4th Order Partial Differential Equations Research Articles

A problem with nonlocal integral 1st kind conditions for 4th order partial differential equation

A problem with nonlocal integral 1st kind conditions for 4th order partial differential equation

Transient magnetized Soret effect on dissipative couple stress convection flow past a cylinder

The current article aims to reveal the thermal characteristics of dissipative magnetized chemically reactive unsteady couple stress convection along a cylinder with radiation, heat source/sink, and Soret influences. Governing partial differential equations (PDEs) are produced based on considered physical situation and geometry. Resultant Navier–Stokes equations (NS-E) are highly non-linear, coupled in nature. Analytical techniques fail to find the solutions of these resultant NS-equations. The discretized algebraic NS equations are solved by applying Crank–Nicolson numerical scheme through pentadiagonal and tridiagonal algorithms. Also, the flow of couple stress fluid with physical effects gives a 4th order PDE which needs pentadiagonal algorithm to solve it. Implementing this algorithm numerically is challenging and it is worthy to investigate to analyze the flow behavior. Hence, the simulated couple stress fluid velocity, energy, and mass diffusion profiles including engineering quantities of interest in the boundary layer region are discussed for different parametric values. Contemporary exploration illustrates that magnifying the buoyancy ratio parameter enhances the velocity field. Thermal field elevated for higher values of energy source/sink number. Also, contour plots of velocity, temperature, and mass diffusion profiles have been drawn in the vicinity of a cylinder. In addition, the range of control parameters performed in this study are = 0.00.6, = 0.2 0.8, = 0.00.3, = 0.10.4, and = 1.04.0. Finally, investigated dimensionless outcomes are verified with available solutions and got good agreement.

Bending of FGM plates under thermal load: Classical thermoelasticity analysis by a meshless method

Abstract Thermal stresses, especially at the interface between two different materials, often play an important role in the failure of laminated composite structures. Thus, it is a strong motivation to replace laminated plate structures by FGM ones if possible. The functional gradation of material coefficients, however, yields new coupling effects between the in-plane deformation and bending modes. Therefore the study of behaviour of FGM plates under thermal loadings has become important. In this paper, the bending of thin and/or thick FGM plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations (PDE) are decomposed into the 2nd order PDEs in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares (MLS) approximation scheme. The attention is paid to the study of the influence of various parameters of gradations of material coefficients on bending of plates.

Isogeometric analysis with strong multipatch C1-coupling

C1 continuity is desirable for solving 4th order partial differential equations such as those appearing in Kirchhoff–Love shell models (Kiendl et al., 2009) or Cahn–Hilliard phase field applications (Gómez et al., 2008). Isogeometric analysis provides a useful approach to obtaining approximations with high-smoothness. However, when working with complex geometric domains composed of multiple patches, it is a challenging task to achieve global continuity beyond C0. In particular, enforcing C1 continuity on certain domains can result in “C1-locking” due to the extra constraints applied to the approximation space (Collin et al., 2016).In this contribution, a general framework for coupling surfaces in space is presented as well as an approach to overcome C1-locking by local degree elevation along the patch interfaces. This allows the modeling of solutions to 4th order PDEs on complex geometric surfaces, provided that the given patches have G1 continuity. Numerical studies are conducted for problems involving linear elasticity, Kirchhoff–Love shells and Cahn–Hilliard equation.

Fuzzy genetic-based noise removal filter for digital panoramic X-ray images

This paper proposed a novel fuzzy genetic-based noise removal filter and surveyed the gain of popular filters for noise removal in the digital orthopantomography (OPG) images. The proposed filter is a non-invasive technique for attaining sub-clinical information from the areas of interest in each tooth, both jaws and maxillofacial.The proposed Poisson removal filter combines 4th-order partial differential equations (PDE), total variation (TV) and Bayes shrink threshold accompanied by fuzzy genetic algorithm (FGA) and the exact unbiased inverse of generalized Anscombe transformation (EUIGAT). Experiments were performed in order to show the effect of noise removal filters on 110 simulated, 106 phantom and 104 panoramic radiographic images for subjects (aged 30–60 years old, 50 males and 54 females). Various noises degraded filters and Canny edge detection was performed separately in three kinds of images. The program measured mean square error (MSE), peak signal to noise ratio (PSNR), image quality index (IQI), structural similarity index metric (SSIM) and figure of merit (FOM).The results verify that the proposed filter enhances physicians’ and dentists’ skill of diagnosing normal and pathological events in the teeth, jaws, temporomandibular joint (TMJ) regions and changeable anatomical panoramic landmarks related to osteoporosis progress in the mandible bone using noise removal and improving images quality. Experimental results show the superiority of this filter over other noise removal filters.

An Augmented Method for 4th Order PDEs with Discontinuous Coefficients

A fast finite difference method is developed for solving 4th order partial differential equations with discontinuous coefficients across an arbitrary interface. The method is based on an augmented approach by introducing an intermediate (augmented variable) boundary condition $$\varDelta u|_{\partial \varOmega }$$ along the boundary so that the problem can be treated as two separated Poisson equations with jumps in the source terms along the interface. Thus a fast Poisson solver can be utilized, which makes the proposed method fast. The augmented variable should be chosen such that the original boundary condition $$\frac{\partial u}{\partial n}|_{\partial \varOmega }$$ is satisfied. In discretization, the augmented variable is solved first using the Schur complement associated with the method. The proposed method is probably the first finite difference method for such an interface problem although the paper is partially motivated by the immersed finite element method (Lin et al in J Comput Appl Math 235(13):3953–3964, 2011) for such a problem. Numerical experiments against analytic solutions show that the computed solution using the proposed method has second order accuracy (convergence) in the maximum norm. Numerical results and analysis are also presented for various jump ratios and arbitrary interfaces.

Bending Analysis of FGM Plates Under Thermal Load

In this paper, the bending of thin functionally graded plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations are decomposed into the 2nd order partial differential equations in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares approximation scheme. The attention is paid to the numerical investigation of the influence of several parameters of gradations of material coefficients on the deflection of the plate.

IGS: An IsoGeometric approach for smoothing on surfaces

We propose a novel approach for smoothing on surfaces. More precisely, we aim at estimating functions lying on a surface, starting from noisy and discrete measurements. The surface is represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional. The latter is equivalent to solving a 4th-order Partial Differential Equation (PDE). In this context, we use Isogeometric Analysis (IGA) for the numerical approximation of such surface PDE, leading to an IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a surface. Indeed, IGA facilitates encapsulating the exact geometrical representation of the surface in the analysis and also allows the use of at least globally C1-continuous NURBS basis functions for which the 4th-order PDE can be solved using the standard Galerkin method. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet of the SOAR space shuttle.

An Image Filter Arithmetic based on GA, PDE and TV

Rician noise pollutes Magnetic Resonance Imaging (MRI) image and makes later work worse. In allusion to remove noise while lessen the loss of details as low as possible, this paper proposed an filter algorithm which comprehensive utilize Genetic Algorithm (GA), PDE and TV, based on 4th order Partial Differential Equations (PDE) and Total Variation (TV) theory. First, it calculates the Total Variation(TV) and the 4th order PDE of kth image, and gives them weight coefficients. Then, it finds the optimal Standard deviation of image of kth image by adjusting weight coefficients based on GA algorithm. Third, it compares PSNR of kth image and PSNR of k+1th image to find whether the algorithm is over. Experimental results show that our new algorithm presented in this paper is more effective in removing Rician noise and giving better Peak Signal Noise Ratio (PSNR) gains without manual intervention in comparison with other traditional filters.

No Drama Quantum Theory?

Is it possible to offer a "no drama" quantum theory? Something as simple (in principle) as classical electrodynamics - a theory described by a system of partial differential equations in 3+1 dimensions, but reproducing unitary evolution of a quantum field theory in the configuration space? The following results suggest an affirmative answer:1. The scalar field can be algebraically eliminated from scalar electrodynamics; the resulting equations describe independent evolution of the electromagnetic field.2. After introduction of a complex 4-potential (producing the same electromagnetic field as the standard real 4-potential), the spinor field can be algebraically eliminated from spinor electrodynamics; the resulting equations describe independent evolution of the electromagnetic field.3. The resulting theories for the electromagnetic field can be embedded into quantum field theories.Another fundamental result: in a general case, the Dirac equation is equivalent to a 4th order partial differential equations for just one component, which can be made real by a gauge transform.Issues related to the Bell theorem are discussed.

The Abreu equation with degenerated boundary conditions

The Abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds. We study the Dirichlet problem of the Abreu equation with degenerated boundary conditions. The solutions provide the Kähler metrics of constant scalar curvature on the complex torus.

A Phase-Field – Finite Element Model for Instabilities in Multilayer Thin Films

ABSTRACTCahn-Hilliard type of phase-field (PF) model coupled with elasticity equations is used to study the instabilities in multilayer thin films. The governing equations of the solid state phase transformation include a 4th order partial differential equation representing the evolution of the conserved PF variable (concentration) coupled to 2nd order partial differential equations representing the mechanical equilibrium. A mixed order Galerkin finite element (FE) model is used including C0 interpolation functions for the displacement, and C1 interpolation functions for the concentration. It is shown that quadratic convergence, expected for conforming elements, is achieved from this coupled mixed-order FE model.Using the PF – FE model, first, we studied the effect of compositional strain on the PF interface thickness and the results of simulations are compared with the analytical solutions of an infinite thin film diffusion couple with a flat interface.Morphological instabilities in binary multilayer thin films are investigated. The alloys with and without intermediate phase are considered, as well as the cases with stable and metastable intermediate phase. Maps of transformations in multilayer systems are carried out considering the effects of initial thickness of layers, compositional strain, and growth of a stable/unstable intermediate phase on the instability of the multilayer thin films. It is shown that at some cases phase transformation, intermediate phase nucleation and growth, or deformation of layers due to high compositional strain can lead to the coarsening of the layers which can result in different mechanical and materials behaviors of the original designed multilayer.

A non-linear Goursat problem for a 4th order polyvibrating equation

SynopsisIn the paper the existence is proved of a solution of a non-linear Goursat problem for a 4th order partial differential equation with the boundary conditions given on four curves emanating from a common point. The problem is reduced to a system of integro-functional equations and then Schauder's fixed point theorem is applied.