Fourth-order Partial Differential Equation Research Articles

Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations

ABSTRACTThis paper proposes a numerical approach to approximate the unknown solution of some high-order fractional partial differential equations. The main idea of this approach is to transform the original problem into an equivalent integral equation that depends only on the boundary values. The linear radial basis functions are used as the main tool for approximating the non-homogeneous terms and time derivative. Also the Caputo's sense is applied to approximate time derivatives. Numerical results demonstrate the order of time steps is and when and , respectively. Finally to overcome the nonlinear terms, predictor–corrector scheme is employed. The efficiency and usefulness of proposed method are demonstrated by some numerical examples.

Monge–Ampére simulation of fourth order PDEs in two dimensions with application to elastic–electrostatic contact problems

We present an efficient moving mesh method for the simulation of fourth order nonlinear partial differential equations (PDEs) in two dimensions using the Parabolic Monge–Ampére (PMA) equation. PMA methods have been successfully applied to the simulation of second order problems, but not on systems with higher order equations which arise in many topical applications. Our main application is the resolution of fine scale behavior in PDEs describing elastic–electrostatic interactions. The PDE system considered has multiple parameter dependent singular solution modalities, including finite time singularities and sharp interface dynamics. We describe how to construct a dynamic mesh algorithm for such problems which incorporates known self similar or boundary layer scalings of the underlying equation to locate and dynamically resolve fine scale solution features in these singular regimes. We find a key step in using the PMA equation for mesh generation in fourth order problems is the adoption of a high order representation of the transformation from the computational to physical mesh. We demonstrate the efficacy of the new method on a variety of examples and establish several new results and conjectures on the nature of self-similar singularity formation in higher order PDEs.

An adaptive fourth-order partial differential equation for image denoising

To overcome the weakness of second order methods such as Perona–Malik model for image denoising, various high order models have been proposed and studied. However, there is not too much analysis of these equations to be found in the literature. In this paper, we propose an adaptive fourth-order partial differential equation, which joints a fourth-order term and a second-order term. The model takes advantage of the fourth-order model’s better image avoiding staircase effect and the second-order model’s better edge preserving effect. By introducing a functional framework and k-bounded partial variation (BPVk) space, we prove the existence of a weak solution of the proposed model. Experimental results show that the proposed model can alleviate the staircase effect and preserve edges accurately.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

Hencky bar-net model for plate buckling

This paper presents the Hencky bar-net model (HBM) for elastic buckling analysis of rectangular plates with any combination of edge conditions. HBM comprises a grid (or net) of rigid bar segments joined by rotational springs as well as a torsional spring system in each grid panel. The HBM is developed from the finite difference plate model that allows researchers to perform plate buckling analysis via solving a set of algebraic equations instead of solving a fourth order partial differential equation. The elastic spring stiffnesses for the HBM that provides the flexibility for deformation are established for the first time for elastic rotationally restrained edges and free edges. Some plate buckling problems were solved by using the developed HBM and the solutions shown to converge to those of the continuum plate counterparts when the grid size becomes very small; thereby verifying the validity of the model.

A semi-analytical Fourier spectral method for the Swift–Hohenberg equation

The Swift–Hohenberg (SH) equation has been widely used as a model for the study of pattern formation. The SH equation is a fourth-order nonlinear partial differential equation and cannot generally be solved analytically. Therefore, computer simulations play an essential role in understanding of nonequilibrium processing. The aim of this paper is to present first- and second-order semi-analytical Fourier spectral methods as an accurate and efficient approach for solving the SH equation. The methods are based on the operator splitting method and are to split the SH equation into linear and nonlinear subequations. The linear and nonlinear subequations have closed-form solutions in the Fourier and physical spaces, respectively. The methods are simple to implement and computationally cheap to achieve high-order time accuracy. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods.

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

A convergence study is presented for a form of gradient elasticity where the enrichment is through the Laplacian of the strain, so that a fourth-order partial differential equation results. Isogeometric finite element analysis is used to accommodate the higher continuity required by the inclusion of strain gradients. A convergence analysis is carried out for the original system of a fourth-order partial differential equation. Both global refinement, using NURBS, and local refinement, using T-splines, have been applied. Theoretical convergence rates are recovered, except for a polynomial order of two, when the convergence rate is suboptimal, a result which also has been found for the (fourth-order) Cahn-Hilliard equation. The convergence analyses have been repeated for the case that an operator split is applied so that a set of two (one-way) coupled partial differential equations results. Differences occur with the results obtained for the original fourth-order equation, which is caused by the boundary conditions, which is the first time this effect has been substantiated.

Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation

In this paper, the local boundary integral equation (LBIE) method based on generalized moving least squares (GMLS) is proposed for solving extended Fisher---Kolmogorov (EFK) equation. Since the local weak forms are directly approximated from nodal data using a GMLS approximation, this method is called direct local boundary integral equation (DLBIE) method. In addition to DLBIE method, three other meshless local boundary integral equation methods such as LBIE method based on moving least squares, LBIE method based on radial point interpolation, and LBIE method based on moving Kriging are applied to find the numerical solution of EFK equation. A comparison between these methods is done from the perspective of accuracy and computational efficiency. The computational efficiency is the most significant advantage of the DLBIE method in comparison with the other local boundary integral equation methods. DLBIE shifts the numerical integrations over low-degree polynomials instead of over complicated shape functions and this reduces the computational costs, significantly. The main purpose of this paper is to show that the local boundary integral equation methods can be used for solving the fourth-order non-linear partial differential equations especially EFK equation. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Modeling and Iterative Learning Control of a Circular Deformable Mirror

An unconditionally stable finite difference scheme for systems whose dynamics are described by a fourth-order partial differential equation is developed using a regular hexagonal grid. The scheme is motivated by the well-known Crank-Nicolson discretization that was originally developed for second-order systems and it is used in this paper to develop a discrete in time and space model of a deformable mirror as a basis for control law design. As one example, the resulting model is used for iterative learning control law design and supporting numerical simulations are given.

Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients

We study the time evolution of a thin liquid film coating the outer surface of a sphere in the presence of gravity, surface tension, and thermal gradients. We derive the fourth-order nonlinear partial differential equation that models the thin film dynamics, including Marangoni terms arising from the dependence of surface tension σ on temperature T. We consider two different imposed temperature distributions with axial or radial thermal gradients. We analyze the stability of a uniform coating under small perturbations and carry out numerical simulations in COMSOL for a range of parameter values. In the case of an axial temperature gradient, we find steady states either with uniform film thickness or with the fluid accumulating at the bottom or near the top of the sphere, depending on the total volume of liquid in the film, dictating whether gravity or Marangoni effects dominate. This suggests a potential method for the indirect measurement of dσ/dT by monitoring the thickness profile of the thin film. In the case of a radial temperature gradient, a stability analysis reveals the most unstable non-axisymmetric modes on an initially uniform coating film.

Critical velocities in fluid-conveying single-walled carbon nanotubes embedded in an elastic foundation

The problem of stability of fluid-conveying carbon nanotubes embedded in an elastic medium is investigated in this paper. A nonlocal continuum mechanics formulation, which takes the small length scale effects into consideration, is utilized to derive the governing fourth-order partial differential equations. The Fourier series method is used for the case of the pinned–pinned boundary condition of the tube. The Galerkin technique is utilized to find a solution of the governing equation for the case of the clamped–clamped boundary. Closed-form expressions for the critical flow velocity are obtained for different values of the Winkler and Pasternak foundation stiffness parameters. Moreover, new and interesting results are also reported for varying values of the nonlocal length parameter. It is observed that the nonlocal length parameter along with the Winkler and Pasternak foundation stiffness parameters exert considerable effects on the critical velocities of the fluid flow in nanotubes.

A fourth order PDE based fuzzy c- means approach for segmentation of microscopic biopsy images in presence of Poisson noise for cancer detection

Background and objectiveFor cancer detection from microscopic biopsy images, image segmentation step used for segmentation of cells and nuclei play an important role. Accuracy of segmentation approach dominate the final results. Also the microscopic biopsy images have intrinsic Poisson noise and if it is present in the image the segmentation results may not be accurate. The objective is to propose an efficient fuzzy c-means based segmentation approach which can also handle the noise present in the image during the segmentation process itself i.e. noise removal and segmentation is combined in one step. MethodsTo address the above issues, in this paper a fourth order partial differential equation (FPDE) based nonlinear filter adapted to Poisson noise with fuzzy c-means segmentation method is proposed. This approach is capable of effectively handling the segmentation problem of blocky artifacts while achieving good tradeoff between Poisson noise removals and edge preservation of the microscopic biopsy images during segmentation process for cancer detection from cells. ResultsThe proposed approach is tested on breast cancer microscopic biopsy data set with region of interest (ROI) segmented ground truth images. The microscopic biopsy data set contains 31 benign and 27 malignant images of size 896 × 768. The region of interest selected ground truth of all 58 images are also available for this data set. Finally, the result obtained from proposed approach is compared with the results of popular segmentation algorithms; fuzzy c-means, color k-means, texture based segmentation, and total variation fuzzy c-means approaches. ConclusionsThe experimental results shows that proposed approach is providing better results in terms of various performance measures such as Jaccard coefficient, dice index, Tanimoto coefficient, area under curve, accuracy, true positive rate, true negative rate, false positive rate, false negative rate, random index, global consistency error, and variance of information as compared to other segmentation approaches used for cancer detection.

Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation

In this article, a fully discrete local discontinuous Galerkin (LDG) method with high-order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time-fractional fourth-order partial differential equation (PDE). In the temporal direction, for approximating the fractional derivative with order α∈(0,1), the weighted and shifted Grünwald difference (WSGD) scheme with second-order convergence rate is introduced and for approximating the integer time derivative, two step backward Euler method with second-order convergence rate is used. For the spatial direction, the LDG method is used. For the numerical theories, the stability is derived and a priori error results are proved. Further, some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed method.

A new spline in compression method of order four in space and two in time based on half-step grid points for the solution of the system of 1D quasi-linear hyperbolic partial differential equations

In this paper, we propose a new three-level implicit method based on a half-step spline in compression method of order two in time and order four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form u_{tt} =A(x,t,u)u_{xx} +f(x,t,u,u_{x},u_{t}). We describe spline in compression approximations and their properties using two half-step grid points. The new method for one-dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function u(x,t) and two half-step points for the known variable ‘x’ in x-direction. The proposed method, when applied to a linear test equation, is shown to be unconditionally stable. We have also established the stability condition to solve a linear fourth-order hyperbolic partial differential equation. Our method is directly applicable to solve hyperbolic equations irrespective of the coordinate system, which is the main advantage of our work. The proposed method for a scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.

A vortex-induced vibration model for the fatigue analysis of a marine drilling riser

ABSTRACTThe objective is to propose a quick and accurate vortex-induced vibration (VIV) model to predict the fatigue life of marine drilling risers. The nonlinear wake oscillator model was applied to represent the lift force, which is described by a fourth-order partial differential equation set. Next, we adopted the generalised integral transform technique (GITT) to transform it into second-order ordinary differential equation system, which was solved by the Runge-Kutta method. Validation of the numerical model was implemented and a good agreement was achieved. Next, an S-N curve was adopted to conduct fatigue prediction. The results demonstrate that under the conditions set forth in this paper, an increase in the current and outer diameter could reduce fatigue life, except when the outer diameter reaches a certain value, at which point the tendency is reversed. Moreover, when the top tension coefficient reaches a certain value, the fatigue life tendency is also reversed.

Geometric Diversity of Living Organisms and Viruses

Recently, it has been celebrated a hundred year anniversary of Einstein's theory of general relativity. Since, underlying geometry of space and time became main source of inspiration of new theories. Though, much attention was paid to the geometric construction of the universe, geometric diversity of living organisms got largely omitted. Of course such reluctant approach has fundamental explanation. General relativity is formulated to explain the gravity, while the role of gravitation on the molecular level in the formation of living organisms is negligibly small. However, mathematical approach used in general relativity can be carried out for two dimensional surfaces. Therefore, scientists started to develop analogical differential geometric approach to predict static shapes of living organisms. As a result associated Euler-Lagrange equations, so called “shape equations” have been formulated. But, shape equations turned to be fourth order partial nonlinear differential equations, and finding a general analytical solution is a difficult problem, even though it has been numerically solved for some specific cases. The numerical solutions generated a large body of beautiful explanation of experimental results especially observed in diversity of lipid membranes shapes. Because of shape equations as well as numerical calculations are not exact equations of motion and analytical solutions, we have decided to employ different approach to the problem. Namely, we have used tensor calculus of moving surfaces to deduce exact equations of surface motion in electromagnetic field and then to analytically solve it for some simplified cases. Simplifications have produced such outstanding equations as Maxwell equations for electrodynamics and Navier-Stokes equations for dynamic fluid are (D. V. Svintradze, (2016), arXiv:1609.07765 [cond-mat.soft]). Also, analytic solutions to simplified equations for homogeneous surfaces, in equilibrium with environment, (D. V. Svintradze, (2016), arXiv:1608.01491 [physics.bio-ph]) have explained why constant mean curvature (CMC) surfaces are such abundant shapes among living organisms and viruses.

High Accuracy Compact Operator Methods for Two-Dimensional Fourth Order Nonlinear Parabolic Partial Differential Equations

Abstract In this study, we develop and implement numerical schemes to solve classes of two-dimensional fourth-order partial differential equations. These methods are fourth-order accurate in space and second-order accurate in time and require only nine spatial grid points of a single compact cell. The proposed discretizations allow the use of Dirichlet boundary conditions only without the need to discretize the derivative boundary conditions and thus avoids the use of ghost points. No transformation or linearization technique is used to handle nonlinearity and the obtained block tri-diagonal nonlinear system has been solved by Newton’s block iteration method. It is discussed how our formulation is able to tackle linear singular problems and it is ensured that the methods retain their orders and accuracy everywhere in the solution region. The proposed two-level method is shown to be unconditionally stable for a class of two-dimensional fourth-order linear parabolic equation. We also discuss the alternating direction implicit (ADI) method for solving two-dimensional fourth-order linear parabolic equation. The proposed difference methods has been successfully tested on the two-dimensional vibration problem, Boussinesq equation, extended Fisher–Kolmogorov equation and Kuramoto–Sivashinsky equation. Numerical results demonstrate that the schemes are highly accurate in solving a large class of physical problems.

Explicit Bézier control net of a PDE surface

The PDE under study here is a general fourth-order linear elliptic Partial Differential Equation. Having prescribed the boundary control points, we provide the explicit expression of the whole control net of the associated PDE Bézier surface.In other words, we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points. The set of scalar coefficients of these combinations works like a mould for PDE surfaces. Thus, once this mould has been computed for a given degree, real-time manipulation of the resulting surfaces becomes possible by modifying the prescribed information.

Vibrations under moving distributed loads of a non-prismatic prestressed Rayleigh beam

The vibrations of a non-prismatic prestressed Rayleigh beam resting on an elastic foundation and traversed by distributed loads moving with constant velocities is investigated. The fourth order partial differential equation with variable coefficients governing vibrations of the non-prismatic prestressed Rayleigh beam is solved using the generalised Galerkin method and a modification of the Struble's asymptotic method. Closed form solutions are obtained for the simply supported and clamped/fixed ends boundary conditions. The theoretical considerations find application in calculations relating to dynamic stresses in railways, bridges and foundation for all types of highways.

Vibrations under moving distributed loads of a non-prismatic prestressed Rayleigh beam

The vibrations of a non-prismatic prestressed Rayleigh beam resting on an elastic foundation and traversed by distributed loads moving with constant velocities is investigated. The fourth order partial differential equation with variable coefficients governing vibrations of the non-prismatic prestressed Rayleigh beam is solved using the generalised Galerkin method and a modification of the Struble's asymptotic method. Closed form solutions are obtained for the simply supported and clamped/fixed ends boundary conditions. The theoretical considerations find application in calculations relating to dynamic stresses in railways, bridges and foundation for all types of highways.