Fourth-order Differential Equation Research Articles

Exponential stability of a non-uniform Euler-Bernoulli beam with axial force

The aim of this paper is to investigate the boundary stabilization of a non-uniform Euler-Bernoulli beam with axial force generated by the equation ρ(x)utt+(σ(x)uxx)xx−(q(x)ux)x=0 defined in (0,1)×(0,∞), where the coefficients ρ(x)>0, σ(x)>0, and q(x)≥0. By adopting a new technique based on a qualitative theory of fourth-order linear differential equations, we prove that the spectrum of the system operator is confined to the open left-half complex plane and that all the associated eigenvalues are geometrically simple. Using this result and the Riesz basis approach, we obtain the exponential stability and the optimal decay rate of the system. This study extends the earlier paper by B.-Z. Guo [SIAM J. Control Optim., 40 (2002), pp. 1905–1923], where no axial force acting on the system (i.e., q≡0).

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).

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.

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

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

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

AbstractThis 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.

Numerical and Theoretical Studies on Axial Compression Performance of Modular Steel Tubular Columns Grouped with Shear-Key Connectors

Shear-keyed inter-modular connections (IMCs) are integral components of high-rise modular steel structures (MSSs), providing robust interconnectivity to support grouped tubular columns across modules, thereby introducing column discontinuities and distinctive structural behavior. This study conducted a comprehensive numerical assessment and theoretical analysis of the axial compression behavior of grouped tubular columns based on a validated finite element model (FEM), which captured the member-to-structural level behavior of steel hollow section (SHS) columns and accommodated geometric imperfections. An FEM was initially developed and validated using 28 axial compression tests documented in the literature, comprising 15 tests on cold-formed and 13 on hot-rolled steel hollow section (SHS) columns. The primary parameters explored in tests included material properties (stainless/carbon), processing methods (cold-formed/hot-rolled), cross-section sizes (D/B), cross-sectional or member slenderness ratios (D/tc, B/tc, or Lc/r), and the number of columns (1, 7, and 11). A comprehensive parametric numerical study involving 103 grouped tubular column FEMs then investigated the influence of initial imperfection, shear-key height (Lt), thickness (tt), steel tube length (D), width (B), thickness (tc), and height (Lc) alongside the effects of space between tube and key, and the gap between tubes. The results indicated that the load-shortening behavior of the grouped columns consists of linear elastic, inelastic, and recession stages. The failure modes observed primarily displayed an S-shaped pair of inward and outward local buckling on the outer sides and double S-shaped local buckling on the interior sides. The buckling arose near the shear key or at 1/4 or 1/2 of the column height. None of the considered models experienced global buckling. Increasing tt, Lt, tc, D, or B enhances strength and stiffness, while Lc or Lc/r linearly affects stiffness and ductility. The columns’ nominal axial strength was reduced because of the shear keys, which decreased compression yielding and caused localized elastic buckling. Subsequently, the theoretical analysis revealed that the design codes do not capture this behavior, and thus, their capacity estimate yields inaccurate findings. This discrepancy renders existing code prediction equations, including those from Indian (IS800), New Zealand (NZS400), European (EC3:1-1), Canadian (CSA S16), American (AISC360-16), and Chinese (GB50017) standards, as well as the model proposed by Li et al., non-conservative. To assure conservative results, the paper recommended modification of existing standards and proposed prediction equations based on a fourth-order differential equation that describes the actual behavior of modular steel columns grouped with shear keys. The proposed design approach accurately predicted the axial compression capacity of modular steel-grouped columns, proving conservative yet effective. This provides valuable data that could transform design and construction techniques for MSSs, extending to various column and IMC forms through adaptable design parameters. This enhancement in structural performance and safety significantly contributes to the advancement of modular construction practices.

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.

Properties of eigenfunctions of a boundary value problem for ordinary differential equations of fourth-order with boundary conditions depending on the spectral parameter

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space Lp,1<p<∞, after removing four functions.

Preoperative diffusion tensor imaging: Fiber-trajectory-distribution-based tractography to identify facial nerve in vestibular schwannoma.

Tractography of the facial nerve based on diffusion MRI is instrumental before surgery for the resection of vestibular schwannoma, but no excellent methods usable for the suppression of motion and image noise have been proposed. The aim of this study was to effectively suppress noise and provide accurate facial nerve reconstruction by extend a fiber trajectory distribution function based on the fourth-order streamline differential equations. Preoperative MRI from 33 patients with vestibular schwannoma who underwent surgical resection were utilized in this study. First, T1WI and T2WI were used to obtain mask images and regions of interest. Second, probabilistic tractography was employed to obtain the fibers representing the approximate facial nerve pathway, and these fibers were subsequently translated into orientation information for each voxel. Last, the voxel orientation information and the peaks of the fiber orientation distribution were combined to generate a fiber trajectory distribution function, which was used to parameterize the anatomical information. The parameters were determined by minimizing the cost between the trajectory of fibers and the estimated directions. Qualitative and visual analyses were used to compare facial nerve reconstruction with intraoperative recordings. Compared with other methods (SD_Stream, iFOD1, iFOD2, unscented Kalman filter, parallel transport tractography), the fiber-trajectory-distribution-based tractography provided the most accurate facial nerve reconstructions. The fiber-trajectory-distribution-based tractography can effectively suppress the effect of noise. It is a more valuable aid for surgeons before vestibular schwannoma resection, which may ultimately improve the postsurgical patient's outcome.

Determination of Temperature Stresses during the Construction of Massive Monolithic Foundation Slabs, Taking into Account the Subgrade Compliance

Background The problem of early cracking caused by the heat of concrete hardening is relevant for massive reinforced concrete structures, including foundation slabs. The purpose of this work is to develop the methodology for determining temperature stresses during the construction of foundation slabs, taking into account the interaction with the subgrade. Methods The Pasternak elastic foundation model with two-bed coefficients is used for the soil. The temperature of the foundation slab is considered a function of only one coordinate z (temperature changes only along the thickness of the slab). As a result, to determine the stress-strain state of the slab, a fourth-order differential equation for deflection was obtained. A technique for numerically solving the resulting equation using the finite difference method is proposed. The calculation of the stress-strain state is preceded by the calculation of the temperature field, which is performed by the finite element method in a simplified one-dimensional formulation. Results The solution to the test problem is presented for a constant modulus of elasticity of concrete over time. The results were compared with finite element calculations in a three-dimensional formulation in the ASNYS software. The calculation was also performed taking into account the dependence of the mechanical characteristics of concrete on its degree of maturity. In this case, the picture of the stress-strain state changes significantly. The proposed method was also successfully tested on experimental data. Conclusion The proposed approach can significantly save calculation time compared to the finite element analysis in a three-dimensional setting.

Dynamics and non-integrability of the double spring pendulum

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system’s dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincaré cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system’s dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits S1 symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilization of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilized a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.

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.

Flexural–torsional buckling with enforced axis of twist and related problems

The classical lateral–torsional instability of a cantilever beam with continuous elastic lateral restraint and a transverse point-load applied at the free end is discussed here through the lens of asymptotic simplifications. One of our main goals is to provide analytical approximations for the critical buckling load, as well as its dependence on various key non-dimensional groups. The first part of this study is concerned with a scenario in which a doubly symmetric beam is constrained to rotate about a fixed axis situated at an arbitrary height above the shear centroidal axis. The second part examines a beam that features a continuous elastic lateral restraint spanning its entire length. Assuming that the stiffness of the constraint, [Formula: see text] (say), is finite, the buckling equations in the second case are described by a system of two coupled fourth-order differential equations in the lateral displacement and the angle of twist. We show that as [Formula: see text], the problem discussed in the first part provides an outer solution for the aforementioned system; the relevant boundary conditions for the next-to-leading-order outer approximation are also derived using matched asymptotics. Our theoretical findings are confirmed by comparisons with direct numerical simulations of the full buckling problem.

Solution of infinite system of fourth order differential equations in $\ell_{p}$ space.

The main objective of this paper is to investigate the solution of infinite system of fourth order differential equations in $\ell_p$ space by using measures of noncompactness. The result is demonstrated with an example.

Analytical solution to a coupled system including tuned liquid damper and single degree of freedom under free vibration with modal decomposition method

This research focuses on employing a linear analytical approach to transform free surface waves and velocities into mode coordinates, with the aim of investigating the free vibration behavior of a coupled system consisting of a Single Degree of Freedom and a sloshing tank. Through a series of manipulations and simplifications of the coupled equations, a fourth-order ordinary differential equation is derived to showcase the overall response of the system, highlighting the contribution of each odd mode. Key concepts explored include system stability, mode-specific natural periods, establishment of initial boundary conditions, and formulation of the complete system response. The analytical method applied to study Tuned Liquid Dampers, a type of elevated sloshing tank, reveals that in higher modes, the lower frequency aligns with the structural natural frequency, while the higher frequency is approximately n times the structural natural frequency (where n is the odd mode number). This approach also elucidates why the system's response does not exhibit a higher-frequency component in higher modes. The study further investigates concepts such as employing an initial perturbation to excite higher frequencies and the potential for approximating the system through the first mode. Additionally, a numerical model was developed using variable separation and modal decomposition methods to complement and validate the analytical approach. Finally, further verification of the model was performed using the Preismann scheme applied to the relevant equations and the central upwind applied to nonlinear equations.

Three solutions for a discrete fourth-order boundary value problem with four parameters

This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others,the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper.Numerical computations of the example confirm our theoretical results.

Exact Solutions of System of Fourth-Order Difference Equations

Exact Solutions of System of Fourth-Order Difference Equations

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.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.