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Boundary-value Problem (boundary-value + problem)
Selected AbstractsWave diffraction by a strip grating: the two-straight line approachMATHEMATISCHE NACHRICHTEN, Issue 1 2004M. A. Bastos Abstract Boundary-value problems of wave diffraction by a periodic strip grating are associated with a Toeplitz operator acting on a space of functions defined on a two-straight line contour. Simple formulas are given for the left inverse of the operator associated with the Neumann boundary-value problem and for the right inverse of the operator associated with the Dirichlet boundary-value problem when the period of the grating is equal to double the width of the strips. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Long-term InHM simulations of hydrologic response and sediment transport for the R-5 catchmentEARTH SURFACE PROCESSES AND LANDFORMS, Issue 9 2007Christopher S. Heppner Abstract The physics-based model known as the Integrated Hydrology Model (InHM) is used to simulate continuous hydrologic response and event-based sediment transport for the R-5 catchment (Oklahoma, USA). For the simulations reported herein the R-5 boundary-value problem was refined, from that reported by Loague et al. (2005), to include (i) an improved conceptualization of the local hydrogeologic setting, (ii) a more accurate topographical representation of the catchment, (iii) improved boundary conditions for surface-water outflow, subsurface-water outflow and evapotranspiration, (iv) improved characterization of surface and subsurface hydraulic parameters and (v) improved initial conditions. The hydrologic-response simulations were conducted in one-year periods, for a total of six years. The sediment-transport simulations were conducted for six selected events. The multi-year water-balance results from the hydrologic-response simulations match the observed aggregate behavior of the catchment. Event hydrographs were generally simulated best for the larger events. Soil-water content was over-estimated during dry periods compared with the observed data. The sediment-transport simulations were more successful in reproducing the total sediment mass than the peak sediment discharge rate. The results from the effort reported here reinforce the contention that comprehensive and detailed datasets are crucial for testing physics-based hydrologic-response models. Copyright © 2007 John Wiley & Sons, Ltd. [source] A dam problem: simulated upstream impacts for a Searsville-like watershedECOHYDROLOGY, Issue 4 2008Christopher S. Heppner Abstract The integrated hydrology model (InHM), a physics-based hydrologic-response model with sediment-transport capabilities, was used to simulate upstream impacts from dam construction/removal for a generalized approximation of the Searsville watershed in Portola Valley, California. Four 10-year simulation scenarios (pre-dam, early dam, current and post-dam) were considered. Each scenario was simulated using the same sequence of synthetically generated rainfall and evapotranspiration. For each scenario the boundary-value problem was constructed based on the available watershed information (e.g. topography, soils, geology, reservoir bathymetry and land use). The results from the simulations are presented in terms of the temporal and spatial characteristics of hydrologic response and sediment transport. The commonalities/differences between the four Searsville-like watershed scenarios are discussed. The effort demonstrates that heuristic physics-based simulation can be a useful tool for the characterization of dam-related impacts at the watershed scale. Copyright © 2008 John Wiley & Sons, Ltd. [source] Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analysesINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2009F. Prunier Abstract The present paper investigates bifurcation analysis based on the second-order work criterion, in the framework of rate-independent constitutive models and rate-independent boundary-value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi-static boundary-value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9:203,274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207,220) and the plasticity limit criterion are revived in order to clarify the application field of the second-order work criterion and to contrast these criteria. The first part of this paper analyses the second-order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (d, = Md,) or in the dual form (d, = Nd,), is discussed. In the second part, the analysis is extended to the boundary-value problems in quasi-static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary-value problem. Copyright © 2008 John Wiley & Sons, Ltd. [source] The response of an elastic half-space under a momentary shear line impulseINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2003Moche Ziv Abstract The response of an ideal elastic half-space to a line-concentrated impulsive vector shear force applied momentarily is obtained by an analytical,numerical computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. The shear force is concentrated along an infinite line, drawn on the surface of the half-space, while being normal to that line as well as to the axis of symmetry of the half-space. An exact loading model is introduced and built into the computational method for this shear force. With this model, a compatibility exists among the prescribed applied force, the geometric decay of the shear stress component at the precursor shear wave, and the boundary conditions of the half-space; in this sense, the source configuration is exact. For the transient boundary-value problem described above, a wave characteristics formulation is presented, where its differential equations are extended to allow for strong discontinuities which occur in the material motion of the half-space. A numerical integration of these extended differential equations is then carried out in a three-dimensional spatiotemporal wavegrid formed by the Cartesian bicharacteristic curves of the wave characteristics formulation. This work is devoted to the construction of the computational method and to the concepts involved therein, whereas the interpretation of the resultant transient deformation of the half-space is presented in a subsequent paper. Copyright © 2003 John Wiley & Sons, Ltd. [source] The formation of dunes, antidunes, and rapidly damping waves in alluvial channelsINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2001L.-H. Huang Abstract Under the effect of a constant current for a long time, a water channel of infinitely long and constant depth interacting with a uniform sandbed of infinite thickness is used to simulate the formation of dunes, antidunes and rapidly damping waves in alluvial channels. The theory of potential flow is applied to the channel flow, while Biot's theory of poroelasticity is adopted to deal with erodible bed material. The governing equations, together with free surface, bed surface, and far field boundary conditions, form a complete boundary-value problem without applying empirical sediment discharge formulas as in conventional researches. The comparison of the present result with Kennedy's (Journal of Fluid Mechanics, 1963; 16: 521,544) instability analysis not only indicates the appropriateness of the present work, but also reveals the advantage of the present study due to its ability to find all kinds of bed forms (including the rapidly damping waves that Kennedy could not find) and of solving for the unclear lagged distance , introduced in Kennedy's work. Copyright © 2001 John Wiley & Sons, Ltd. [source] Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous wallsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010M. Babaelahi Abstract Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two-point boundary-value problem for the similarity function into an initial-value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd. [source] Parallel numerical simulation for the coupled problem of continuous flow electrophoresisINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2007M. Chau Abstract The performance of parallel subdomain method with overlapping is analysed in the case of the 3D coupled boundary-value problem of continuous flow electrophoresis which is governed by Navier,Stokes equations coupled with convection,diffusion and potential equations. Convergence of parallel synchronous and asynchronous iterative algorithms is studied. Comparison between implemented explicit and implicit schemes for the transport equation is made using these algorithms and shows that both methods provide similar results and comparable performances. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical simulation of vortical ideal fluid flow through curved channelINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2003N. P. Moshkin Abstract A numerical algorithm to study the boundary-value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co-ordinate system. The convergence of the finite-difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka,Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two-dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd. [source] Non-self-adjoint boundary-value problem with discontinuous density functionMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2010Murat Ad Abstract We determine spectrum and principal functions of the non-self-adjoint differential operator corresponding to 1-D non-self-adjoint Schrödinger equation with discontinuous density function, provide some sufficient conditions guaranteeing finiteness of eigenvalues and spectral singularities, and introduce the convergence properties of principal functions. Copyright © 2009 John Wiley & Sons, Ltd. [source] Solvability of singular integral system connected with seismic waves around the boreholeMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2006Leïla Alem Abstract We study in this article a boundary-value problem arising in the propagation of waves in an elastic half-space covered by a layer with a vertical borehole. We first show a uniqueness theorem under some restrictions on the solution. For the existence, we use the direct integral equation method. We obtain a singular integral system on the half-line. For the solvability, we reduce this system to an elliptic pseudodifferential equation and establish the Fredholm property. Finally, we compute the index of the associated operator for various values of Poisson's ratio. Copyright © 2006 John Wiley & Sons, Ltd. [source] Asymptotic and spectral properties of operator-valued functions generated by aircraft wing modelMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004A. V. Balakrishnan Abstract The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution,convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. More precisely, the generalized resolvent is a finite-meromorphic function on the complex plane having a branch-cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotical approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd. [source] Complex-distance potential theory, wave equations, and physical waveletsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002Gerald Kaiser Potential theory in ,n is extended to ,n by analytically continuing the Euclidean distance function. The extended Newtonian potential ,(z) is generated by a (non-holomorphic) source distribution ,,(z) extending the usual point source ,(x). With Minkowski space ,n, 1 embedded in ,n+1, the Laplacian ,n+1 restricts to the wave operator ,n,1 in ,n, 1. We show that ,,(z) acts as a propagator generating solutions of the wave equation from their initial values, where the Cauchy data need not be assumed analytic. This generalizes an old result by Garabedian, who established a connection between solutions of the boundary-value problem for ,n+1 and the initial-value problem for ,n,1 provided the boundary data extends holomorphically to the initial data. We relate these results to the physical avelets introduced previously. In the context of Clifford analysis, our methods can be used to extend the Borel,Pompeiu formula from ,n+1 to ,n+1, where its riction to Minkowski space ,n, 1 provides solutions for time-dependent Maxwell and Dirac equations. Copyright © 2002 John Wiley & Sons, Ltd. [source] Wave diffraction by a strip grating: the two-straight line approachMATHEMATISCHE NACHRICHTEN, Issue 1 2004M. A. Bastos Abstract Boundary-value problems of wave diffraction by a periodic strip grating are associated with a Toeplitz operator acting on a space of functions defined on a two-straight line contour. Simple formulas are given for the left inverse of the operator associated with the Neumann boundary-value problem and for the right inverse of the operator associated with the Dirichlet boundary-value problem when the period of the grating is equal to double the width of the strips. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Numerical methods for fourth-order nonlinear elliptic boundary value problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2001C. V. Pao Abstract The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth-order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence-comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two-point boundary-value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347,368, 2001 [source] Paraxial ray methods for anisotropic inhomogeneous mediaGEOPHYSICAL PROSPECTING, Issue 1 2007Tijmen Jan Moser ABSTRACT A new formalism of surface-to-surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user-selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface-to-surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non-eikonal and ray-tangent solutions of the dynamic ray-tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface-to-surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface-to-surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary-value problems of a four-parametric system of paraxial rays, connecting the two surfaces, similarly as the well-known surface-to-surface matrices in isotropic media in ray-centred coordinates. Applications of such boundary-value problems include the two-point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more. [source] Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analysesINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 9 2009F. Prunier Abstract The present paper investigates bifurcation analysis based on the second-order work criterion, in the framework of rate-independent constitutive models and rate-independent boundary-value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi-static boundary-value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9:203,274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207,220) and the plasticity limit criterion are revived in order to clarify the application field of the second-order work criterion and to contrast these criteria. The first part of this paper analyses the second-order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (d, = Md,) or in the dual form (d, = Nd,), is discussed. In the second part, the analysis is extended to the boundary-value problems in quasi-static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary-value problem. Copyright © 2008 John Wiley & Sons, Ltd. [source] An interpolation-based local differential quadrature method to solve partial differential equations using irregularly distributed nodesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2008Hang Ma Abstract To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations: small shifts of mixed type with rapid oscillationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2004M. K. Kadalbajoo Abstract We study the boundary-value problems for singularly perturbed differential-difference equations with small shifts. Similar boundary-value problems are associated with expected first-exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54: 249,283; 1982; 42: 502,531; 1985; 45: 687,734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The effect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method. Copyright © 2004 John Wiley & Sons, Ltd. [source] Multiscale Galerkin method using interpolation wavelets for two-dimensional elliptic problems in general domainsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004Gang-Won Jang Abstract One major hurdle in developing an efficient wavelet-based numerical method is the difficulty in the treatment of general boundaries bounding two- or three-dimensional domains. The objective of this investigation is to develop an adaptive multiscale wavelet-based numerical method which can handle general boundary conditions along curved boundaries. The multiscale analysis is achieved in a multi-resolution setting by employing hat interpolation wavelets in the frame of a fictitious domain method. No penalty term or the Lagrange multiplier need to be used in the present formulation. The validity of the proposed method and the effectiveness of the multiscale adaptive scheme are demonstrated by numerical examples dealing with the Dirichlet and Neumann boundary-value problems in quadrilateral and quarter circular domains. Copyright © 2003 John Wiley & Sons, Ltd. [source] A structural observation for linear material laws in classical mathematical physicsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2009Rainer Picard Abstract A class of linear material laws is considered, which covers a number of diverse initial boundary-value problems of classical mathematical physics. The claim that this class is indeed to a large extent sufficiently general is exemplified for a number of specific models from classical physics. Copyright © 2009 John Wiley & Sons, Ltd. [source] On the diffraction of Poincaré wavesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2001P. A. Martin Abstract The diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two-dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary-value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary-value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non-empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd. [source] On a third-order Newton-type method free of bilinear operatorsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2010S. Amat Abstract This paper is devoted to the study of a third-order Newton-type method. The method is free of bilinear operators, which constitutes the main limitation of the classical third-order iterative schemes. First, a global convergence theorem in the real case is presented. Second, a semilocal convergence theorem and some examples are analyzed, including quadratic equations and integral equations. Finally, an approximation using divided differences is proposed and used for the approximation of boundary-value problems. Copyright © 2009 John Wiley & Sons, Ltd. [source] A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2008J. K. Kraus Abstract We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant , in the strengthened Cauchy,Bunyakowski,Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright © 2007 John Wiley & Sons, Ltd. [source] | ||||||||||||||||