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Ordinary and partial differential equations with operators containing the p -Laplace operator, the so called ϕ -Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered. [ABSTRACT FROM AUTHOR]

Abstract: In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum forces solutions to have higher integrability as well. In the present paper, we consider -obstacle problems of the nonhomogeneous anisotropic elliptic equations under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe. [Copyright &y& Elsevier]

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation ∂ t u − Δ e − Δ u + e − Δ u − 1 = 0. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects [6]. Our investigations reveal that we must control the size of both ‖ e − Δ u (x , 0) ‖ W 2 , 2 (Ω) and ‖ e Δ u (x , 0) ‖ ∞ , Ω suitably to achieve our results. Related results in [8,10] were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity. [ABSTRACT FROM AUTHOR]

• A system consisting of the Euler-Bernoulli beam subjected to tendon-induced axial loading is studied. • The system is modelled using a set of partial differential equations derived by Hamilton's principle. • The model is thoroughly experimentally validated for a wide range of loadings and configurations. • The effect of the tendon tension and its mass on the modal properties and frequency loci veering are studied. • Further numerical studies show the effect of the tendon on the torsional modes and structural stability. Theoretical and experimental modal analysis of the system consisting of the Euler-Bernoulli beam axially loaded by a tendon is studied in the paper. The beam-tendon system is modelled using a set of partial differential equations derived by Hamilton's principle and the coupling between the beam and the tendon is ensured by the boundary conditions. Theoretical modal analysis is conducted using a boundary value problem solver and the results are thoroughly experimentally validated using a bench-top experiment. In particular, the effect of the tendon tension on the modal properties of the system is studied. It is found that by increasing the tension, the natural frequencies of the beam decrease while the natural frequencies of the tendon increase. It is also shown that these two sets of modes interact with each other through frequency loci veering. The effect of the tendon mass is also experimentally and numerically studied and it is shown that lighter tendon produces fewer vibration modes in the studied frequency region. Two further numerical studies are conducted to demonstrate the effect of the tendon on the torsional modes of the beam, and to study the structural stability. Overall, an excellent agreement between the numerical and experimental results is obtained, giving the confidence in the derived theoretical model. [ABSTRACT FROM AUTHOR]

• The discrete LS method is developed for two-dimensional time-fractional PDEs on irregular domains. • No mesh division is needed with easy implementation and high accuracy. • The LS method is easy to be extended to three-dimensional problems. • Regularization is applied to reduce the condition number of the coefficient matrix. In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we mainly investigate the critical points associated to solutions u of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain Ω in R 2. Based on the fine analysis about the distribution of connected components of a super-level set { x ∈ Ω : u (x) > t } for any min ∂ Ω u (x) < t < max ∂ Ω u (x) , we obtain the geometric structure of interior critical points of u. Precisely, when Ω is simply connected, we develop a new method to prove Σ i = 1 k m i + 1 = N , where m 1 , ⋯ , m k are the respective multiplicities of interior critical points x 1 , ⋯ , x k of u and N is the number of global maximal points of u on ∂Ω. When Ω is an annular domain with the interior boundary γ I and the external boundary γ E , where u | γ I = H , u | γ E = ψ (x) and ψ (x) has N local (global) maximal points on γ E. For the case ψ (x) ≥ H or ψ (x) ≤ H or min γ E ψ (x) < H < max γ E ψ (x) , we show that Σ i = 1 k m i ≤ N (either Σ i = 1 k m i = N or Σ i = 1 k m i + 1 = N). [ABSTRACT FROM AUTHOR]

In this paper, we propose the boundary control for undesired vibrations suppression with output constraint of the flapping-wing robotic aircraft (FWRA). We also present the dynamics of the flexible wing of FWRA with governing equations and boundary conditions, which are partial differential equations (PDEs) and ordinary differential equations (ODEs), respectively. An energy-based barrier Lyapunov function is introduced to analyze the system stability and prevent violation of output constraint. With the effect of the proposed boundary controller, distributed states of the system remain in the constrained spaces. Then the IBLF-based boundary controls are proposed to assess the stability of the FWRA in the presence of output constraint. [ABSTRACT FROM AUTHOR]

In this paper, we firstly introduce a complicated system obtained by mixing a nonlinear evolutionary partial differential equation and a mixed variational inequality in infinite dimensional Banach spaces in the case where the set of constraints is not necessarily bounded and the problem is driven by nonlocal boundary conditions, which is called partial differential variational inequality ((PDVI), for short). Then, we show that the solution set of the mixed variational inequality involved in problem (PDVI) is nonempty, bounded, closed and convex. Moreover, the upper semicontinuity and measurability properties for set-valued mapping U : [ 0 , T ] × E 2 → C b v ( E 1 ) (see (3.7) , below) are also established. Finally, several existence results for (PDVI) are obtained by using a fixed point theorem for condensing set-valued operators and theory of measure of noncompactness. [ABSTRACT FROM AUTHOR]

This paper concerns a continuous-time, finite horizon, optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty. We assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity. The objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon. Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint. The problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies, respectively. We investigate behaviors of free boundaries, study regularities of the value function, and give optimal investment and maintenance policies. As we know, this is a first integral result for an investment–maintenance problem with a finite time horizon due to use of partial differential equation (PDE) technique. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry–exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when “formal” canard solutions exist. [Copyright &y& Elsevier]

This paper is concerned with the asymptotic behavior of solution to the Cahn–Hilliard equation subject to the following dynamic boundary conditions: and the initial condition where Ω is a bounded domain in Rn (n⩽3) with smooth boundary Γ , and Γs>0, σs>0, gs>0, hs are given constants; Δ|| is the tangential Laplacian operator, and ν is the outward normal direction to the boundary.This problem has been considered in the recent paper by Racke and Zheng (Adv. Differential Equations 8 (1) (2003) 83) where the global existence and uniqueness were proved. In a very recent manuscript by Prüss, Racke and Zheng (Konstanzer Schrift. Math. Inform. 189 (2003)) the results on existence of global attractor and maximal regularity of solution have been obtained. In this paper, convergence of solution of this problem to an equilibrium, as time goes to infinity, is proved. [Copyright &y& Elsevier]

The objective of this paper is the extension and application of a newly-developed Green's function Monte Carlo (GFMC) algorithm to the estimation of the derivative of the solution of the one-dimensional (1D) Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional GFMC approach for the solution of partial differential equations subject to these boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. My work, inspired by the work of K.K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. My past work has involved the application of this algorithm to the estimation of the solution of the 1D Laplace equation, the Helmholtz equation and the modified Helmholtz equation. In this work, this algorithm has been adapted to the estimation of the derivative of the solution which is a very important development. In the traditional approach involving reflection, to estimate the derivative at a certain number of points, one has to a priori estimate the solution at a larger number of points. In the case of a one-dimensional problem for instance, to obtain the derivative of the solution at a point, one has to obtain the solution at two points, one on each side of the point of interest. These points have to be close enough so that the validity of the first-order approximation for the derivative operator is justified and at the same time, the actual difference between the solutions at these two points has to be at least an order of magnitude higher than the statistical error in the estimation of the solution, thus requiring a significantly larger number of random-walks than that required for the estimation of the solution. In this new approach, identical amount of computational resources is needed irrespective of if we are trying to estimate the solution or the derivative. This becomes very significant in electromagnetic problems where the scalar/vector potential is the unknown in the PDE of interest, but the quantity of interest is the electric/magnetic field or in heat conduction problems where temperature of an object is the unknown variable in a PDE, but the quantity of interest is the spatial/temporal variation of the temperature. In this work, this algorithm is applied to the estimation of the derivative of the solution of the 1D Helmholtz equation which is the frequency domain version of both Maxwell's equations and the heat conduction equation. As a result the algorithm is an important first step in the development of computationally efficient GFMC algorithms for Neumann and mixed boundary condition problems. The numerical results have been validated by an exact, analytical solution and very good agreement has been observed. The long-term goal of this research is the application of this methodology to the numerical solution of the F region ionization problem in space plasma modeling. [ABSTRACT FROM AUTHOR]

In this paper, we establish the second order estimates for solutions of the first initial–boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can work for a wide range of fully nonlinear PDEs under very general conditions. [ABSTRACT FROM AUTHOR]

An alternative vector potential formulation is used to solve the Navier–Stokes (N–S) equations in 3D incompressible viscous flow problems with and without through-flow boundaries. Difficulties of the vector potential formulation include the implementation of boundary conditions for through-flow boundaries and the numerical treatment of fourth-order partial differential equations. The advantages on the other hand are the automatic satisfaction of the continuity equation; and pressure is decoupled from the velocity. The objective of this paper is to introduce the appropriate gauge and boundary conditions on the vector potential formulation by a localized meshless method. To handle the divergence-free property, a Coulomb gauge condition is enforced on the vector potential to ensure its existence and uniqueness mathematically. We further improve the algorithm to through-flow problems for the boundary conditions of vector potential by introducing the concept of Stokes' theorem. Based on this innovation, there is no need to include an additional variable to tackle the through-flow fields. This process will greatly simplify the imposition of boundary conditions by the vector potential approach. Under certain conditions, the coupled fourth-order partial differential equations can be easily solved by using this meshless local differential quadrature (LDQ) method. Due to the LDQ capability to deal with the high order differential equations, this algorithm is very attractive to solve this fourth-order vector potential formulation for the N–S equations as comparing to the conventional numerical schemes such as finite element or finite difference methods. The proposed vector potential formulation is simpler and has improved accuracy and efficiency compared to other pressure-free or pressure-coupled algorithms. This investigation can be regarded as the first complete study to obtain the N–S solutions by vector potential formulation through a LDQ method. Two classic 3D benchmark problems, lid-driven cavity and backward-facing step flows, are numerically solved to examine the feasibility of the improved algorithm. Results show the flexibility of the proposed vector potential formulation for the 3D Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Atherosclerosis is a leading cause of death in the United States and worldwide; it originates from a plaque which builds up in the artery. In this paper, we consider a simplified model of plaque growth involving LDL and HDL cholesterols, macrophages and foam cells, which satisfy a coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow. We prove that there exist small radially symmetric stationary plaques and establish a sharp condition that ensures their stability. We also determine necessary and sufficient conditions under which a small initial plaque will shrink and disappear, or persist for all times. [ABSTRACT FROM AUTHOR]

It is known that an analytical expression of solutions to nonlinear wave equations in higher dimensions may be helpful to recognizing new nonlinear physical phenomena and developing novel numerical integrators. Therefore, in this paper, we present an analytical expression for the initial–boundary value problem of higher-dimensional nonlinear wave equations U t t ( X , t ) − a 2 Δ U ( X , t ) = f ( U ( X , t ) , U t ( X , t ) ) , U ( X , t 0 ) = U 0 ( X ) , U t ( X , t 0 ) = U 1 ( X ) , posed on a bounded domain Ω ⊂ R d for d ≥ 1 and equipped with the Robin boundary conditions ∇ U ⋅ n + λ U = β ( X , t ) , X ∈ ∂ Ω , where n is the unit outward normal vector at the boundary ∂Ω, and λ is a constant. [ABSTRACT FROM AUTHOR]

This paper focuses on designing an observer for a flexible-link manipulator based on the partial differential equation (PDE) dynamic model. The main contribution of this observer׳s novel design is its ability to estimate vibratory, infinite dimensional states requiring only the practical measurement values of boundary positions. The proposed observer is proven asymptotically stable by theoretical analysis. The vibration of the flexible link is eliminated with a PD control. Favorable performance of the closed-loop system is demonstrated by simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time–space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann–Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters θ and ϑ . Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a characteristics-based approach for solving elastic wave problems with time-dependent traction boundary conditions. A generalized mathematical model for this important class of problems is expressed as a set of first-order, linear, hyperbolic partial differential equations. We analyze the mathematical structure of this first-order linear system, verify its hyperbolicity, derive its characteristic form, and deduce its eigenvalues, eigenvectors, and Riemann invariants. The eigenvalues correspond to the wave speeds, while the Riemann invariants are used to construct a solution by the method of characteristics. We benchmark the method of characteristics against several popular modal approaches. Two of these, which we refer to as the concentrated body force method (CBFM) and the homogeneous eigenfunction expansion method (HEEM), were developed to simplify the well-established but tedious Mindlin–Goodman method. To homogenize the boundary conditions and enable modal analysis, the CBFM and HEEM forgo the usual formalism of linear transformations (à la Mindlin–Goodman) in favor of intuitive modeling assumptions and postulated solution structures. We find, however, that these approaches introduce an artificial stress discontinuity at the forced boundary in their reformulated problems. When these reformulated problems are solved by modal analysis, spurious oscillations and significant overshoot, similar to the Gibbs phenomenon, emerge in the stress profile at the artificial discontinuity. We demonstrate that these oscillations and overshoot are physical manifestations of a series solution for stress, obtained from term-by-term differentiation, that is not uniformly convergent, as required by the formalism of mathematical analysis. The method of characteristics solution, on the other hand, is exact to within machine precision, yielding no artificial discontinuities, spurious oscillations, or unphysical overshoot. Unlike the modal approaches, the method of characteristics solves the first-order problem with time-dependent boundary conditions ‘as is’ without any reformulation, restructuring, or postulated solution structures. Further, its solutions require no post-processing: no a posteriori solution treatment like l'Hôpital's rule to accommodate resonance (resonant behavior is inherently captured without the emergence of singularities), no term-by-term differentiation to deduce stress (stress is primitive in a first-order velocity–stress formulation of elastodynamics), and no convergence tests. [Copyright &y& Elsevier]

Abstract: We develop mathematical framework and computational tools for calculating frequency responses of linear time-invariant PDEs in which an independent spatial variable belongs to a compact interval. In conventional studies this computation is done numerically using spatial discretization of differential operators in the evolution equation. In this paper, we introduce an alternative method that avoids the need for finite-dimensional approximation of the underlying operators in the evolution model. This method recasts the frequency response operator as a two point boundary value problem and uses state-of-the-art automatic spectral collocation techniques for solving integral representations of the resulting boundary value problems with accuracy comparable to machine precision. Our approach has two advantages over currently available schemes: first, it avoids numerical instabilities encountered in systems with differential operators of high order and, second, it alleviates difficulty in implementing boundary conditions. We provide examples from Newtonian and viscoelastic fluid dynamics to illustrate utility of the proposed method. [Copyright &y& Elsevier]

Abstract: Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both and norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order to for the solution itself in norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order to in the norm for or Lipschitz continuous interfaces associated with a or continuous solution. [Copyright &y& Elsevier]

Abstract: The method of difference potentials generalizes the method of Calderon’s operators from PDEs to arbitrary difference equations and systems. It offers several key advantages, such as the capability of handling boundaries/interfaces that are not aligned with the discretization grid, variable coefficients, and nonstandard boundary conditions. In doing so, the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid. Previously, we have applied the method of difference potentials to solving several variable coefficient interior Helmholtz problems with fourth and sixth order accuracy. We have employed compact finite difference schemes as a core discretization methodology. Those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself. Numerical experiments corroborate the high order accuracy of our method for variable coefficients, regular grids, and non-conforming boundaries. In the current paper, we extend the previously developed methodology to exterior problems. We present a complete theoretical analysis of the algorithm, as well as the results of a series of numerical simulations. Specifically, we study the scattering of time-harmonic waves about smooth shapes, subject to various boundary conditions. We also solve the transmission/scattering problems, in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside. In all the cases, our methodology guarantees high order accuracy for regular grids and non-conforming boundaries and interfaces. [Copyright &y& Elsevier]

Abstract: Let be the unit ball centered at the origin in . In this paper we study the following problem where , . We prove that for any , if is small enough, the above problem has a positive solution concentrating at distinct points tending to the boundary of as goes to . [Copyright &y& Elsevier]

Abstract: In this paper we consider the second-order parabolic partial differential systems and elliptic systems with space domains. We prove the existence and uniqueness results in the Sobolev spaces with weights. In this solution spaces the derivatives of solutions are allowed to blow up near the boundary and also the coefficients of the systems may oscillate to a great extent or blow up near the boundary. [Copyright &y& Elsevier]

Abstract: In this paper, the nonexistence of parabolic boundary points of infinite type of certain domains in is showed. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the 1-D compressible Euler–Poisson equations with moving physical vacuum boundary condition. It is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local well-posedness of classical solutions is established in the case of the adiabatic index . [Copyright &y& Elsevier]

Abstract: This paper concerns a singular control problem whose value function is governed by a time-dependent HJB equation with gradient constraints. The method is to transform a two-dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies. Moreover we analyze the behaviors of the free boundary surfaces. The main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter. [Copyright &y& Elsevier]

Abstract: In this paper, the dynamic instability of a shear deformable composite plate subjected to periodic non-uniform in-plane loading is studied for four sets of boundary conditions. The static component and the dynamic component of the applied periodic in-plane loading are assumed to vary according to either parabolic or linear distributions. Initially, the plate membrane problem is solved using the Ritz method to evaluate the plate in-plane stress distributions within the prebuckling range due to the applied non-uniform in-plane edge loading. Subsequently using the evaluated stress distribution within the plate, the equations governing the plate instability boundaries are formulated via Hamilton''s variational principle. Employing Galerkin''s method, these partial differential equations are reduced into a set of ordinary differential equations (Mathieu type of equations) describing the plate dynamic instability behaviour. Following Bolotin''s method, the instability regions are determined from the boundaries of instability, which represents the periodic solution of the differential equations with period T and 2T to the Mathieu equations. The instability regions are determined for uniform, linear and parabolic dynamic in-plane loads using first-order and second-order approximations. Numerical results are also presented to bring out the effects of span to thickness ratio, shear deformation, aspect ratio, boundary conditions and static load factor on the instability regions. [Copyright &y& Elsevier]

Abstract: The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency–response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail. [Copyright &y& Elsevier]

Abstract: The paper addresses the problem of constructing non-reflecting boundary conditions for two types of one dimensional evolution equations, namely, the cubic nonlinear Schrödinger (NLS) equation, with , and the equation obtained by letting . The usual restriction of compact support of the initial data is relaxed by allowing it to have a constant amplitude along with a linear phase variation outside a compact domain. We adapt the pseudo-differential approach developed by Antoine et al. (2006) for the NLS equation to the second type of evolution equation, and further, extend the scheme to the aforementioned class of initial data for both of the equations. In addition, we discuss efficient numerical implementation of our scheme and produce the results of several numerical experiments demonstrating its effectiveness. [Copyright &y& Elsevier]

Abstract: The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of Coron et al. (2010) , the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in . We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the -sense with , the difference between the actual out-flux and a forecast demand over a fixed time period. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a class of generalized nonlinear parabolic partial differential equations, which are related to the Malik–Perona model in image analysis. [Copyright &y& Elsevier]

Abstract: This paper is devoted to general balance laws (with a possibly non-local source term) with a non-characteristic boundary. Basic well posedness results are obtained. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of a size-structured model with Ricker type birth function as well as random fluctuation in the growth process. The complete model takes the form of a reaction–diffusion equation with a nonlinear and nonlocal boundary condition. We study some dynamical properties of the model by using the theory of integrated semigroups. It is shown that Hopf bifurcation occurs at a positive steady state of the model. This problem is new and is related to the center manifold theory developed recently in [P. Magal, S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age-structured models, Mem. Amer. Math. Soc., in press] for semilinear equation with non-densely defined operators. [Copyright &y& Elsevier]

Abstract: This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena. [Copyright &y& Elsevier]

Abstract: We develop in this paper a moving mesh spectral method for the phase-field model of two-phase flows with non-periodic boundary conditions. The method is based on a variational moving mesh PDE for the phase function, coupled with efficient semi-implicit treatments for advancing the mesh function, the phase function and the velocity and pressure in a decoupled manner. Ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider one-dimensional nonlinear Schrödinger equation on under the boundary conditions , , , for . It is proved that for a prescribed and analytic positive potential , the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the mathematical analysis of a thermo-mechanical model describing phase transitions in terms of the entropy and order structure balance law. We consider a macroscopic description of the phenomenon and make a presentation of the model. Then, the initial and boundary value problem is addressed for the related PDE system, which contains some nonlinear and singular terms with respect to the temperature variable. Existence of the solution is shown along with the boundedness of both phase variable χ and absolute temperature ϑ. Finally, uniqueness is proved in the framework of a source term depending Lipschitz continuously on ϑ. [Copyright &y& Elsevier]

Abstract: In this paper was considered the scattering problem for the nonstationary Dirac-type systems of n () equations on the half-plane when the system has () incident and () scattered waves. In case is divisible by , we formulate the inverse scattering problem for a nonstationary Dirac-type system when considering m () scattering problems on the half-plane with the same incident waves but different boundary conditions. Moreover, the scattering operator for the nonstationary Dirac-type system on half-plane was defined and unique restoration of the potential with respect to the scattering operator was proved. [Copyright &y& Elsevier]

Abstract: In this paper we consider the nonlinear viscoelastic equation with initial conditions and Dirichlet boundary condition. Under some appropriate assumption on g, by introducing a potential well we establish the global existence result for the equation above. Furthermore, we investigate explicit exponential energy decay of the viscoelastic wave equation under the potential well. [Copyright &y& Elsevier]

[Display omitted] • A system consisting of the Euler-Bernoulli beam subjected to tendon loading is studied. • The tendon is attached to the beam at the tip and in several locations • Computational free vibration analysis is conducted and validated. • The effect of the number and location of the attachment points is evaluated. • Structural stability of the beam-tendon system is investigated. Free vibration and stability analysis of the system consisting of the Euler–Bernoulli beam axially loaded by a tendon is studied in the paper. The tendon is attached to the cantilever beam at the tip as well as in several spanwise locations using mechanical attachment points. This novel beam-tendon system is modelled using a set of partial differential equations and the coupling between the beam and the tendon is ensured by the boundary and continuity conditions. Computational free vibration analysis is conducted using a boundary value problem solver and the results are thoroughly experimentally validated using a bench-top experiment. In particular, the effect of the number of the attachment points and their location on the frequency-loading diagram of the beam-tendon system is investigated for the first time. It is found that the applied axial load causes a frequency shift of beam-dominated natural frequencies, and frequency loci veering between beam-dominated and tendon-dominated modes. Both of these features are shown to be dependent on a number and location of the attachment points. In addition, the effect of the attachment points on the structural stability of the system is numerically studied and it is observed that the critical force of the system with the intermittently attached tendon is always higher than for the system with no attachment points. [ABSTRACT FROM AUTHOR]

Solving approximate energy flow partial differential equations for structural elements can be done analytically (energy flow analysis—EFA) or using finite element approximations (energy finite element method—EFEM). In this paper, energy equations are solved using the spectral element method (SEM). This new approach, which is called the energy spectral element method (ESEM), can be applied to predict the distribution of the energy flow and energy density of built-up structures at high frequencies. Energy spectral element method is a matrix formulation based on the general solution of the partial differential equations for energy density in structural vibrations such as longitudinal and transversal vibrations of frames. A spectral energy element can be shown to be equivalent to an infinite number of energy finite elements. In this work, numerical models involving coupled rods and beams are generated by energy spectral element method, and the results obtained are compared with energy densities computed from the displacement fields predicted by the SEM to solve the conventional boundary value problem. [Copyright &y& Elsevier]

Abstract: In this paper we prove that the initial value problem of the OST equation (, ), where and denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space when , and globally well-posed in when . [Copyright &y& Elsevier]

Abstract: This paper proposes a unified analytical solution of the wave equation governing the propagation of the longitudinal stress wave in an elastic rod. Within a single formula derived by using the Laplace transform and inverse transform, the solution covers the contributions of the external excitations, the nonzero initial conditions and the inhomogeneous boundary conditions altogether, including such boundary conditions that the dependent variables at the ends of the rod are restricted with an equation. The proposed formula, particularly suitable for transient problems, could be regarded as an exact interpolation function in the time domain, provided the rod works as a component in a complex system, etc. Four examples are presented to show the applications of the solution. [Copyright &y& Elsevier]

Abstract: This paper is a continuation of [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2007) 1423–1449] and [N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl. (2008), in press], where we analyzed nonlinear parabolic problem on a bounded domain Ω of with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. Here u is modeled to describe dynamic deflection of the elastic membrane. When a voltage—represented here by λ—is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) must occur when it exceeds a certain critical value (pull-in voltage), creating a so-called “pull-in instability” which greatly affects the design of many devices. In an effort to achieve better MEMS design, the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile . In this work, some a priori estimates of touchdown behavior are established, based on which the refined touchdown profiles are obtained by adapting self-similar method and center manifold analysis. Applying various analytical and numerical techniques, some properties of touchdown set—such as compactness, location and shape—are also discussed for different classes of varying permittivity profiles. [Copyright &y& Elsevier]

Abstract: The aim of this paper consists in to give sufficient conditions to ensure the existence and location of the solutions of a nonlinear fully fourth-order equation with functional boundary conditions. The arguments make use of the upper and lower solutions method, a ϕ-Laplacian operator and a fixed point theorem. An application of the beam theory to a nonlinear continuous model of the human spine allows to estimate its deformation under some loading forces. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the local side of the bifurcation from infinity at the first eigenvalue of several elliptic operators. We underline that the key to decide the local behavior of the bifurcation lies in the sign of certain integral involving all the values of the nonlinearity. [Copyright &y& Elsevier]

Abstract: In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55–81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of . This paper extends this study to the resolution of a mixed exterior Laplace''s problem. Here, we give existence, unicity and regularity results in ''s theory with , in weighted Sobolev spaces. [Copyright &y& Elsevier]

Abstract: This paper presents a partial classification for type-changing symplectic Monge–Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems. [Copyright &y& Elsevier]