Fourth-order Partial Differential Equation Research Articles

In-plane and out-of-plane dynamics of curved pipes conveying fluid by integral transform method

This study presents a new solution for the dynamic behavior of in-plane and out-of-plane motion of a curved pipe conveying fluid, using the generalized integral transform technique (GITT). The system of sixth- and fourth-order partial differential equations governing three-dimensional motion of the curved pipe is integral transformed into three coupled systems of second-order ordinary differential equations, which are solved numerically by using the NDSolve routine of Mathematica. Excellent convergence behavior of the GITT solution is shown by comparing the vibration deflections at different points along the curved pipe centerline obtained with of different truncation orders. The results of natural frequencies and vibration responses compare well with available results in the literature obtained by other methods. Furthermore, we investigate the dynamic behavior of the curved pipe with different axial forces, flow velocities and other vibration parameters. A relationship between boundary conditions and axial forces is provided, and a series of vibration displacement and velocity trajectories is obtained. It is shown that the GITT is a convenient and effective mathematical method for vibration analysis of the curved pipe conveying fluid and can be used for further study of the dynamics of variable-curvature curved pipes.

Simulation of Generalized Nonlinear Fourth Order Partial Differential Equation with Quintic Trigonometric Differential Quadrature Method

This paper aims to focus on the implementation of a new approach, quintic trigonometric B-spline basis functions in differential quadrature method to find numerical solution of generalized fourth order partial differential equations with nonlinearity involved. The obtained results using this approach are presented in comparison with available exact and numerical solutions obtained by other researchers. The obtained solutions are in agreement with the available exact solutions and even better than the solutions proposed by the other schemes in the literature. The applicability of the scheme are demonstrated by different eight test problems for the discussed equations that are simulated, for calculating errors like L2, $${{L}_{\infty }}$$, RMS and GRE. To visualize the same, solutions are also presented graphically along with the exact solutions. Scheme is shown to be unconditionally stable with the help of eigenvalues, which demonstrates the consistency of the proposed numerical scheme.

Faster R-CNN, fourth-order partial differential equation and global-local active contour model (FPDE-GLACM) for plaque segmentation in IV-OCT image

The accurate segmentation of plaque in intravascular optical coherence tomography (IV-OCT) image plays an important role in coronary atherosclerotic heart disease (CAD) diagnosis. To effectively provide information of coronary artery stenosis, we propose a novel hybrid framework which includes the faster R-CNN, fourth-order partial differential equation and global-local active contour model (FPDE-GLACM). This framework can efficiently detect and segment the plaque area in Speckle noise-contaminated IV-OCT images. We first detect plaque area by faster R-CNN and set bounding-box as the initial contour for active contour model. And then we minimize the joint energy functional of PDE-GLACM part to achieve the segmentation and denoising of IV-OCT images by gradient descent and finite difference scheme. Specifically, by using the Gaussian image minus original image to get the edge guide image, GLACM part obtains accurate plaque segmentation results. We perform experiments on 5000 IV-OCT images and set clinical manual segmentation results as ground truth. As expected, the results illustrate that the proposed FPDE-GLACM can provide better performance on plaque detection and segmentation. And these results may assist doctor in CAD diagnosis and treatment.

A Computational Framework for Design and Optimization of Flexoelectric Materials

We combine isogeometric analysis (IGA), level set (LS) and pointwise density-mapping techniques for design and topology optimization of piezoelectric/flexoelectric materials. We use B-spline elements to discretize the fourth-order partial differential equations of flexoelectricity, which require at least [Formula: see text] continuous approximations. We adopt the multiphase vector LS model which easily copes with various numbers of material phases and multiple constraints. In case studies, we first confirm the accuracy of the IGA model and then provide numerical examples for both pure and composite flexoelectric structures. The results demonstrate the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization and particularly by multi-material topology optimization for flexoelectric composites.

Longitudinal–Flexural–Torsional Dynamic Behavior of Liquid-Filled Pipelines: An Analytic Solution

Abstract The 14 partial differential equation system describing the longitudinal–flexural–torsional dynamic behavior of liquid-filled pipelines contains coupled equations due to mutual boundary conditions and Poisson contraction ratio terms. Solutions of the above system are available in the frequency-domain or in the time-domain (method of characteristics (MOC)). In this paper, an analytic solution in the domain of time and space is achieved. Double integral transform, namely, finite sine Fourier transform (FSFT) and Laplace transform, is applied to the derived system of the 14 fourth-order partial differential equations, yielding an algebraic system in terms of the transformed variables. The inversion of the FSFT is an easy task, but the analytic inversion of the Laplace transforms is very challenging. Both integral transform inversions of the 14 transformed variables are successfully performed, and an analytic matrix formula in the domain of time and space along with numerical results is obtained.

Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains

A new numerical scheme has been developed based on the fast and efficient meshless local weak form i.e direct meshless local Petrov–Galerkin (DMLPG) method for solving the fractional fourth-order partial differential equation on computational domains with complex shape. The fractional derivative is the Riemann–Liouville fractional derivative. At first, a finite difference scheme with the second-order accuracy has been employed to discrete the time variable. Then, the DMLPG technique is employed to achieve a full-discrete scheme. The time-discrete scheme has been studied in terms of unconditional stability and convergence order by the energy method in the L2 space. Also, some numerical results are presented to show the efficiency and accuracy of the proposed technique on the simple and complex domains with the irregular and non-regular grid points.

A high order numerical method and its convergence for time-fractional fourth order partial differential equations

This paper is concerned with design and analysis of a high order numerical approach based on a uniform mesh to approximate the solution of time-fractional fourth order partial differential equations. In this approach, we first approximate the time-fractional derivative appearing in the governing equation by means of Caputo’s definition and then construct a sextic B-spline collocation method for solving the resulting equation. It is proved that the present method is unconditionally stable. Convergence analysis of the method is discussed. We prove that the method is (2−α)th order convergence with respect to time variable and fourth order convergence with respect to space variable. Three test problems are considered to demonstrate the applicability and efficiency of the new method. It is shown that the rate of convergence predicted theoretically is the same as that obtained experimentally. Numerical results have been compared with those reported previously in literature. It is shown that our method yields more accurate results when compared to the existing methods.

Local existence of a fourth-order dispersive curve flow on locally Hermitian symmetric spaces and its application

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrödinger flow equation to the one considered in the present paper.

An efficient MAPS for solving fourth order partial differential equations using trigonometric functions

In this paper, we apply the method of approximate particular solutions (MAPS) based on the trigonometric functions to solve the fourth order partial differential equations (PDEs) through the use of particular solutions of two second order differential equations. The derivation of the closed-form particular solutions for higher order PDEs is a challenge and could be tedious. Such task can be achieved by a simple algebraic procedure to alleviate the difficulty of the derivation of particular solutions for fourth order PDEs. Since the closed-form particular solutions for the second order PDEs with constant coefficients are known, the proposed solution procedure is simple and direct. Five numerical examples are illustrated to demonstrate the feasibility and effectiveness of the proposed method.

Consistent integration schemes for meshfree analysis of strain gradient elasticity

Integration schemes with nodal smoothed derivatives, which meet integration constraint conditions, are robust and efficient for use in meshfree Galerkin methods, however, most of them are focussed on the classical elasticity determined by a second-order partial differential equation. In this paper, arbitrary-order integration constraint conditions are derived for strain gradient elasticity in a fourth-order partial differential equation. These integration constraint conditions provide the discrete forms of nodal shape functions and their first- and second-order derivatives. Furthermore, to meet the integration constraint conditions, consistent integration schemes are designed with nodal smoothed (but not standard) derivatives at evaluating points. It is shown that such nodal smoothed derivatives are able to satisfy the differentiation of approximation consistency. Finally, several case studies are given and the results demonstrate that, based on convergence, accuracy and efficiency, the numerical performance of consistent integration in meshfree analysis of strain gradient elasticity is superior to the standard Gaussian one.

$$C^1$$-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

In this talk, we propose a C¹-conforming quadrilateral spectral element method for fourth order partial differential equations. The approximation space is constituted by those globally C¹-continuous functions, which are essentially rational polynomials on each physical quadrilateral mapped from polynomials on the reference square by using bilinear transformations. So we first concentrate on the construction the shape functions of an arbitrary order on arbitrary quadrilateral meshes. The quadrilateral spectral element approximation scheme is then established with a brief explanation of the implementation. Numerical experiments are illustrated to demonstrate the accuracy and efficiency of our proposed method.

Triangulation-based isogeometric analysis of the Cahn–Hilliard phase-field model

This paper presents triangulation-based Isogeometric Analysis of the Cahn–Hilliard phase-field model. The Cahn–Hilliard phase-field model is governed by a time-dependent fourth-order partial differential equation. The corresponding primal variational form involves second-order operators, making it difficult to be directly analyzed with traditional C0 finite element analysis. In this paper, we construct C1 Bernstein–Bézier simplicial elements through macro-element techniques, including various triangle-split based macro-elements in both 2D and 3D space. We extend triangulation-based isogeometric analysis to solving the primal variational form of the Cahn–Hilliard equation. We validate our method by convergence analysis, showing the nodal and degree-of-freedom advantages over C0 Finite Element Analysis. We then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two-dimensional squares and three-dimensional cubes. We incorporate an adaptive time-stepping scheme in these numerical experiments. Our numerical study demonstrates that triangulation-based isogeometric analysis offers optimal convergence and time step stability, is applicable to complex geometry and allows local refinement.

The field of tensions of a rotating anisotropic disc of a variable thickness loaded with undistracted forces on the outer contour

The work gives a solution of the plane elasticity problem for rotating polar-orthotropic annular disks of a variable thickness. The disk is loaded with a system of equal focused forces on the outer contour applied evenly along the rim and symmetric concerning the diameter. The disk is seated with an interference fit on the flexible shaft so that a constant contact pressure acts on the interior contour. The stresses and deformations arising in such a rotating anisotropic annular disk will be non-axisymmetric. A conclusion of a fourth-order partial differential equation for the effort function is drawn. Its general solution is searched out in the form of a Fourier series of cosines with even numbers. As a result, an infinite system of ordinary differential equations is solved for the coefficients of the series. These differential equations correspond to the linear Volterra integral equations of the 2 nd kind, which are solved using resolvents. Constants of integration are determined from the border conditions. Expressions for the stress components are written through the effort function by the well-known formulas. We find the components of the displacement vector in the disk by the integration of the Hooke’s law equations for the polar-orthotropic plate. We calculate the deformation components in a ring anisotropic disk by Cauchy differential relations if we know the displacements. The solved formulas for stresses, deformations and displacements completely describe the stress-deformed state in a rotating polar-orthotropic disc of variable thickness with a system of focused forces on the outer contour. The results of the work can be used in the design of working disks of turbomachines and turbo compressors, as well as rotors of centrifugal stands.

Image smoothing via adaptive fourth‐order partial differential equation model

To overcome the staircase effects and simultaneously avoid edge blurring, here the authors proposed a novel algorithm of adaptive fourth-order partial differential equation (PDE) for image smoothing. The proposed method is based on You and Kaveh (YK) model that first performs a pre-processing using coherence-enhancing diffusion filter before the smoothing. Then, the authors select an adaptive edge-stopping function founded on the Tansig function to effectively remove the noise and insignificant details without destruction of important edges and textures in the image. MATLAB simulation results validate the efficiency of the proposed model quantitatively and visually, when compared with other state-of-the-art methods.

Comparative study of analytical and numerical evaluation of the dynamic response of buried pipelines to road-cut excavation blasting

ABSTRACTThis study develops a mathematical model of buried pipelines subjected to surface blast loading based on the theory of beam on elastic foundation. The Fourier transform, a mathematical formula that converts the time domain of the problem to the frequency domain, was used in order to solve a fourth-order non-homogeneous partial differential equation. Transforming the solution back to the time domain, the blast-induced Peak Particle Velocity (PPV) of the pipeline can be calculated. In addition to the mathematical model, a three-dimensional finite element model has been established, thereby drawing a comparison between analytical and numerical results. It can be concluded that the analytically calculated PPV values are found to be higher than the corresponding numerical values. Lastly, the safe distance from the pipeline to blast source and the maximum allowable ANFO explosive weight for two types of rock have been presented in the form of graphs by imposing a limit of 50 mm/s for PPV. This comparative study has investigated the effect of road-cut excavation blasting on pipelines buried under only two types of rock mass. However, it can be used for different types of rock and explosives, mainly thanks to the comprehensiveness of the analytical solution.

Intravascular optical coherence tomography image segmentation based on Gaussian mixture model and adaptive fourth-order PDE

The accuracy of the fibrotic plaque segmentation is vital in identifying the coronary artery stenosis. In this paper, we address an automated approach (APDE-GMM) for separating the fibrotic plaque area of intravascular optical coherence tomography (IV-OCT) images. Under this approach, an objective function consisting of a new energy functional with Rayleigh distribution and the negative log-likelihood function of Gaussian mixture model (GMM) is developed. Also, the study presents an adaptive diffusivity function where the gradient threshold can be associated to suppress the effect of speckle noise. The parameter estimation is carried out by the expectation–maximization technology. In addition, this paper derives a fourth-order partial differential equation (PDE) via Euler–Lagrange equation to obtain the optimal solutions. It has been compared to other segmentation approaches on synthetic and clinical IV-OCT images. The results demonstrate that APDE-GMM segmentates more accurately.

Application of the Onsager-Machlup integral in solving dynamic equations in nonequilibrium systems.

In 1931, Onsager proposed a variational principle which has become the base of many kinetic equations for nonequilibrium systems. We have been showing that this principle is useful in obtaining approximate solutions for the kinetic equations, but our previous method has a weakness that it can be justified, strictly speaking, only for small incremental time. Here we propose an improved method which does not have this drawback. The improved method utilizes the integral proposed by Onsager and Machlup in 1953, and can tell us which of the approximate solutions is the best solution without knowing the exact solution. The improved method has an advantage that it allows us to determine the steady state in nonequilibrium system by a variational calculus. We demonstrate this using three examples, (a) simple diffusion problem, (b) capillary problem in a tube with corners, and (c) free boundary problem in liquid coating, for which the kinetic equations are written in second or fourth-order partial differential equations.

On The Free Vibration Analysis of Simply Supported Beam with Shear Deformation

This paper considered the free vibration analysis of a simply supported beam with shear deformation. The governing equation of fourth order partial differential equation was reduced to an ordinary differential equation using series solution which is the solution of the governing equation. Numerical result was presented and plotted against x, it was found that the response amplitude of the beam increases with increase in the length of the beam and radius of gyration but reduces with an increase in foundation modulus and shear modulus. To keep the deflection of the beam in check, there’s need to keep the length of the beam and the radius of gyration in check to avoid over deflection of the beam which can cause damage in building structures and so on.

The Approximation of the Fourth-Order Partial Differential Equations in the Class of the Piecewise Polynomial Functions on the Triangular Grid

The present work determines the deviation of the piecewise cubic almost-solution of the biharmonic equation and derives the general formula (3) for its calculation. Based on this concept, we obtained an approximation of the equation. A number of numerical calculations were carried out in order to confirm the obtained formula (see pictures 1 and 2) experimentally. In general, for all selected biharmonic functions, the deviation value "ΔΔ(𝑢) turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with 25 ≤ 𝑁 ≤ 35, the absolute error was about 0, 0001, which gives an approximately asymptotic estimate of 𝑂(ℎ2) when the partitioning step is ℎ → 0.

Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations

This paper presents a novel approach to numerical solution of a class of fourth-order time fractional partial differential equations (PDEs). The finite difference formulation has been used for temporal discretization, whereas the space discretization is achieved by means of non-polynomial quintic spline method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered, and the computational outcomes are compared with those found in the exiting literature. It is revealed that the presented scheme is more accurate as compared to current variants on the topic.