Terms Of Boundary Values Research Articles

A serendipity fully discrete div-div complex on polygonal meshes

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré–Korn-type inequalities for hybrid fields.

On Some Class of Normal Differential Operators for First Order

In this work, we construct the minimal and maximal operators generated by linear differential-operator expression for first order in the Hilbert space of vector-functions on finite symmetric interval. Then, deficiency indices of the minimal operator will be calculated and the space of boundary values of this operator will be constructed. By using of Calkin-Gorbachuk method, the general representation of all normal extensions of the formally normal minimal operator in terms of boundary values will also be established. Moreover we explore the spectrum structure of these extensions.

Evolution equations on time-dependent intervals

Abstract We study initial boundary value problems for linear evolution partial differential equations posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterize the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schrödinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.

A family of Dirichlet–Morrey spaces

ABSTRACTTo each weighted Dirichlet space , 0<p<1, we associate a family of Morrey-type spaces , , constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.

An iterative spatial-stepping numerical method for linear elliptic PDEs using the Unified Transform

A method for solving boundary value problems usually referred to as the unified transform or the Fokas method was introduced in the late nineties. A crucial role in this method is played by the so-called global relation, which characterizes the associated generalized Dirichlet-to-Neumann map, which can be used to determine the unknown boundary values in terms of the given boundary data. Then, the solution can be expressed via an integral formulated in the complex Fourier plane. This method can be considered as the spectral analogue of the classical Boundary Integral Method, which is formulated in the physical plane. Recently, it has been realized that the numerical implementation of the unified transform leads to a collocation-type method: this involves expanding the unknown boundary values in terms of appropriate basis functions and choosing a suitable set of complex collocation points. Herewith, an iterative spatial-stepping algorithm for computing the solution of a linear PDE in the interior of a convex polygon, is presented. The starting point of this algorithm is the analysis of the approximate global relation and its numerical solution. In more details, the interior of the polygon is partitioned into smaller concentric polygonal domains where the solution is computed recursively starting from the boundaries and proceeding towards the center. The solution of each inner polygon is computed by a spatial-stepping scheme using the Dirichlet and Neumann values of the respective outer polygon, which are computed via the approximate global relation.

The Ablowitz–Ladik system on a finite set of integers

We show how to solve initial-boundary value problems for integrable nonlinear differential–difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The implementation of this method to the Ablowitz–Ladik system yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem, which has a jump matrix with explicit -dependence involving certain functions referred to as spectral functions. Some of these functions are defined in terms of the initial value, while the remaining spectral functions are defined in terms of two sets of boundary values. These spectral functions are not independent but satisfy an algebraic relation called global relation. We analyze the global relation to characterize the unknown boundary values in terms of the given initial and boundary values. We also discuss the linearizable boundary conditions.

Linear Elliptic PDEs in a Cylindrical Domain with a Polygonal Cross‐Section

Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well‐posed boundary value problem (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late 1990s. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified Helmholtz, and Helmholtz equations, formulated in a three‐dimensional cylindrical domain with a polygonal cross‐section.

Initial‐Boundary Value Problem for Integrable Nonlinear Evolution Equation with 3 × 3 Lax Pairs on the Interval

We present an approach for analyzing initial‐boundary value problems which are formulated on the finite interval (, where L is a positive constant) for integrable equation whose Lax pairs involve 3 × 3 matrices. Boundary value problems for integrable nonlinear evolution partial differential equations (PDEs) can be analyzed by the unified method introduced by Fokas and developed by him and his collaborators. In this paper, we show that the solution can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem (RHP). The relevant jump matrices are explicitly given in terms of the three matrix‐value spectral functions , and , which in turn are defined in terms of the initial values, boundary values at , and boundary values at , respectively. However, these spectral functions are not independent; they satisfy a global relation. Here, we show that the characterization of the unknown boundary values in terms of the given initial and boundary data is explicitly described for a nonlinear evolution PDE defined on the interval. Also, we show that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained for the case of boundary value problems formulated on the half‐line.

On the spectrums of some class of selfadjoint singular differential operators

In this work, based on the Everitt-Zettl and Calkin-Gorbachuk methods in terms of boundary values all self adjoint extensions of the minimal operator generated by some linear singular multipoint symmetric differential operator expression for first order in the direct sum of Hilbert spaces of vector functions on the right semi-axis are described. Later structure of the spectrumof these extensions is investigated

Improved boundary values of ocean wave fields using a data assimilation scheme

The objective of this study was to assess the accuracy of numerical wave forecasts through data assimilation of boundary values. A sequential data assimilation scheme was adopted to utilise altimeter data. The performance of the system in terms of operational applications, specifically for typhoon waves, was investigated. Two typhoons that occurred in 2006 around Taiwan (Kaemi and Shanshan) were used as case studies. The proposed data assimilation increased the forecast accuracy of the boundary values in terms of the wave parameters, such as the wave heights and periods. The results showed that after the assimilation, the assimilation model was significantly improved, especially the peak value of the wave and time of occurrence of the peak value.

A numerical technique for linear elliptic partial differential equations in polygonal domains.

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

The Poisson and the Biharmonic Equations in the Interior of a Convex Polygon

A new method for analyzing linear elliptic partial differential equations in the interior of a convex polygon was developed in the late 1990s. This method does not rely on the classical approach of separation of variables and on the use of classical integral transforms and therefore is well suited for the investigation of the biharmonic equation. Here, we present a novel integral representation of the solution of the biharmonic equation in the interior of a convex polygon. This representation contains certain free parameters and therefore is more general than the one presented in [1]. For a given boundary value problem, by choosing these free parameters appropriately, one can obtain the simplest possible representation for the solution. This representation still involves certain unknown boundary values, thus for this formula to become effective it is necessary to characterize the unknown boundary values in terms of the given boundary conditions. This requires the investigation of certain relations refereed to as the global relations. A general approach for analyzing these relations is illustrated by solving several problems formulated in the interior of a semistrip. In addition, for completeness, similar results are presented for the Poisson equation by employing an integral representation for the Laplace equation which is more general than the one derived in the late 1990s.

Обратимость линейных отношений, порожденных интегральным уравнением с операторными мерами

We investigate linear relations generated by an integral equation with operator measures on a segment in the infinite-dimensional case. In terms of boundary values, we obtain necessary and sufficient conditions under which these relations S possess the properties: S is a closed relation; S is an invertible relation; the kernel of S is finite-dimensional; the range of S is closed; S is a continuously invertible relation and others. The results are applied to a system of integral equations becoming a quasi-differential equation whenever the operator measures are absolutely continuous as well as to an integral equation with multi-valued impulse action.

Selfadjoint extensions of a singular differential operator

In this work, firstly in the Hilbert space of vector-functions all selfadjoint extensions of the minimal operator generated by linear singular symmetric differential expression with a selfadjoint operator coefficient A in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.

Dynamic Response of Unlined Circular Tunnels Subjected to Internal Explosion

Due to the development of urban areas, underground structures are increasingly needed. Therefore, not only it is likely to have two tunnels located close to each other, but also it happens frequently.Because of the complexity of realistic problems, closed form analytical solutions are not accessible for them. However, latest advances in computational techniques have made numerical approaches more practical for realistic problems. Boundary element method (BEM) is one of such approaches. This technique formulates the problem in terms of boundary values. The main advantage of BEM, especially in comparison with finite element and finite difference methods, is that the discretization is only applied to the boundary, thus reducing the volume of modeling and the number of unknown variables. Moreover, the radiation condition at infinity is exactly satisfied in this method which is very striking for wave propagation problems. First the formulation of boundary element method in the time domain is presented and then a parametric study on the dynamic response of unlined circular tunnels subjected to internal explosion is done. Depth of tunnels, vertical and horizontal distance between two tunnels, diameter of tunnels, and other parameters related to geometry are important factors which can affect displacements and stresses of tunnels under explosive loading. In the present study, effect of two parameters has been investigated: changing the depth of the tunnel that is subjected to internal explosion (case A) and changing the horizontal distance between two shallow tunnels subjected to explosion (case B). Radial displacements and circumferential stresses are the results obtained. Case A- Tunnel depth: The tunnel behavior in depth of H=50R is alike to behavior of same tunnel in infinite depth. Also, radial displacement in upper point of tunnel in depth of H=2R is about 2.7 times of the same tunnel in infinite depth. Circumferential stress in points placed on horizontal diameter of tunnel in depth of H=2R is about 1.8 times of the same tunnel in infinite depth. Case B- Tunnels distance: Two tunnels with horizontal distance of greater than L=50R don’t affect each other. When horizontal distance parameter of two tunnels is L=2.5R then the maximum normalized displacement of the side point of the loaded tunnel, is approximately 2.15 times of the same tunnels with distance of infinite. Also, when horizontal distance of two tunnels is L=2.5R, circumferential stress in lower point of the loaded tunnel, which is normalized with static stress, is approximately 1.2 times of the same tunnels with distance of infinite. Normalized results calculated here display the importance of the geometric parameters on the behavior of unlined tunnels subjected to internal explosion and provide suitable criteria for displacement design of shallow tunnels that are subjected to internal explosion. The accuracy of the present study for the computation of radial displacements and circumferential stresses is verified by solving several problems.

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q t and q xxx have the same sign (KdVI) or two boundary conditions if q t and q xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q x (0,t),q xx (0,t)} and {q xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ (t)(t,k), where Φ (t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ (t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ (t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ (t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.

3D diffraction modeling of singly scattered acoustic wavefields based on the combination of surface integral propagators and transmission operators

We present an improved method for modeling 3D acoustic wavefields scattered at smooth curved interfaces. The approach is based on a high-frequency approximation of surface integral propagators and a correct description of their boundary values in terms of transmission operators. The main improvement is a uniform local approximation of these operators in the form of effective reflection and transmission coefficients. We show that the effective coefficients represent a generalization of the plane-wave coefficients widely used in conventional seismic modeling, even for the case of curved reflectors, nonplanar wavefronts, and finite frequencies. The proposed method is capable of producing complex wave phenomenas, such as caustics, edge diffractions, and head waves. Seismograms modeled for even simple models reveal significant errors implicit in the plane-wave approximation. Comparison of modeling based on effective coefficients with the analytic solution reveals errors less than 4% in peak amplitude at seismic frequencies.

Computational study of the effect of finger width and aspect ratios for the electrostatic levitating force of MEMS combdrive

Since the width ratio between movable and fixed fingers, and the aspect ratio between the height and width of fingers, can play very important roles for combdrive levitation control, computational study of variations in those parameters for electrostatic levitating force acting on the movable finger is indispensable for MEMS performance. For diverse finger width and aspect ratios of MEMS combdrive design, the BEM has become a better method than the domain-type FEM because BEM can provide a complete solution in terms of boundary values only, with substantial saving in modeling effort. DBEM still has some advantages over conventional BEM for singularity, so the DBEM was used to simulate the fringing of field around the edges of the fixed finger and movable finger of MEMS combdrive for diverse finger width and aspect ratios. Results show that the less the finger width ratio is, the larger the levitating force acting is. Furthermore, the levitating force becomes more dominant as the aspect ratio increases, but it will be kept constant while the aspect ratio becomes larger.

Numerical studies of variations in the gap and finger width ratio and travelled distance for the driving force of a radio-frequency microelectromechanical system device using the dual boundary element method

For the comb-drive design of microelectromechanical systems (MEMSs), the driving force due to the electrostatic field is very important, and an accurate electrostatic analysis is essential and indispensable. For various gaps, finger width ratios and travelled distances of the comb drive of MEMSs, the dual boundary element method (DBEM) has become a better method than the domain-type finite element method because the DBEM can provide a complete solution in terms of boundary values, with substantial saving in modelling effort. In this article, the DBEM is used to simulate the fringing field around the edges of the fixed and movable fingers of the comb drive of an MEMS for diverse design cases, and many electrostatic problems for typical comb drive designs of MEMSs are analysed, investigated and compared with a widely used approximate method. Results show that the driving force is obviously dependent on the travelled distance, and the approximate method cannot work well for all travelled positions because there is an apparent error (not less than 10 per cent), especially at the beginning and ends of the range of travel. In addition, the smaller the gap between movable and fixed fingers, the larger the driving force is, and the error of approximate method also becomes more and more predominant as the gap decreases. The results also demonstrate that the difference between the DBEM and the approximate method effect due to finger width ratio is very small. Using the DBEM presented in this article, an accurate and reasonable electrostatic field can be obtained, and the follow-up control method of driving force for the comb drive of an MEMS can be implemented more precisely.

FEM versus BEM - Efficaciously modeling exterior electrostatic problems with singularity for electron devices

For the performance of electron devices, an accurate electrostatic analysis is essential, and the boundary element method (BEM) has become a better method than the domain-type finite element method (FEM) because BEM can provide a complete solution in terms of boundary values only with substantial savings in modeling effort for the variable design stage. But, for exterior problems with singularity arising from the degenerate boundary (e.g., the edge of parallel-plate capacitor), the dual BEM becomes one of the most efficacious and robust tools for simulating the fringing field near the edge of electron devices because no laborious artificial boundary technique was needed like conventional BEM. After the fringing field is known well, the charge and capacitance of electron devices can be accurately calculated, and we can also understand the minimum allowable data of dielectric strength for keeping off dielectric breakdown. Electrostatics, as used here, involves charges in motion as well as at rest. Generally, there are five fundamental quantities (voltage, charge, current, capacitance, and resistance) in electrostatics that are involved in almost all applications. For most electrical engineers, voltage, or electromotive force (EMF) is the most important one. Electrostatic problems generally play a very important role in improving the performance and reliability of electron devices in the design stage (Cheng, 1989). Although we all understand that the beginning of electrostatic theory is believed to have occurred several centuries before, the first meaningful application, the commercially electrostatic precipitator, was just installed by Cottrell in 1907. Besides two major present technologies from 1907 to now, electrostatic precipitation and electrostatic coating, Castle also suggested that there would be several new industrial applications to come from developments in the fields of micro electromechanical systems (MEMS), biotechnology, ultrafine particles, nanotechnology, and space for the future applications of electrostatics. Because electrostatics still affects the performance of MEMS and electron devices critically nowadays, how to accurately obtain the electric potential V and electric field intensity E becomes especially important for engineers. We all know that scientists and engineers usually use several techniques in solving continuum or field problems. Loosely speaking, these techniques can be classified as experimental, theoretical (or analytical), or numerical.