a rational spectral collocation method for solving a class of parameterized singular perturbation problems a new kind of numerical method based on rational spectral collocation with the sinh transformation is presented for solving parameterized singularly perturbed two point boundary value problems with one boundary layer by means of the sinh transformation the original chebyshev points are mapped onto the transformed ones clustered near the singular points of the problem the results from asymptotic analysis as regards the singularity of the solution are employed to determine the parameters in the transformation numerical experiments including several nonlinear cases illustrate the high accuracy and efficiency of our method

utd diffraction coefficients for higher order wedge diffracted fields an asymptotic expansion of a multiple integral for knife edge diffraction is derived the expansion expresses the field in terms of single edge diffraction by considering this expression a similar expansion for higher order utd diffracted fields is proposed by means of a set of transition functions the result removes some of the shortcomings of the original set out of the utd when the incident field is not a ray optical field

fourth order finite difference method for boundary value problems with two small parameters abstract we present a finite difference scheme for a class of linear singularly perturbed boundary value problems with two small parameters the problem is discretized using a bakhvalov type mesh it is proved under certain conditions that this scheme is fourth order accurate and that its error does not increase when the perturbation parameter tends to zero numerical examples are presented which demonstrate computationally the fourth order of the method

inverse mode problems for the finite element model of a vibrating rod the inverse mode problems for the finite element model of an axially vibrating rod are formulated and solved it is known that for the finite element model based on linear shape functions of the rod the mass and stiffness matrices are both tridiagonal it is shown that the finite element model of the rod can be constructed from two eigenvalues their corresponding eigenvectors and the total mass of the rod the necessary and sufficient conditions for the construction of a physically realizable rod with positive mass and stiffness elements from two eigenpairs and the total mass of the rod are established if these conditions are satisfied then the construction of the model is unique

structured quadratic inverse eigenvalue problem i serially linked systems quadratic pencils arising from applications are often inherently structured factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters for physical feasibility structural constraints must be respected consequently they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications however the issue of solvability is problem dependent and has to be addressed structure by structure this paper considers one particular structure where the elements of the physical system if modeled as a mass spring system are serially linked the discussion recasts both undamped and damped problems in a framework of inequality systems that can be adapted for numerical computation some open questions are described

comparison of lossy wedge diffraction coefficients with application to mixed path propagation loss prediction a heuristic lossy wedge diffraction coefficient is compared to the exact maliuzhinets diffraction coefficient for a 180 degrees wedge neglecting surface waves the comparison shows good agreement even for high incidence angles and greatly dissimilar wedge face materials this agreement provides justification for using the heuristic coefficient for the calculation of mixed path propagation loss since wedge angles are typically near 180 degrees for this application the heuristic coefficient has the advantages of being continuous at the reflection boundary and of being relatively simple to evaluate especially at angles other than 180 degrees

a numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location a class of nonlinear singularly perturbed interior layer problems is examined in this paper solutions exhibit an interior layer at an a priori unknown location a numerical method is presented that uses a piecewise uniform mesh refined around numerical approximations to successive terms of the asymptotic expansion of the interior layer location the first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered denote the perturbation parameter as e and the number of mesh intervals to be used as n the method is shown to converge point wise to the true solution with a first order convergence rate overlooking a logarithmic factor for sufficiently small e n 1 a numerical experiment is presented to demonstrate the convergence rate established

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

a heuristic utd slope diffraction coefficient for rough lossy wedges heuristic wedge diffraction coefficients for computing propagation path loss over finitely conducting earth are extended to include slope diffraction with the assumption that propagation of energy through the wedge is negligible the slope diffraction terms for the lossy wedge are obtained in an analogous manner as for the perfectly conducting case except that special care must be taken with the factor multiplying the incident field for grazing incidence results given show that the slope diffraction term produces continuous results that behave reasonably when compared with results for perfectly conducting wedges

a modification of the shishkin discretization mesh for one dimensional reaction diffusion problems in this paper we consider a modification of the shishkin discretization mesh designed for the numerical solution of one dimensional singularly perturbed reaction diffusion problems the modification consists of a slightly different choice of the transition points between the fine and coarse parts of the mesh we prove that this change does not affect the order of convergence of the numerical solution obtained by using the central finite difference scheme however due to a better layer resolving mesh numerical results show an improvement in the accuracy of the computed solution when compared to the results on the standard shishkin mesh

a slope utd solution for a cascade of multishaped canonical objects the case of multiple diffraction from a cascade of two dimensional multi shaped canonical objects such as wedges knife edges and cylinders is investigated a heuristic slope uniform theory of diffraction utd solution is presented that can give accurate and uniform field predictions at any area point and for any kind of environment the solution is then compared against measurements taken in an anechoic chamber and a very good agreement is observed furthermore an algorithm for implementing the proposed utd solution to rural area field predictions is also presented based on a novel algorithm for optimum fit of canonical shapes to the terrain irregularities comparisons between the model and measurements taken in real terrain profile show a very good agreement and uniform field predictions the importance of the choice of the used canonical object to terrain modeling is also discussed since different diffraction patterns are observed for different combination of the canonical objects

boundary layer resolving pseudospectral methods for singular perturbation problems pseudospectral methods are investigated for singularly perturbed boundary value problems for ordinary differential equations odes which possess boundary layers it is well known that if the boundary layer is very small then a very large number of spectral collocation points is required to obtain accurate solutions we introduce here a new effective procedure based on coordinate stretching and the chebyshev pseudospectral method to resolve the boundary layers stable and accurate results are obtained for very thin boundary layers with a fairly small number of spectral collocation points

wide angle shift map pe for a piecewise linear terrain a finite difference approach a wide angle parabolic wave equation solution is presented using shift map and finite difference techniques the corresponding split step fourier solution is well known the solution using finite difference technique where the standard parabolic wave equation is modified into the so called claerbout equation allowing propagation angles up to 45deg from the paraxial direction is also well known here we present an extension to that solution in which the shift map technique is incorporated into the finite difference scheme allowing a varying terrain to be considered the result is a solution that corresponds to the well known split step solution which is believed to perform well for terrain slopes up to 10deg 15deg and discontinuous slope changes on the order of 15deg 20deg this solution is a first order one with respect to the terrain slope however when using the finite difference technique it is also possible to find a second order solution with respect to the terrain slope this new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg which is an improvement

an inverse eigenvalue problem for the finite element model of a vibrating rod an inverse eigenvalue problem for the finite element model of a longitudinally vibrating rod whose one end is fixed and the other end is supported on a spring is considered it is known that the mass and stiffness matrices are both tridiagonal for the finite element model of the rod based on linear shape functions it is shown that the cross section areas can be determined from the spectrum of the rod the inverse vibration problem can be recast into an inverse eigenvalue problem of a special jacobi matrix the necessary and sufficient conditions for the construction of a physically realizable rod with positive cross section areas are established a numerical method is presented and an illustrative example is given

a numerical method for singularly perturbed turning point problems with an interior layer the objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer the method is based on the asymptotic expansion technique and the reproducing kernel method rkm the original problem is reduced to interior layer and regular domain problems the regular domain problems are solved by using the asymptotic expansion method the interior layer problem is treated by the method of stretching variable and the rkm four numerical examples are provided to illustrate the effectiveness of the present method the results of numerical examples show that the present method can provide very accurate approximate solutions

analysis of a finite difference scheme for a singularly perturbed problem with two small parameters abstract we study a model linear convection diffusion reaction problem where both the diffusion term and the convection term are multiplied by small parameters e d and e c respectively depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain sharp bounds for the derivatives of the solution are derived using a barrier function technique these bounds are applied in the analysis of a simple upwind difference scheme on shishkin meshes this method is established to be almost first order convergent independently of the parameters e d and e c

rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer abstract we are concerned with using high accuracy numerical methods for solving singularly perturbed problems whose solutions exhibit an interior layer a rational spectral collocation method in barycentric form with sinh transformation is firstly applied to find the approximate solution to the given problem the sinh transformation consists of two parameters the location and the width of the interior layer with the tool of the asymptotic analysis the interior layer can be easily located then in determining the width of the interior layer we design a nonlinear unconstrained optimization problem which is solved by using the basic differential evolution de algorithm the feasibility and effectiveness of the proposed method is verified by numerical results the proposed method can also be applied to other singularly perturbed problems

generalization of utd double diffraction to the case of impenetrable wedges with relatively general boundary conditions in the framework of the uniform geometrical theory of diffraction utd a novel high frequency solution is provided which allows the description of the rays doubly diffracted by a pair of impenetrable straight wedges with relatively general boundary conditions under a spherical wave illumination such a utd solution is heuristically derived by imposing the continuity of the total field at the shadow boundaries of the first order i e geometrical optics utd wedge single diffracted ray field utd solution it is the extension to the case of double diffraction of the first order solutions proposed by luebbers and by burnside and burgener it allows a uniform description of the field at any observation point including those aspects where transition regions overlap and the cascading of utd single wedge diffraction coefficients fails the proposed solution does not consider surface wave effects and for the special case of two perfectly conducting wedges it reduces to the formulation present in the literature of which it can be considered a generalization indeed it requires the same special functions involved in utd double diffraction by two perfectly conducting wedges the double diffracted ray field is expressed in analytical closed form providing a deeper understanding of the physics involved in double diffraction and a new powerful tool to be employed in any modern ray based code

b spline collocation method for a two parameter singularly perturbed convection diffusion boundary value problems in this paper we have presented a b spline collocation method for solving a class of two parameter singularly perturbed boundary value problems we use b spline collocation method on piecewise uniform shishkin mesh which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order numerical results are presented in the end which support the theoretical results

a rational spectral collocation method for third order singularly perturbed problems a new numerical method is developed for solving a class of third order singularly perturbed boundary value problems first of all the given problem is transformed into a system of two ordinary differential equations odes subject to suitable initial and boundary conditions then the rational spectral collocation method in barycentric form with sinh transformation is applied to solve the system of odes according to the asymptotic analysis the location and width of boundary layer of the given problem which are chosen as parameters in the sinh transformation can be determined ample numerical experiments are presented to illustrate the computational efficiency and accuracy of the our method

noise attenuation by a hard wedge shaped barrier this paper is concerned with the problem of sound screening by a wedge like barrier the sound source is assumed to be point like and the receiver is located in the shadow of the source sound field so that according to geometrical optics only the field diffracted by the edge of the barrier is considered first for the hard wedge in space three models are used for calculating the amplitude of the edge diffracted field these are the uniform theory of diffraction utd the hadden pierce model both in the frequency domain and the biot tolstoy theory of diffraction which is a time domain formulation it is first shown that even at relatively low frequencies the frequency domain models perform quite satisfactorily as compared to the exact time domain theory hence and due to its relative simplicity the utd is proposed as an accurate calculation scheme for solving problems with edge diffraction by hard wedges it is also proved from theoretical calculations that the amplitude of the edge diffracted field increases for an increasing angle of the wedge and consequently the hard half plane gives the lowest field amplitude in the shadow zone some applications are then considered for evaluating the performance of a barrier on a flat ground either completely hard or with mixed homogeneous boundary conditions an improvement of the scheme for calculating the sound field in the all hard case is achieved through considering the multiple diffraction in this case only to the second order between the top of the wedge barrier and its base the results show that for usually occurring situations increasing the angle of the hard wedge barrier affects negatively its efficiency through diminishing its insertion loss these conclusions are also supported by the results of some experimental measurements conducted at a scale model level

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

a rational spectral collocation method with adaptively transformed chebyshev grid points a spectral collocation method based on rational interpolants and adaptive grid points is presented the rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed chebyshev points the locations of the grid points are adapted to singularities of the underlying solution and the locations of these singularities are approximated by the locations of poles of chebyshev pade approximants numerical experiments on two time dependent problems one with finite time blow up and one with a moving front indicate that the method far outperforms the standard chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to 1 1

calculation of higher order diffracted fields for multiple edge transition zone diffraction a well known problem in conventional uniform theory of diffraction utd is multiple edge transition zone diffraction here higher order diffracted fields have to be used in order to improve the result and these fields are added by means of a series unfortunately this series will converge slowly for many edges which means that a good enough result might require a large number of terms however by summing up the series properly the computation time can be kept reasonable which will be shown for knife edges the summation technique is though valid for wedges as well