Fourth Order Linear 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).

Асимптотики спектральных данных краевых задач четвертого порядка

In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods rely on the recent results concerning regularization and asymptotic analysis for higher-order differential operators with distribution coefficients. The results of this paper have applications to the theory of inverse spectral problems as well as a separate significance.

General solutions’ laws of linear partial differential equations II

This paper uses Z transformations to obtain the general solutions of a large number of second-order, third-order and fourth-order linear partial differential equations for the first time, including one-dimensional inhomogeneous wave equation. Using general solutions, we have obtained plenty of exact solutions of the typical definite solution problems, which proves the important value of general solutions. We propose the Z4 transformation for the first time and initially use it to solve a specific case. We successfully obtain the Fourier series solution using the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famous unresolved debate in the history of mathematics.

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.

Sharp oscillation theorem for fourth-order linear delay differential equations

In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation x(4)(t)+p(t)x(τ(t))=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x^{(4)}(t) + p(t)x\\bigl(\ au (t)\\bigr) = 0 $$\\end{document} by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.

Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

The Fourier integral method was used in this work to determine the stress fields in a two dimensional (2D) elastic soil mass of semi-infinite extent subject to line and strip loads of uniform intensity acting on the boundary. The two dimensional plane strain problem was formulated using stress-based method. The Fourier integral was used to transform the biharmonic stress compatibility equation to a fourth order linear ordinary differential equation (ODE) in terms of the stress function. The ODE was solved subject to the boundedness condition to obtain the bounded stress function. Cartesian stress components were obtained using the Love stress functions. Application of the stress boundary conditions for the case of line load of uniform intensity and the cases of uniformly distributed load on a strip of finite width gave the respective unknown constants of the Love stress functions; and hence the complete determination of the Cartesian stress components for the two cases considered. Inversion of the Fourier integral expressions obtained for the normal and shear stresses in the Fourier parameter gave respective expressions for the normal and shear stress fields for line and finite strip loads of finite width in the physical domain variables. The results obtained agreed with the results from previous studies which used displacement based methods.

A Comparison between the fourth order linear differential equation with its boundary value problem

In this paper, we study a fourth order linear differential equation. We found an upper bound for the solutions of this differential equation and also, we prove that all the solutions are in L4(0, ∞). By comparing these results we obtain that all the eigenfunction of the boundary value problem generated by this differential equation are bounded and in L4(0, ∞).

Closed-form solutions of non-uniform axially loaded beams using Lie symmetry analysis

In this paper, the governing differential equation of a beam with axial force is studied using the Lie symmetry method. Considering the inhomogeneous beam and non-uniform axial load, the governing equation is a fourth-order linear partial differential equation with variable coefficients with no closed-form solution. We search for a favourable coordinate system where the governing equation has a simpler-form or a closed-form solution. A favourable coordinate transformation is found using the Lie transformation group method. The system of determining equations for the governing equation of a beam with non-uniform axial load is derived and then solved to find a favourable coordinate system dependent on the spatially variable stiffness, mass, and axial force. The class of non-uniform axially loaded beams which have a closed-form solution is determined. The fixed-free boundary condition is imposed to find the invariant closed-form solution. A comparison between the analytical solution derived by the Lie symmetry method and the numerical solution is presented. Lie symmetry analysis yields hitherto undiscovered closed-form solutions for non-uniform axially loaded beams.

Mixed Meshless Local Petrov-Galerkin Methods for Solving Linear Fourth-Order Differential Equations

Mixed Meshless Local Petrov-Galerkin Methods for Solving Linear Fourth-Order Differential Equations

VOAs and Rank-Two Instanton SCFTs

We analyze the mathcal {N}=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases.

VARIATIONAL METHODS TO THE FOURTH-ORDER LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES

In this paper, the existence of solutions for the fourth-order linear and nonlinear differential equations with non-instantaneous impulses is studied by applying variational methods. The interesting point lies in that the variational structures corresponding to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses are established for the first time.

Minimax Optimization for a System of Line-of-Sight Stabilization

The minimax problem of stabilization is solved for a line of sight near a programmed trajectory. The motion of this line is described by a system of fourth-order linear differential equations. In this problem, the perturbations are considered as the deviations of the initial position from zero and as time-varying perturbations. The stabilization is performed by a linear feedback. The feedback coefficients are obtained as optimal for the worst possible perturbations.

Refinement Asymptotic Formulas of Eigenvalues and Eigenfunctions of a Fourth Order Linear Differential Operator with Transmission Condition and Discontinuous Weight Function

In this paper, we promote the refinement method for estimating asymptotic expression of the fundamental solutions of a fourth order linear differential equation with discontinuous weight function and transmission conditions. These refinement solutions utilize more accurate asymptotic formulas for the eigenvalues and eigenfunctions for the problem.

$$L^p$$ L p -solutions of a nonlinear third order differential equation and the Poincaré–Perron problem

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory $$L^p$$ -solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (1) the characteristic polynomial associated with the linear part has simple and real roots, (2) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (3) the polynomial coefficients are in $$L^p([t_0,\infty [)$$ . These results are applied to study a fourth order linear differential equation of Poincare type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems cannot be applied.

Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type

Size-dependent bending behavior of Bernoulli-Euler nano-beams is investigated by elasticity integral theory. Eringen’s strain-driven nonlocal integral formulation, providing the bending moment interaction field as output of the convolution between elastic curvature and bi-Helmholtz averaging kernel, is examined. Corresponding higher-order constitutive boundary conditions are established for the first time, showing inapplicability of strain-driven nonlocal integral law of bi-Helmholtz type to continuous nano-structures defined on bounded domains. As a remedy, an innovative stress-driven nonlocal integral elastic model of bi-Helmholtz type is presented for inflected Bernoulli-Euler nano-beams, by swapping input and output of Eringen’s nonlocal integral elastic law. The proposed stress-driven nonlocal integral constitutive law is proven to be equivalent to a fourth-order linear differential equation in the elastic curvature, fulfilling suitable homogeneous constitutive boundary conditions. The corresponding elastic equilibrium problem of an inflected nano-beam is well-posed and numerically solved for statics schemes of nano-technological interest, under standard loading and kinematic boundary conditions. New benchmarks for nonlocal computational mechanics are also detected.

Semicommuting and Commuting Operators for the Heun Family

We derive the most general families of differential operators of first and second degree semi-commuting with the differential operators of the Heun class. Among these families we classify all those families commuting with the Heun class. In particular, we discover that a certain generalized Heun equation commutes with the Heun differential operator allowing us to construct the general solution to a complicated fourth order linear differential equation with variable coefficients which Maple 16 cannot solve.

Lyapunov-type inequalities for fourth-order boundary value problems

This paper is concerned with some two-point boundary value problems for fourth-order linear differential equations. Our study is based on the absolute maximums of Green’s functions corresponding to two-point boundary value problems. Thus, the best constants of Lyapunov-type inequalities for the problems are proved. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for two-point boundary value problems in this direction. In addition, some applications of the obtained inequalities are given.

Symmetries of Fundamental Solutions and Their Application in Continuum Mechanics

An application of the symmetries of fundamental solutions in continuum mechanics is presented. It is shown that the Riemann function of a second-order linear hyperbolic equation in two independent variables is invariant with respect to the symmetries of fundamental solutions, and a method is proposed for constructing such a function. A fourth-order linear elliptic partial differential equation is considered that describes the displacements of a transversely isotropic linear elastic medium. The symmetries of this equation and the symmetries of the fundamental solutions are found. The symmetries of the fundamental solutions are used to construct an invariant fundamental solution in terms of elementary functions.

Uniqueness Results for Higher Order Elliptic Equations in Weighted Sobolev Spaces

We prove some uniqueness results for the solution of two kinds of Dirichlet boundary value problems for second- and fourth-order linear elliptic differential equations with discontinuous coefficients in polyhedral angles, in weighted Sobolev spaces.