axial compressive load in thermal environments. In this study, the shear deformation theory for homogeneous laminated shells is extended to laminated shells consisting of functionally graded (FG) nanocomposite layers. The nanocomposite plies of laminated cylindrical shells (LCSs) are arranged in a piecewise FG distribution along the thickness direction. Temperature-dependent material properties of FG-nanocomposite plies are estimated through a micromechanical model, and CNT efficiency parameters are calibrated based on polymer material properties obtained from molecular dynamics simulations. After mathematical modeling, second-order time-dependent and fourth-order coordinate-dependent partial differential equations are derived within SDT, and a closed-form solution for the dimensionless frequency parameter and critical axial load is obtained for first time. After the accuracy of the applied methodology is confirmed by numerical comparisons, the unique influences of ply models, the number and sequence of plies and the temperature on the critical axial load and vibration frequency parameter within SDT and Kirchhoff–Love theory (KLT) are presented with numerical examples.

Free and forced vibrations of beams under distributed load

Introduction. Many years of experience in design and operation of structures prove that the reliability of structures cannot be ensured by strength calculations alone. In addition to strength and stiffness of structures, their vibrations are often required to be considered. These types of calculations appear highly comprehensive, involving a large number of different factors to be taken into account. At present, as structures are becoming more and more complex, great attention is paid to seismic design, which is especially relevant for buildings subjected to high seismic loads. Engineering structures experience increased loads associated with the above-mentioned and other reasons. Free and forced vibrations of various elastic structures obtain a significant relevance among researchers as manifested by numerous publications on this issue.Aim. To present a complete mathematical problem for the two most common methods of end restraint, to determine natural frequency spectrum and eigenforms of beam vibrations.Materials and methods. The present study involves a beam of variable cross-section made of homogeneous material, subjected to transverse distributed load, which makes bending vibrations. Free and forced vibrations are described by differential equations. The homogeneous equation is solved first, then – the nonhomogeneous equation. The study involves application of D’Alembert’s principle, variable separation method, and test checks.Results. The authors obtained a fourth-order partial differential equation with constant coefficients and determined a natural frequency spectrum and eigenforms of vibrations. A high accuracy of the obtained results enables the characteristics of free and forced vibrations of beams to be determined in a shorter way with fewer calculations. Notably, the amplitude and eigenform of forced vibrations of beams appear dependent on the proximity of the disturbing frequency to the eigenvalues and the phase change of components in vector disturbance process.Conclusions. The authors advanced hypotheses about a dependence between the damping and linear viscous friction coefficients as well as about the constancy of damping coefficient for all eigenvalues.

A New Solution to the Grain Boundary Grooving Problem in Polycrystalline Thin Films When Evaporation and Diffusion Meet in Power Electronic Devices.

This paper constituted an extension of two previous studies concerning the mathematical development of the grain boundary grooving in polycrystalline thin films in the cases of evaporation/condensation and diffusion taken separately. The thermal grooving processes are deeply controlled by the various mass transfer mechanisms of evaporation-condensation, surface diffusion, lattice diffusion, and grain boundary diffusion. This study proposed a new original analytical solution to the mathematical problem governing the grain groove profile in the case of simultaneous effects of evaporation-condensation and diffusion in polycrystalline thin films by resolving the corresponding fourth-order partial differential equation ∂y∂t=C∂2y∂x2-B∂4y∂x4 obtained from the approximation ∂y∂x2≪1. The comparison of the new solution to that of diffusion alone proved an important effect of the coupling of evaporation and diffusion on the geometric characteristics of the groove profile. A second analytical solution based on the series development was also proposed. It was proved that changes in the boundary conditions of the grain grooving profile largely affected the different geometric characteristics of the groove profile.

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.

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

In this study, we consider the inverse problem of determining the time-dependent beam stiffness coefficient q(t) and transverse bending vibrations u(x,t) of a homogeneous beam using the finite difference method. The problem is governed by a fourth-order partial differential equation, which is discretized in space and time using numerical techniques to obtain an accurate and stable solution. An additional condition in the form of an integral is also included to determine q(t). The results obtained for different discretization grid points are compared, and a simple test example is presented to demonstrate the accuracy and agreement of the numerical solutions with analytical solutions. The paper provides a comprehensive overview of the methodology used to solve the inverse problem, and presents a detailed analysis of the results obtained.

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.