2nd Order Partial Differential Equations Research Articles

Functional gradation of material coefficients and size –effects in heat conduction: Numerical simulations

It is well known that the classical theory of heat conduction is scale invariant. On the other hand, it is experimentally evident that size-dependent effects are observable in small samples of micro/nano-scale dimensions. Incorporation of higher-order gradients of primary field variables into constitutive relationships yields a qualitative explanation of size-effects. Form the mathematical point of view, the governing equations are given by the partial differential equations (PDE) with higher-order derivatives in higher-grade continuum theories. As compared with classical theory of continua, the other complication of governing equations occurs in case of continuous media with functional gradation of material coefficients, when the problem is described by the PDE with variable coefficients. The traditional weak formulations are considered in global sense, hence the whole analysed domain/boundary is to be discretized into finite size elements. On the other hand, the strong formulations bring better computational efficiency because of elimination of integrations, but the price which should be paid is the need to approximate higher order derivatives of field variables. One of the most criticized shortcoming of the finite element (FE) approximation is its limited continuity on element intersections. The C0 continuity is insufficient for calculation of gradients of field variables on element intersections as well as for numerical solution of problems with governing equations of higher than 2nd order partial differential equations (PDE). Recently widely spread and elaborated mesh-free approximations utilize the higher order continuous shape functions. Another advantage is elimination of discretization of the analysed domain into finite elements, since only the nodes are scattered on the domain and its boundary. Both the strong and weak formulations are applicable in combination with mesh-free approximations. The Moving Finite Element Approximation (MFEA) and its utilization in mesh-free formulations for heat conduction problems is presented in this paper.

Approximating the Navier-Stokes Equations with the Explicit Finite Difference Method to Solve the Lid-driven Cavity Flow Problem

The Navier-Stokes equations, one of the seven Millennium Problems, are a set of 2nd-order partial differential equations that are near impossible to solve. Instead of solving it, this paper uses the finite difference method to convert the differential terms in the Navier-Stokes Equations into algebraic equations to generate approximate solutions. The procedure to use the explicit finite difference method is outlined and proved in the context of solving the lid-driven cavity flow problem. The solution discussed is then applied in Mat lab and its accuracy and speed is evaluated. The finite difference method is then evaluated and its applications are outlined.

Boundary Integral Technique of 2nd Order Partial Differential Equation by Using Radial Basis

The solution of second order partial differential equation, with continuous change in coefficients by the formation of integral equation and then using radial basis function approximation (RBSA), has been developed in this paper. Use of boundary element method (BEM), which gives the solution of heat or mass diffusion in non-homogenous medium with function varying smoothly in space is also the part of this article. Discretization of boundary of integral domain instead of entire domain of the problem concerned is also the distinction of the recent work. The numerical solution of some problems with known value of the variable has also been included at the end.

Factorized One-Way Wave Equations

The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the definition of the wave propagation direction by the geometric position of the transmitter and receiver.

The second-order problem for k-presymplectic Lagrangian field theories: application to the Einstein\u2013Palatini model

In general, the system of 2nd-order partial differential equations made of the Euler–Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of this work is to develop a fully geometric constraint algorithm which allows us to find a submanifold where the Euler–Lagrange equations have solution, and split the constraints into two kinds depending on their origin. We do so using k-symplectic geometry, which is the simplest intrinsic description of classical field theories. As a second aim, the Einstein–Palatini model of General Relativity is studied using this algorithm.

3D asymmetric dynamic Green’s functions of a thermoelastic transversely isotropic solid by a method of potentials

A theoretical formulation is presented for the displacements, temperature and stresses in a 3 D transversely isotropic thermoelastic solid under arbitrary mechanical loads and heat flux. By using two scalar potential functions, 3 D patch-load, patch-heat source, point-load and point-heat source Green’s functions are given for the displacements, temperature and stresses. Basic governing equations of coupled thermoelasticity in transversely isotropic solids are presented in the cylindrical coordinates system. By means of two scalar potential functions, the four coupled governing equations for this class of problems are uncoupled into a 6th order and a 2nd order partial differential equations. Applying the Fourier expansions and Hankel transforms, the angular and radial variables are suppressed and 6th and 2nd order ordinary differential equations are obtained. By solving the ordinary differential equations, the potential functions, displacements, temperature and stresses are obtained in the Fourier-Hankel transformed space. By applying the Hankel inversion theorem, all responses are presented in the physical time-spatial space. To prove the validity of the analytical and numerical solutions, our results are compared with existing solutions reported in the literature. Numerical results are also included to show the effects of material anisotropy, elastic and thermal properties; and frequency of excitation on the responses.

Combining the Runge Approximation and the Whitney Embedding Theorem in Hybrid Imaging

Abstract This paper addresses enforcing non-vanishing constraints for solutions to a 2nd-order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension $d\geq 2$, under suitable regularity assumptions, the family of $2d$ solutions such that their Jacobian has maximal rank in the domain is both open and dense. The case of less regular coefficients is also addressed, together with other constraints, which are relevant for applications to recent hybrid imaging modalities. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [ 44]. The method is very general and can be used in other settings.

A Weibull Distribution: Flow and Heat Transfer of Nanofluids Containing Carbon Nanotubes with Radiation and Velocity Slip Effects

In this study, the Tiwari and Das model is numerically studied, in case of a moving plate containing both single-walled and multiwalled carbon nanotubes (SWCNTs and MWCNTs, respectively), in the presence of thermal radiation and the slip effect. Employing the similarity transformation, a set of 2nd-order partial differential equations (which are used to model the flow and heat transfer) are solved numerically using the boundary value problem with 4th-order accuracy (BVP4C) method. The effects of related parameters, such as the volume fraction of nanoparticles, moving, slip, and radiation parameter on the heat transfer performance are analysed and discussed. Results indicate that a unique solution was placed when the plate travels in assisting flow conditions. Additionally, as the nanoparticle volume fraction (φ) rises at φ = 0.2, the skin friction and heat transfer rate decrease. It is also observed that when the slip parameter (β) increases at β = 0.4, the skin friction decreases, whereas the heat transfer rate increases. Meanwhile, the heat transfer rate decreases when the thermal radiation (NR) increases to 0.7. Moreover, it is found that the SWCNTs are more efficient when the skin friction coefficient and the Nusselt number are considered. It is found that the Weibull distribution is more suitable in fitting the skin friction data.

Preconditioned Subspace Descent Method for Nonlinear Systems of Equations

AbstractNonlinear least squares iterative solver is considered for real-valued sufficiently smooth functions. The algorithm is based on successive solution of orthogonal projections of the linearized equation on a sequence of appropriately chosen low-dimensional subspaces. The bases of the latter are constructed using only the first-order derivatives of the function. The technique based on the concept of the limiting stepsize along normalized direction (developed earlier by the author) is used to guarantee the monotone decrease of the nonlinear residual norm. Under rather mild conditions, the convergence to zero is proved for the gradient and residual norms. The results of numerical testing are presented, including not only small-sized standard test problems, but also larger and harder examples, such as algebraic problems associated with canonical decomposition of dense and sparse 3D tensors as well as finite-difference discretizations of 2D nonlinear boundary problems for 2nd order partial differential equations.

One-Way Wave Equation Derived from Impedance Theorem

The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathematically more convenient than the 2nd order PDE. Furthermore the 1st order wave equation being valid for three-dimensional wave propagation in an inhomogeneous continuum is derived.

Numerical study of size effects in micro/nano plates by moving finite elements

Bending of plates subjected to stationary transversal loading is studied within the strain-gradient elasticity. The complete formulation including the governing equations and boundary conditions is re-derived for the thin elastic functionally graded as well as homogeneous plate starting from the unified formulation admitting the assumptions of three plate bending theories [46]. The second-order strain gradient theory of elasticity (proposed by Mindlin) with using one microstructural length-scale parameter is employed instead of classical elasticity. Although the derived formulation for thin plate is simpler than for thick plates, it involves the 6th order derivatives of deflections instead of the 4th order derivatives in the classical elasticity. The original system of governing equations is decomposed into the system of 2nd order partial differential equations and new approximation method (MFE – moving finite elements) is proposed for numerical implementation of the derived formulation. The numerical experiments are performed for study the stability, convergence and efficiency of the method as well as for study of the size-effect in micro/nano plates.

Conditions for translation and scaling invariance of the neutron diffusion equation

Lie group methods are applied to the time-dependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an exterior differential system so as to leverage the isovector symmetry analysis approach. Some of the advantages of this method as compared to traditional symmetry analysis approaches are revealed through its use in the context of a 2nd order PDE. In this context, various material properties appearing in the mathematical model (e.g., a diffusion coefficient and macroscopic cross section data) are left as arbitrary functions of space and time. The symmetry analysis that follows is restricted to a search for translation and scaling symmetries; consequently the Lie derivative yields specific material conditions that must be satisfied in order to maintain the presence of these important similarity transformations. The principal outcome of this work is thus the determination of analytic material property functions that enable the presence of various translation and scaling symmetries within the time-dependent, monoenergetic neutron diffusion equation. The results of this exercise encapsulate and generalize many existing results already appearing in the literature. While the results contained in this work are primarily useful as phenomenological guides pertaining to the symmetry behavior of the neutron diffusion equation under certain assumptions, they may eventually be useful in the construction of exact solutions to the underlying mathematical model. The results of this work are also useful as a starting point or framework for future symmetry analysis studies pertaining to the neutron transport equation and its many surrogates.

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.

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Huygens triviality – a concept invented by Jacques Hadamard – describes an equivalence class connecting those 2nd order partial differential equations which are transformable into the wave equation. In this work it is demonstrated, that the Schrödinger equation with the time-independent Hamiltonian belongs to such an equivalence class. The wave equation is the equation for which Huygens’ principle (HP) holds. The HP was a subject of confusion in both physics and mathematics literature for a long time. Not surprisingly, the role of this principle was obscured from the beginnings of quantum mechanics causing some theoretical and experimental misunderstandings. The purpose of this work is to bring the full clarity into this topic. By doing so, we obtained a large amount of new results related to uses of Lie sphere geometry, of twistors, of Dupin cyclides, of null electromagnetic fields, of AdS-CFT correspondence, of Penrose limits, of geometric algebra, etc. in physical problems ranging from the atomic to high energy physics and cosmology.

Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential

The experimental techniques have evolved to a stage where various examples of nanostructures with non-trivial shapes have been synthesized, turning the dynamics of a constrained particle and the link with geometry into a realistic and important topic of research. Some decades ago, a formalism to deduce a meaningful Hamiltonian for the confinement was devised, showing that a geometry-induced potential (GIP) acts upon the dynamics. In this work we study the problem of prescribed GIP for curves and surfaces in Euclidean space R3, i.e., how to find a curved region with a potential given a priori. The problem for curves is easily solved by integrating Frenet equations, while the problem for surfaces involves a non-linear 2nd order partial differential equation (PDE). Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries of R3, which turns the PDE into an ordinary differential equation (ODE) and leads to cylindrical, revolution, and helicoidal surfaces. Helicoidal surfaces are particularly important, since they are natural candidates to establish a link between chirality and the GIP. Finally, for the family of helicoidal minimal surfaces, we prove the existence of geometry-induced bound and localized states and the possibility of controlling the change in the distribution of the probability density when the surface is subjected to an extra charge.

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.

From Conformal Group to Symmetries of Hypergeometric Type Equations

We show that properties of hypergeometric type equations become transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, we deduce the symmetries of the hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and 3-dimensional Laplace equation. We also derive the symmetries of the confluent and Hermite equation from the so-called Schr\"odinger symmetries of the heat equation in 2 and 1 dimension. Finally, we also describe how properties of the ${}_0F_1$ equation follow from the Helmholtz equation in 2 dimensions.

Geometric gradient estimates for solutions to degenerate elliptic equations

We establish new, optimal geometric gradient estimates for solutions to a class of 2nd order partial differential equations, \(\mathcal {L}(X, \nabla u, D^2 u) = f\), whose diffusion properties (ellipticity) degenerate along the a priori unknown set of critical points of an existing solution, \(\mathcal {S}(u) := \{ X : \nabla u(X) = 0 \}\). Such a quantitative information plays a decisive role in the analysis of a number of analytic and geometric problems. The results proven in this work are new even for simple equation \(|\nabla u | \cdot \Delta u = 1\). Under natural nondegeneracy condition on the source term \(f\), we further establish the precise asymptotic behavior of solutions at interior gradient vanishing points. These new estimates are then employed in the study of some well known problems in the theory of elliptic PDEs.

Transient response of a thermoelastic half‐space to mechanical and thermal buried sources

With the aid of a complete set of two scalar potential functions, the transient responses of an isotropic thermoelastic half‐space subjected to time dependent tractions and heat flux applied to a finite patch at an arbitrary depth below a free surface are derived. Using the displacements‐ and temperature‐potential function relationships, the coupled equations of motion and energy equation are uncoupled, resulting in two (6th and 2nd order) partial differential equations in the cylindrical coordinate system, which are solved with the aid of Fourier series expansion and joint Hankel‐Laplace integral transforms. The solutions are also investigated in details for tractions varying with time in terms of a Heaviside step function and heat flux as a Dirac delta function, which may be used as a kernel in any integral based method for more complicated thermoelastodynamic initial‐boundary value problems. Due to the complexity of the integrands involved in the general case, the integrals cannot be resolved analytically and thus an appropriate numerical algorithm is used for the inversion of the Laplace and Hankel integral transforms. To demonstrate the pattern of deformations as well as the distribution of change of temperature at the free surface of the half‐space, numerical evaluations for these functions are presented for an isotropic material.

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.