Fourth-order Differential Equation Research Articles

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

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

Asymptotic and Oscillatory Analysis of Fourth-Order Nonlinear Differential Equations with p-Laplacian-like Operators and Neutral Delay Arguments

This paper delves into the asymptotic and oscillatory behavior of all classes of solutions of fourth-order nonlinear neutral delay differential equations in the noncanonical form with damping terms. This research aims to improve the relationships between the solutions of these equations and their corresponding functions and derivatives. By refining these relationships, we unveil new insights into the asymptotic properties governing these solutions. These insights lead to the establishment of improved conditions that ensure the nonexistence of any positive solutions to the studied equation, thus obtaining improved oscillation criteria. In light of the broader context, our findings extend and build upon the existing literature in the field of neutral differential equations. To emphasize the importance of the results and their applicability, this paper concludes with some examples.

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

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

High-Order, Accurate Finite Difference Schemes for Fourth-Order Differential Equations

High-Order, Accurate Finite Difference Schemes for Fourth-Order Differential Equations

Full operator preconditioning and the accuracy of solving linear systems

Abstract Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.

Vibration of Non-Uniform Bernoulli-Euler Beam Under Moving Distributed Masses Resting on Pasternak Elastic Foundation Subjected to Variable Magnitude

This paper investigates the dynamic response of a clamped-clamped non-uniform Bernoulli-Euler beam resting on a Pasternak elastic foundation to variable magnitude moving distributed masses. The predicament is dictated by a partial differential equation of fourth order, which features coefficients that are both variable and singular. The primary aim is to derive an analytical solution for this category of a dynamic problem. To achieve this, we employ the method of Galerkin with a series representation of the Heaviside function to reduce the equation to second-order ordinary differential equations with variable coefficients. We simplify these transformed equations using (i) the Laplace transformation technique in conjunction with convolution theory for solving moving force problems, and (ii) finite element analysis in conjunction with the Newmark method for solving analytically unsolvable moving mass problems due to their harmonic nature. We first solve the moving force problem using the finite element method and compare it against analytical solutions as validation for its accuracy in solving analytically unsolvable moving mass problems. The numerical solution obtained from the finite element method is shown to be comparable favorably against analytical solutions of our moving force problem. Lastly, we calculate displacement response curves for both moving distributed force and mass models at various time t for our dynamical problem presentation purposes.

Generalized RKM methods for solving classes of fourth-order fractional partial differential equation

This paper introduces new techniques for solving some fractional partial differential equations of fourth order (FPDEs). Combining numerical RKM techniques with the method of lines (MOL) led to the development of innovative methods; we apply the developed approach to some cases involving fourth-order FPDEs and compare the numerical and analytical answers.

Static term structure of interest rate construction with tension interpolation splines

<abstract><p>Traditional theories of term structure of interest rate consist of four major classical theories, including Pure Expectation Theory, Liquidity Preference Theory, Preferred Habitat Theory and Market Segmentation Theory. However, they cannot be well interpreted by the traditional static term structure of interest rate methods such as polynomial spline and exponential spline. To address problems on low precision and weak stability of traditional methods in constructing static interest rate term structure curve, in this paper, we introduce the tension interpolation spline based on a fourth-order differential equation with local tension parameters calculated by Generalized Reduced Gradient (GRG) algorithm. Our primary focus is to illustrate its better prediction effect and stability with an empirical study conducted using datum of treasury bonds. Then, we divided the datum into intra-sample datum for estimating tension parameters and out-of-sample datum for evaluating their robustness of predicting stochastics collected from Shanghai Stock Exchange on $ {2^{{\rm{nd}}}} $ February, 2019. According to the principle of total least squares and total least absolute deviations, the result shows that the tension interpolation spline model has better precision and stronger stability in prediction of out-of-sample treasury bonds prices compared with the model established by polynomial spline and exponential spline. In addition, it can better explain the Liquidity Preference Theory, which confirms that it is suitable for constructing the static term structure of interest rates in the securities exchange market.</p></abstract>

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.

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

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

EXISTENCE RESULTS AND NUMERICAL SOLUTION OF FULLY FOURTH ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a fully fourth-order nonlinear functional differential equation which contains all lower derivatives of proportional delay arguments.By the reduction of the problem to operator equation for the right-hand side nonlinear function, we establish the existence and uniqueness of the solution and construct iterative methods on both continuous and discrete levels for solving it. We obtain the total error estimate for the discrete iterative solution. Many examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order

A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results.

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

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

Bending theory of composite pressure vessels: A closed-form analytical approach

Composite pressure vessels are strategic for the present and future of the aerospace, transportation, and energy industries. These shell structures are utilized to store and transport liquids and gases and are principally composed of a cylindrical shell and two elliptical heads. The current rules for the structural design of composite pressure vessels make use of the membrane theory that excludes the bending of the structure. Nevertheless, the cylindrical shell and the elliptical heads present different curvature radii. These changes generate a localized bending deformation that determines an increase in the stress state as an effect of a local bending moment and shear force distributions that arise in the geometric transition zone. The stresses related to the bending effects die out rapidly, moving away from the junction plane, but they are sensibly higher than the membrane stress and, therefore, cannot be neglected in the mechanical assessment. Here, a closed-form analytical solution for the bending theory of composite shells is proposed. It can assist in the design of composite pressure vessels and drive the design towards reliable configurations for structural integrity. Using the Donnell–Mustari–Vlasov theory for thin-walled composite shells, the governing fourth-order differential equation is deduced and solved to obtain the displacements and stresses acting in the junction. The analytical results are successfully compared with those of reference finite element models for validation; their accuracy is proved considering different thicknesses, lay-ups, and flattening factors of the vessel

A New Discriminant Method for the Stability of Fourth-Order Differential Equation

The stability of nonlinear differential equations has been widely studied based on various methods. In this paper, we transform a class of fourth-order differential equations into a perturbation system and study the stability and asymptotic stability of zero solutions. By utilizing Lyapunov’s second method, we give sufficient conditions for zero-solution stability and asymptotic stability and provide the construction method of the zero-solution asymptotically stable Lyapunov function of such fourth-order differential equations. In addition, this paper removes the requirement of negative definite when the general differential equation is zero solution stable and obtains a looser stability condition. This paper provides an alternative approach to the research of stability of differential systems.

Oscillation results for nonlinear weakly canonical fourth-order delay differential equations via canonical transform

By using canonical transformation technique, we convert the nonlinear weakly canonical fourth-order delay differential equations into strongly canonical equations. Then we apply oscillation results for canonical equations to get similar results for the studied equation. The significance and novelty of our results are illustrated via numerical examples.

A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions

This article encounters the use of two wavelet methods, namely the collocation method based on Haar wavelets (CMHW) and the higher-order collocation method based on Haar wavelets (HCMHW), to solve linear and nonlinear fourth-order differential equations with different forms of given data such as two-point boundary conditions and two-point integral boundary conditions. Managing these types of boundary conditions can be challenging in numerical methods. However, in this study, these types of equations are handled in a simple manner using the Haar wavelet expressions, as provided in the given information. In the case of nonlinear problems, the quasi-linearization technique is introduced to linearize the equation. Nonlinear fourth-order differential equations are transformed into a simple linear system of algebraic equations using the quasi-linearization technique and Haar wavelets. These equations are then solved very easily to find the solution of the differential equations. The convergence rate and stability of both the methods are studied in details. The convergence rate of the proposed HCMHW is faster than the CMHW (2+2s>2,s=1,2…). Some of the examples are given to indicate the better performance and accuracy of the proposed HCMHW.

Application of fixed point theorems in triple bipolar controlled metric space to solve cantilever beam problem

In this article, we introduce the notion of triple bipolar controlled metric space and prove some new fixed point theorems on this space, reformulate the concept of α-admissible mapping with respect to θ in triple bipolar controlled metric space, and introduce the concept of α-admissible crooked mapping with respect to θ in triple bipolar controlled metric space. We supported our results with suitable examples. We also present an application of the derived result to prove the existence and uniqueness solution of the cantilever beam problem, which is in fourth-order differential equation form.

Stochastic stability of solutions for a fourth-order stochastic differential equation with constant delay

In this paper, we present sufficient conditions to ensure the stochastic asymptotic stability of the zero solution for a specific type of fourth-order stochastic differential equation (SDE) with constant delay. By reducing the fourth-order SDE to a system of first-order SDEs, we utilize a fourth-order quadratic function to derive an appropriate Lyapunov functional. This functional is then employed to establish standard criteria for the nonlinear functions present in the SDE. The stability result obtained in this study is novel and extends the existing findings on stability in fourth-order differential equations. Additionally, we provide an illustrative example to demonstrate the significance and accuracy of our main result.