absolute continuity and existence of solutions to dynamic inclusions in time scales we provide some properties for absolutely continuous functions in time scales then we consider a class of dynamical inclusions in time scales and extend to this class a convergence result of a sequence of almost inclusion trajectories to a limit which is actually a trajectory of the inclusion in question we also introduce the so called euler solution to dynamical systems in time scales and prove its existence a combination of the existence of euler solutions with the compactness type result described above ensures the existence of an actual trajectory for the dynamical inclusion when the setvalued vector field is nonempty compact convex and has closed graph

existence and uniqueness of positive solutions to a class of singular m point boundary value problems abstract this paper investigates the existence and uniqueness of positive solutions to a class of singular m point boundary value problems of second order ordinary differential equations a sufficient condition for the existence and uniqueness of c 0 1 positive solutions as well as c 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t u may be singular at u t 0 and or t 1

comparison theorems for linear dynamic equations on time scales we obtain several comparison theorems for second order linear dynamic equations on a time scale these results extend comparison theorems for the continuous case and provide some new results in the discrete case as well as other more general situations

chain rule and invariance principle on measure chains in this note we prove a chain rule for mappings on abstract measure chains and apply our result to deduce an invariance principle for non autonomous dynamic equations

solving singular second order three point boundary value problems using reproducing kernel hilbert space method this paper investigates the numerical solutions of singular second order three point boundary value problems using reproducing kernel hilbert space method it is a relatively new analytical technique the solution obtained by using the method takes the form of a convergent series with easily computable components however the reproducing kernel hilbert space method cannot be used directly to solve a singular second order three point boundary value problem so we convert it into an equivalent integro differential equation which can be solved using reproducing kernel hilbert space method four numerical examples are given to demonstrate the efficiency of the present method the numerical results demonstrate that the method is quite accurate and efficient for singular second order three point boundary value problems

new kamenev type oscillation criteria for second order nonlinear advanced dynamic equations this paper presents a new kamenev type oscillation criterion for a second order advanced dynamic equation on a time scale two examples are included to show that the result obtained complements and improves those reported in the literature

on existence theorems of positive solutions to nonlinear singular differential equations abstract suppose i f t u 0 1 0 0 is continuous and is increasing on u ii there exists a function g 1 0 g b b and g b b 2 is integrable on 1 such that f t bu g b f t u t u 0 1 0 consider the singular boundary value problem u t f t u t 0 0 t 1 u 0 au u 1 0 then a necessary and sufficient condition for eq to have c 0 1 positive solutions is that 0 0 1 s 1 s f s 1 d s a necessary and sufficient condition for eq to have c 1 0 1 positive solutions is that 0 0 1 f s 1 s d s also the uniqueness iterative methods and computational methods of c 1 0 1 positive solutions have been studied our nonlinearity may be singular at t 0 and or t 1

accurate and efficient numerical integration of weakly singular integrals in galerkin efie solutions a galerkin descretization of the electric field integral equation for perfectly conducting surfaces using rao wilton glisson 1982 basis functions requires the numerical evaluation of integrals with singular kernels over triangular regions these singularities have been traditionally handled by utilizing a singularity extraction procedure to produce a regular integral and an analytic function to replace the original singular integral a new approach is presented here in which the four dimensional 4 d weakly singular integrals unique to the galerkin rao wilton glisson electric field integral equation solution for perfectly conducting surfaces are transformed into integrals with regular integrands the transformations allow some of the integrations to be performed analytically in some cases reducing the original 4 d integral into a 1 d numerical integration the accuracy and convergence properties of the new method are demonstrated by evaluating the scalar potential function over a unit triangle

on the calculation of potential integrals for linear source distributions on triangular domains present analytical formulas for the singular field contributions of linear source distributions on triangular domains the formulas can be used for near field calculations of electric and magnetic source currents especially for the calculation of method of moments mom matrix elements due to the solution of surface integral equations for mom self coupling terms involving the scalar green function of free space the authors present expressions comprising analytical solutions of both the source integrals and the testing integrals

oscillation for nonlinear second order dynamic equations on a time scale abstract we obtain some oscillation criteria for solutions to the nonlinear dynamic equation x q t x p t f x 0 on time scales in particular no explicit sign assumptions are made with respect to the coefficients p t q t we illustrate the results by several examples including a superlinear emden fowler dynamic equation

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

oscillation criteria for second order nonlinear neutral variable delay dynamic equations abstract this paper is concerned with oscillation of second order nonlinear neutral dynamic equations on time scales with a variable delay by using the generalized riccati technique and integral averaging technique new oscillation criteria are obtained for all solutions of the equation some results extend known results for difference equations when the time scale is the set z of positive integers and for differential equations when the time scale is r several examples are given to illustrate the results of the paper

gronwall bellman type nonlinear delay integral inequalities on time scales abstract in this paper some gronwall bellman type nonlinear delay integral inequalities on time scales are established which provide a handy tool in deriving boundedness of solutions of certain delay dynamic equations on time scales our results generalize some of the main results in lipovan 2006 1 pachpatte 2000 2 ferreira and torres 2009 3 zhang and meng 2008 4 cheung and ren 2006 5 kim 2009 6 and some of our results unify continuous and discrete analysis in the literature

positive solutions of singular sublinear emden fowler boundary value problems abstract this paper is about the dirichlet boundary value problem of singular sublinear emden fowler equations a necessary and sufficient condition for the existence of positive solutions to this problem has been obtained by using the method of lower and upper solutions the uniqueness of positive solutions has also been discussed

oscillation criteria for second order matrix dynamic equations on a time scale we obtain oscillation criteria for a second order self adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function we illustrate our results with some nontrivial examples

oscillatory behavior of second order half linear damped dynamic equations by refining the generalized riccati transformation technique we obtain new oscillation criteria for a class of second order half linear delay dynamic equations with damping on a time scale assumptions in our theorems are less restrictive these results complement and improve related contributions to the subject in particular the results obtained amend related results reported in the literature

theoretical and numerical treatment of surface integrals involving the free space green s function this paper deals with the problem of the calculation of surface integrals for electromagnetic scattering in the case of the widely popular double triangular basis functions first introduced by rao wilton and glisson 1982 an entire set of formulas is obtained which overrides the difficulties inherent to the singularity of the integrands and results showing the stability accuracy and efficiency of the methods developed are reported in an application of the method of moments to the case of perfectly conducting surfaces and computation of near field in domains including the surface itself furthermore the authors provide insight as regards the capability of triangular basis functions to model near field patterns

some new oscillation results for a nonlinear dynamic system on time scales we study the oscillation of a system of two first order nonlinear equations on time scales this form includes the classical emden fowler differential equation and many of its extensions we generalize some well known results of atkinson bohner erbe peterson and others we illustrate the results by several examples including a superlinear emden fowler dynamic system

generalized gronwall bellman type delay dynamic inequalities on time scales and their applications abstract in this paper we establish some new gronwall bellman type delay dynamic inequalities on time scales which can be used as a handy tool in the investigation of qualitative as well as quantitative analysis of solutions of delay dynamic equations on time scales our results generalize the main results of ou iang l ou iang the boundedness of solutions of linear differential equations y a t y 0 shuxue jinzhan 3 1957 409 415 jiang f c jiang f w meng explicit bounds on some new nonlinear integral inequality with delay j comput appl math 205 2007 479 486 and yuan z l yuan x w yuan f w meng some new delay integral inequalities and their applications appl math comput 208 2009 231 237

static and dynamic potential integrals for linearly varying source distributions in two and three dimensional problems a technique for the evaluation of static and dynamic potentials due to source distributions defined on domains with simple shape is presented the domains considered are polyhedral regions and in two dimensional problems plane polygons on which uniform or linearly varying source distributions are defined it is shown how three dimensional two dimensional potential integrals are always reducible to surface line integrals with nonsingular kernel by use of a nonlinear transformation for the integration variables that permits analytic integration in the static case the integration on the boundary is performed analytically and closed form results are given in the dynamic case the expressions of the boundary integrals are given in a form suitable for numerical integration the use of matrix notation allows for very compact expressions readily translatable into computer programs

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

positive solutions of semi positone hammerstein integral equations and applications abstract existence of solutions for semi positone integral equations of hammerstein type is obtained by using the fixed point index method many boundary value problems of differential equations can be transformed into these integral equations a specific example is given

some pachpatte type inequalities on time scales in this paper using the comparison theorem we investigate some pachpatte type integral inequalities on time scales which provide explicit bounds on unknown functions our results extend some known dynamic inequalities on time scales unify and extend some continuous inequalities and their corresponding discrete analogues some applications of the main results are given in the end of this paper

singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra a numerical solution of integral equations typically requires calculation of integrals with singular kernels the integration of singular terms can be considered either by purely numerical techniques e g duffy s method polar co ordinate transformation or by singularity extraction in the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically this method has been well established for linear and constant shape functions in this paper we extend the method for polynomial shape functions of arbitrary order we present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the helmholtz equation or its gradient and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra the presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high order basis functions numerical experiments show that the developed method leads to a more accurate and robust integration scheme and in many cases also a faster method than for example duffy s transformation copyright 2003 john wiley sons ltd

oscillation of second order superlinear dynamic equations with damping on time scales this paper concerns the oscillation of solutions to the second order superlinear dynamic equation with damping r t x d t d p t x d t q t f x s t 0 on a time scale t which is unbounded above no sign conditions are imposed on r t p t and q t the function f c r r is assumed to satisfy xf x 0 and f x 0 for x 0 we illustrate the results by several examples

a necessary and sufficient condition for the existence of positive solutions to singular sublinear three point boundary value problems abstract this paper investigates the existence of positive solutions for second order singular sublinear three point boundary value problems a necessary and sufficient condition for the existence of c 0 1 as well as c 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem also the uniqueness iterative methods convergence rate and computational methods of the c 1 0 1 positive solutions are studied our nonlinear term may be singular at t 0 and or t 1

some delay integral inequalities on time scales in this paper using gronwall s inequality we investigate some delay integral inequalities on time scales which provide explicit bounds on unknown functions our results unify and extend some delay integral inequalities and their corresponding discrete analogues the inequalities given here can be used as tools in the qualitative theory of certain classes of delay dynamic equations on time scales

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

on the existence of positive solutions for 2 n order singular boundary value problems by applying upper and lower solutions method the existence of c2n 2 0 1 and c2n 1 0 1 positive solution for a class of 2n order singular differential equations boundary value problems is obtained

on the numerical integration of the linear shape functions times the 3 d green s function or its gradient on a plane triangle formulas for the static and dynamic potentials due to linearly varying source distributions defined on a planar triangle are presented the general linear variation of the source is represented in terms of three linear basis functions each one associated with a different vertex of the triangular domain the singular kernels considered are given by the 3 d green s function and its gradient both for the static and dynamic case in the static case the evaluation of potentials is performed analytically and compact form results are given in the dynamic case the results are given in a form suited for numerical integration

efficient evaluation of weakly singular integrals arising from electromagnetic surface integral equations solving electromagnetic em problems by integral equation methods relies on the accurate evaluation of singular integrals related to the green s function in the method of moments mom with the rao wilton glisson rwg basis function for solving surface integral equations sies the gradient operator on the scalar green s function can be moved onto the basic function and testing function resulting in a 1 r weak singularity in the integral kernel where r is the distance between an observation point and a source point the weakly singular integral can be evaluated with the well known duffy s method but it requires a two fold numerical integration in this work we develop a novel approach to evaluate the singular integral by using a local polar coordinate system the approach can automatically cancel the singularity and reduce the integral to a one fold numerical integration by deriving a closed form expression for the integral over the polar coordinate numerical examples are presented to demonstrate the effectiveness of the approach

wong s comparison theorem for second order linear dynamic equations on time scales we obtain wong type comparison theorems for second order linear dynamic equations on a time scale the results obtained extend and are motivated by wong s comparison theorems as a particular application of our results we show that the difference equation

an accurate method for the calculation of singular integrals arising in time domain integral equation analysis of electromagnetic scattering an accurate method for the evaluation of the cauchy principal value integrals arising in time domain electromagnetic wave scattering is presented this is applied to a boundary integral equation bie method employing quadratic curvilinear surface elements where such singularities do not vanish as they generally do when simpler but less accurate discretizations are employed the technique involves weakening the singularity in the original kernel to a degree where conventional integration methods may be employed and transforming the strong singularity to a line integral in a form which allows cancellation of its singular components the effectiveness of the method is demonstrated by scattering calculations on a variety of targets and the computational cost of the approach is trivial

nonnegative solutions to nonlocal boundary value problems for systems of second order differential equations dependent on the first order derivatives abstract in this paper we consider nonlocal boundary value problems for systems of second order differential equations with dependence on the first order derivatives and deviating arguments by using a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three nonnegative solutions to such problems we investigate our problem both for delayed and advanced arguments i i and also for the case when i t i t t t 0 1 in all cases arguments i i can change the character on 0 1 so in some subinterval i of 0 1 they can be delayed in i and advanced in 0 1 i some remarks concern also the case when differential equations do not depend on the first order derivatives examples illustrate some results

machine precision evaluation of singular and nearly singular potential integrals by use of gauss quadrature formulas for rational functions a new technique for machine precision evaluation of singular and nearly singular potential integrals with 1 r singularities is presented the numerical quadrature scheme is based on a new rational expression for the integrands obtained by a cancellation procedure in particular by using library routines for gauss quadrature of rational functions readily available in the literature this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions the rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed and several numerical examples are provided

some new dynamic inequalities on time scales in this paper based on some known dynamic inequalities we investigate certain new dynamic inequalities on time scales which provide explicit bounds on unknown functions our results unify and extend some continuous inequalities and their corresponding discrete analogues