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Showing papers on "Boundary value problem published in 2019"


Book
04 Nov 2019
TL;DR: In this paper, the authors consider linear initial value problems, Sturm-Liouville problems and related inequalities in several independent variables, including difference inequalities and boundary value problems for linear systems and nonlinear systems.
Abstract: Preliminaries linear initial value problems miscellaneous difference equations difference inequalities qualitative properties of solutions of difference systems qualitative properties of solutions of higher order difference equations qualitative properties of solutions of neutral difference equations boundary value problems for linear systems boundary value problems for nonlinear systems miscellaneous properties of solutions of higher order linear difference equations boundary value problems for higher order difference equations Sturm-Liouville problems and related inequalities difference inequalities in several independent variables.

939 citations


Journal ArticleDOI
TL;DR: A method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy that increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth.
Abstract: We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy. In this procedure, a coarse grid of training points is used at the initial training stages, while more points are added at later stages based on the value of the residual at a larger set of evaluation points. This method increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth. Numerical results are presented for benchmark problems for scalar-valued PDEs, namely Poisson and Helmholtz equations, as well as for an inverse acoustics problem.

450 citations



Journal ArticleDOI
TL;DR: In this article, a general approach is provided for the free vibration analysis of rotating functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells with arbitrary boundary conditions.

229 citations


Journal ArticleDOI
TL;DR: In this paper, the unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid is studied, and the governing equations of the problem are transformed to the similarity equations by using similarity transformation technique.

219 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions at infinity by the nonlinear steepest descent method of Deift and Zhou.

180 citations


Journal ArticleDOI
TL;DR: In this paper, transfer matrices are used to explain the properties of non-Hermitian Hamiltonians and predict their topological properties, including their sensitivity to boundary conditions and a piling up of all states at the boundary, signalling a breakdown of the conventional bulk boundary correspondence.
Abstract: Non-Hermitian Hamiltonians---proposed to describe dissipative systems where the particles have a finite lifetime---exhibit many puzzling properties, strongly contrasting their Hermitian counterparts. In particular, their spectra exhibit extreme sensitivity to boundary conditions and a piling up of all states at the boundary, signalling a breakdown of the conventional bulk-boundary correspondence. The authors study these systems using transfer matrices, whereby they can explain as well as predict the aforementioned features without resorting to numerical computations. This analytical approach allows for a deeper understanding of the topological properties of non-Hermitian systems.

166 citations



Journal ArticleDOI
TL;DR: For the first time, buckling behavior of functionally graded (FG) nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) is investigated and the size-dependent behavior of nanostructured systems is studied for buckling response of the graded anisotrop material.
Abstract: For the first time, buckling behavior of functionally graded (FG) nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) is investigated. Also, it is the first time that the size-dependent behavior of nanostructured systems is studied for buckling response of the graded anisotropic material. The properties of graded material are assumed vary exponentially through the z-direction. Nonlocal strain gradient theory is utilized to predicate the size-dependent buckling behavior of the nanoplate. The nanoplate is modeled by a higher order shear deformation refined plate theory in which any shear correction factor not used. Governing equations and boundary conditions are obtained using a virtual work of variational approach. To solve the buckling problem for different boundary conditions, Galerkin’s approach is utilized. Finally, the influences of different boundary conditions, small-scale parameters, geometry parameters and exponential factor are studied and discussed in detail. It is hoped that the present numerical results can help the engineers and designers to understand and predict the buckling response of FG anisotropic materials.

121 citations


Journal ArticleDOI
TL;DR: In this paper, a modified periodic boundary condition for non-Hermitian topological systems is proposed, and a topological number characterizing the system is defined in the same way as in the corresponding Hermitian system.
Abstract: A modified periodic boundary condition adequate for non-Hermitian topological systems is proposed. Under this boundary condition, a topological number characterizing the system is defined in the same way as in the corresponding Hermitian system, and hence, at the cost of introducing an additional parameter that characterizes the non-Hermitian skin effect, the idea of bulk-edge correspondence in the Hermitian limit can be applied almost as it is. We develop this framework through the analysis of a non-Hermitian Su-Schrieffer-Heeger model with chiral symmetry and prove the bulk-edge correspondence in a generalized parameter space. A finite region in this parameter space with a nontrivial pair of chiral winding numbers is identified as topologically nontrivial, indicating the existence of a topologically protected edge state under an open boundary.

119 citations


Book
19 Sep 2019
TL;DR: A Primer on Line Integral Methods A general framework Geometric integrators Hamiltonian problems Symplectic methods s-stage trapezoidal methods Runge-Kutta line integral methods Examples of Hamiltonian Problems Nonlinear pendulum Cassini ovals Henon-Heiles problem N-body problem Kepler problem Circular restricted three-body problems Fermi-Pasta-Ulam problem Molecular dynamics Analysis of the Hamiltonian Boundary Value Methods (HBVMs) Derivation and analysis of the methods runge-kutta formulation Properties of HBVMs
Abstract: A Primer on Line Integral Methods A general framework Geometric integrators Hamiltonian problems Symplectic methods s-stage trapezoidal methods Runge-Kutta line integral methods Examples of Hamiltonian Problems Nonlinear pendulum Cassini ovals Henon-Heiles problem N-body problem Kepler problem Circular restricted three-body problem Fermi-Pasta-Ulam problem Molecular dynamics Analysis of Hamiltonian Boundary Value Methods (HBVMs) Derivation and analysis of the methods Runge-Kutta formulation Properties of HBVMs Least square approximation and Fourier expansion Related approaches Implementing the Methods and Numerical Illustrations Fixed-point iterations Newton-like iterations Recovering round-off and iteration errors Numerical illustrations Hamiltonian Partial Differential Equations The semilinear wave equation Periodic boundary conditions Nonperiodic boundary conditions Numerical tests The nonlinear Schrodinger equation Extensions Conserving multiple invariants General conservative problems EQUIP methods Hamiltonian boundary value problems Appendix: Auxiliary Material Bibliography Index

Journal ArticleDOI
TL;DR: In this article, the authors studied non-Bloch bulk-boundary correspondence in a non-Hermitian Su-Schieffer-Heeger model in a domain wall configuration where the left and right bulks have different parameters.
Abstract: We study non-Bloch bulk-boundary correspondence in a non-Hermitian Su-Schieffer-Heeger model in a domain wall configuration where the left and right bulks have different parameters. Focusing on the case where chiral symmetry is still conserved, we show that non-Hermitian skin effects of bulk states persist in the system, while the definition of the non-Bloch winding number of either bulk depends on parameters on both sides of the boundary. Under these redefined non-Bloch topological invariants, we confirm non-Bloch bulk-boundary correspondence under the domain wall configuration, which exemplifies the impact of boundary conditions in non-Hermitian topological systems.

Journal ArticleDOI
TL;DR: In this article, a modified couple stress theory and isogeometric analysis (IGA) was used to simulate the small-scale effects on bending and buckling on composite laminate microplate under complex boundary conditions in thermal environment.

Journal ArticleDOI
16 Apr 2019
TL;DR: The immersed boundary method is a methodology for dealing with boundary conditions at fluid-fluid and fluid-solid interfaces as discussed by the authors, which has been attracting growing attention in the field of fluid dynamics.
Abstract: The immersed boundary method is a methodology for dealing with boundary conditions at fluid–fluid and fluid–solid interfaces. The immersed boundary method has been attracting growing attention in t...

Journal ArticleDOI
TL;DR: In this paper, the authors presented both effective analytic and numerical solutions to MHD flow and heat transfer past a permeable stretching/shrinking sheet in a hybrid nanofluid with suction/injection and convective boundary conditions.
Abstract: The purpose of this study is to present both effective analytic and numerical solutions to MHD flow and heat transfer past a permeable stretching/shrinking sheet in a hybrid nanofluid with suction/injection and convective boundary conditions. Water (base fluid) nanoparticles of alumina and copper were considered as a hybrid nanofluid.,Proper-similarity variables were applied to transform the system of partial differential equations into a system of ordinary (similarity) differential equations. Exact analytical solutions were then presented for the dimensionless stream and temperature functions. Further, the authors introduce a very nice analytic and numerical solutions for both small and large values of the magnetic parameter.,It was found that no/unique/two equal/dual physical solutions exist for the investigated boundary value problem. The physically realizable practice of these solutions depends on the range of the governing parameters. For a stretching/shrinking sheet, it was deduced that a hybrid nanofluid works as a cooler on increasing some of the investigated parameters. Moreover, in the case of a shrinking sheet, the first solutions of hybrid nanofluid are stable and physically realizable rather than the nanofluid, while those of the second solutions are not for both hybrid nanofluid and nanofluid.,The present results for the hybrid nanofluids are new and original, as they successfully extend (generalize) the problems previously considered by different authors for the case of nanofluids.

Journal ArticleDOI
TL;DR: In this article, the free vibration of porous square plate, circular plate, and rectangle plate with a central circular hole in the framework of isogeometric analysis (IGA) was studied.

Journal ArticleDOI
TL;DR: A dynamic procedure for the slip coefficients is formulated, providing a dynamic slip wall model free of a priori specified coefficients that alleviates the well-known problem of the wall-stress under-estimation by current subgrid-scale (SGS) models.
Abstract: Wall modelling in large-eddy simulation (LES) is necessary to overcome the prohibitive near-wall resolution requirements in high-Reynolds-number turbulent flows. Most existing wall models rely on assumptions about the state of the boundary layer and require a priori prescription of tunable coefficients. They also impose the predicted wall stress by replacing the no-slip boundary condition at the wall with a Neumann boundary condition in the wall-parallel directions while maintaining the no-transpiration condition in the wall-normal direction. In the present study, we first motivate and analyse the Robin (slip) boundary condition with transpiration (non-zero wall-normal velocity) in the context of wall-modelled LES. The effect of the slip boundary condition on the one-point statistics of the flow is investigated in LES of turbulent channel flow and a flat-plate turbulent boundary layer. It is shown that the slip condition provides a framework to compensate for the deficit or excess of mean momentum at the wall. Moreover, the resulting non-zero stress at the wall alleviates the well-known problem of the wall-stress under-estimation by current subgrid-scale (SGS) models (Jimenez & Moser, AIAA J., vol. 38 (4), 2000, pp. 605-612). Second, we discuss the requirements for the slip condition to be used in conjunction with wall models and derive the equation that connects the slip boundary condition with the stress at the wall. Finally, a dynamic procedure for the slip coefficients is formulated, providing a dynamic slip wall model free of a priori specified coefficients. The performance of the proposed dynamic wall model is tested in a series of LES of turbulent channel flow at varying Reynolds numbers, non-equilibrium three-dimensional transient channel flow and a zero-pressure-gradient flat-plate turbulent boundary layer. The results show that the dynamic wall model is able to accurately predict one-point turbulence statistics for various flow configurations, Reynolds numbers and grid resolutions.

Journal ArticleDOI
TL;DR: In this article, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions are investigated by means of an improved Fourier series method (IFSM). And the porosity coefficients are determined by the porosa coefficients.
Abstract: In this paper, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions is investigated by means of an improved Fourier series method (IFSM). It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions was expressed as the simple cosine. The size of the pore in a rectangular plate is determined by the porosity coefficients. Using the first-order shear deformation theory(FSDT), the energy expression of FGP rectangular plate is created. In order to obtain the admissible function of displacement for functionally graded porous rectangular plate, the IFSM is employed. Then, the Rayleigh-Ritz method is used to solve coefficients in the Fourier series which determine natural frequencies and modal shapes. Convergence and comparative research are performed to prove the convergence, reliability and accuracy of the current method. On this foundation, some new results covering the influence of the geometrical parameters subject to classical and elastic boundary condition are presented, and the parametric studies are also investigated in detail, which can provide a reference for future research by other researchers.

Journal ArticleDOI
TL;DR: In this article, the authors extended the stress-driven nonlocal integral model of elasticity for 1D nano-structures to Kirchhoff axisymmetric nano-plates.

Journal ArticleDOI
TL;DR: The paper presents the results from 2-D and 3-D Poiseuille flows showing convergence rates typical for weakly compressible SPH and a new correction is proposed to the popular density diffusion term treatment to correct for pressure errors at the boundary.

Journal ArticleDOI
TL;DR: In this paper, a refined plate theory was proposed to describe the response of anti-symmetric cross-ply laminated plates subjected to a uniformly distributed nonlinear thermo-mechanical loading.
Abstract: The present paper addresses a refined plate theoryin order to describe the response of anti-symmetric cross-ply laminated plates subjected to a uniformlydistributed nonlinear thermo-mechanical loading. In the present theory, the undetermined integral terms are used and the variables number is reduced to four instead of five or more in other higher-order theories. The boundary conditions on the top and the bottom surfaces of the plate are satisfied; hence the use of the transverse shear correction factors isavoided. The principle of virtual work is used to obtain governing equations and boundary conditions. Navier solution for simply supported plates is used to derive analytical solutions. For the validation of the present theory, numerical results for displacements and stressesare compared with those of classical, first-order, higher-order and trigonometricshear theories reported in the literature.

Journal ArticleDOI
TL;DR: In this paper, variational formulations and governing equations with boundary conditions are derived for a pair of Euler-Bernoulli beam bending models following a simplified version of Mindlin's...
Abstract: As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s ...

Journal ArticleDOI
TL;DR: In this paper, the size-dependent nonlinear primary resonance of periodic soft excited micro/nano-beams made of bi-directional functionally graded materials (2D-FGMs) is studied.
Abstract: With the aid of advanced design techniques, functionally graded materials as promising new materials can be fabricated into various micro/nano-structures to acquire stronger mechanical performance. In this work, the size-dependent nonlinear primary resonance of periodic soft excited micro/nano-beams made of bi-directional functionally graded materials (2D-FGMs) is studied. To accomplish this end, the nonlocal strain gradient theory of elasticity is utilized within the framework of the refined hyperbolic shear deformation beam theory to construct a size-dependent beam model. On the basis of the variational approach using the principle of Hamilton, the non-classical differential equations of motion are achieved. Thereafter, a discretization scheme based numerical solving process via generalized differential quadrature method (GDQM) together with the pseudo-arclength continuation technique, the nonlocal strain gradient frequency response and amplitude response associated with the nonlinear primary resonance of 2D-FGM micro/nano-beams with different boundary conditions are obtained. It is displayed that the nonlocal size dependency makes a reduction in the oscillation amplitudes associated with both of the bifurcation points, but the strain gradient size effect causes to increase them. These patterns are more significant for the second bifurcation point and for the simply supported-simply supported boundary conditions in comparison with the clamped-clamped ones. Also, it is observed that by increasing the value of both of the axial and lateral material property gradient indexes, the peak of the oscillation amplitude and its associated excitation frequency increase.

Journal ArticleDOI
TL;DR: In this article, the bending, vibration and buckling characteristics of functionally graded porous graphene-reinforced nanocomposite curved beams are studied based on a trigonometric shear deformation theory.

Journal ArticleDOI
TL;DR: In this article, the free motion of a fractional capacitor microphone is investigated and the Euler-Lagrange equations are established, and numerical simulations are obtained and dynamical behaviors are numerically discussed.
Abstract: Free motion of a fractional capacitor microphone is investigated in this paper. First, a capacitor microphone is introduced and the Euler-Lagrange equations are established. Due to the fractional derivative's the history independence, the fractional order displacement and electrical charge are used in the equations. Fractional differential equations involve in the right- and left-hand-sided derivatives which is reduced to a boundary value problem. Finally, numerical simulations are obtained and dynamical behaviors are numerically discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors introduce immersed boundary (IB) methods for fluid-structure interactions (FSIs) of rigid and elastic bodies, which impose momentum forcing on an Eulerian mesh to satisfy boundary conditions on the interface between fluid and structure, which enables us to use a non-body conforming grid system for complex-shaped moving bodies.

Journal ArticleDOI
TL;DR: In this paper, a new class of dynamic boundary conditions for the Cahn-Hilliard equation in a rather general setting is proposed, based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation.
Abstract: The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn–Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager’s principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints: conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of the asymptotic limit as $${t\to+\infty}$$ and characterize the stability of local energy minimizers for the system.

Journal ArticleDOI
TL;DR: In this paper, a Keller-Segel type parabolic-elliptic system involving nonlinear diffusion and chemotaxis in a smoothly bounded domain was studied under no-flux boundary conditions.
Abstract: This paper deals with a Keller–Segel type parabolic–elliptic system involving nonlinear diffusion and chemotaxis in a smoothly bounded domain , , under no-flux boundary conditions. The system contains a Fokker–Planck type diffusion with a motility function , . The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when , or , . In addition, we find the conditions on parameters, and , that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.

Journal ArticleDOI
TL;DR: In this paper, a new class of hybrid type fractional differential equations and inclusions via some non-local three-point boundary value conditions are investigated and some examples to illustrate their results are provided.
Abstract: We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our results.

Journal ArticleDOI
TL;DR: Based on the nonlocal strain gradient theory and surface elasticity theory, a unified size-dependent plate model is developed for buckling analysis of rectangular nanoplates in this article, which is capable of capturing nonlocal effect, strain gradient effect as well as surface energy effects simultaneously.