elastic and sound orthonormal beams and localized fields in linear mediums i basic equations basic definitions and general relations for elastic and sound fields defined by a given set of orthonormal functions on a real manifold are presented the proposed mathematical formalism makes it possible to obtain families of orthonormal beams and localized fields in both isotropic and anisotropic linear elastic mediums as well as the similar sound fields in an ideal liquid all these fields are described as superpositions of plane waves whose intensities and phases are specified by a set of orthonormal scalar functions on a two or three dimensional manifold by way of illustration the fields defined by the spherical harmonics are considered

a preparation theorem for weierstrass systems it is shown that lion and rolin s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a weierstrass system

a new analytical and numerical treatment for singular two point boundary value problems via the adomian decomposition method based on the adomian decomposition method a new analytical and numerical treatment is introduced in this research to investigate linear and non linear singular two point bvps the effectiveness of the proposed approach is verified by several linear and non linear examples

non polynomial spline solution of singularly perturbed boundary value problems a generalized scheme based on quartic non polynomial spline functions is proposed this scheme is designed for numerical solution of singularly perturbed two point boundary value problems arising in the study of science and engineering the scheme leads to a five diagonal linear system of equations convergence analysis of the method is briefly discussed two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method

polynomial spline approach for solving second order boundary value problems with neumann conditions abstract in this paper a new difference scheme based on quartic splines is derived for solving linear and nonlinear second order ordinary differential equations subject to neumann type boundary conditions the scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary which is superior to the well known numerov s scheme with the accuracy being fourth order convergence analysis of the present method for linear cases is discussed finally numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method

diffusion problems with a mixed nonlinear boundary condition abstract existence results are established for a class of singular diffusion problems with a general nonlinear boundary condition the proofs are based on a priori bounds for all solutions and a shooting technique applied to a corresponding initial value problem various uniqueness results are obtained by assuming either the variables in the diffusion coefficient or in the source term are separable

an intrinsic tensor technique in minkowski space with applications to boundary value problems this article describes an intrinsic tensor technique for solving boundary value problems in electrodynamics of moving media which entails considerable notational simplification and calculational advantages on the basis of generalized impedance and characteristic matrix methods the reflection and transmission operators of a multilayer structure consisting of anisotropic media uniformly moving with different velocities are found a general solution to the problem of reflection and transmission at the interface between two moving isotropic media is obtained and explicit expressions for doppler shifts snell s laws critical and brewster angles reflection and transmission coefficients and the radiation pressure force are found the material tensor of a uniformly moving linear medium is expressed through its reflection and transmission operators

a spline method for second order singularly perturbed boundary value problems we consider a difference scheme based on cubic spline in compression for second order singularly perturbed boundary value problem of the form ey p x y q x y r x y a 0 y b 1 the method is shown to have second and fourth order convergence depending on the choice of parameters 1 and 2 involved in the method the method is tested on an example and the results found to be in agreement with the theory

a new spline in compression approximation for one space dimensional quasilinear parabolic equations on a variable mesh in this paper we propose a new two level implicit method of order two in time and four in space directions based on spline in compression approximation for the numerical solution of one space dimensional quasi linear parabolic partial differential equation on a uniform mesh the derivation and the stability of the proposed method are discussed in details we have extended the method to non uniform mesh numerical results are given to illustrate the usefulness of the proposed method

high order accuracy nonpolynomial spline solutions for 2μth order two point boundary value problems sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2 th order two point bvps where is a positive integer is presented by transforming the problem into a system of second order two point bvps the nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem the proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost convergence analysis of the proposed method is discussed numerical examples are included to illustrate the practical usefulness of our method

a fourth order finite difference method based on non uniform mesh for a class of singular two point boundary value problems abstract we present a three point finite difference method based on non uniform mesh for a class of singular two point boundary value problem x y f x y y 0 a y 1 b 0 we show that the method based on non uniform mesh provides o h4 convergent approximations this method is illustrated by two numerical examples one is linear and the other is non linear

quasi analytic solutions of analytic ordinary differential equations and o minimal structures it is well known that the non spiraling leaves of real analytic foliations of codimension 1 all belong to the same o minimal structure naturally the question arises if the same statement is true for non oscillating trajectories of real analytic vector fields we show under certain assumptions that such a trajectory generates an o minimal and model complete structure together with the analytic functions the proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi analyticity result from which the main theorem follows as applications we present an infinite family of o minimal structures such that any two of them do not admit a common extension and we construct a non oscillating trajectory of a real analytic vector field in dimension 5 that is not definable in any o minimal extension of the reals

taylor series method for solving a class of nonlinear singular boundary value problems arising in applied science abstract taylor series method is applied to a class of nonlinear singular boundary value problems without requiring any specific technique in handling the singularity at the origin for the problems with exponential nonlinearity a variable transformation is introduced to accelerate the convergence of the solution by converting this exponential nonlinearity into a polynomial nonlinearity the efficiency and applicability of the algorithm are assessed and tested on several problems arising in applied science pade approximants are used to verify that the series solutions to these problems converge within the boundaries an error analysis is presented and the results show that the analytical approximate solutions obtained are highly accurate

complements of subanalytic sets and existential formulas for analytic functions we show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra this result has an equivalent formulation in logic consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers such an expression is equivalent to an existential expression involving functions from the same class provided that the variables approach neither infinity nor the boundary of the domain

cubic spline solution of singularly perturbed boundary value problems with significant first derivatives abstract we develop a numerical technique for a class of singularly perturbed two point singular boundary value problems on an uniform mesh using polynomial cubic spline the scheme derived in this paper is second order accurate the resulting linear system of equations has been solved by using a tri diagonal solver numerical results are provided to illustrate the proposed method and to compared with the methods in r k mohanty urvashi arora a family of non uniform mesh tension spline methods for singularly perturbed two point singular boundary value problems with significant first derivatives appl math comput 172 2006 531 544 m k kadalbajoo v k aggarwal fitted mesh b spline method for solving a class of singular singularly perturbed boundary value problems int j comput math 82 2005 67 76

on the convergence of finite difference methods for weakly regular singular boundary value problems the second order finite difference methods m 1 based on a non uniform mesh and m 2 based on an uniform mesh developed by chawla and katti finite difference methods and their convergence for a class of singular two point boundary value problems numer math 39 1982 341 350 for weakly regular singular boundary value problems p x y f x y 0

maxwell s square roots from the metric tensors of wave surfaces and branches of solutions of the photon and phonon wave equations for the wave equations of optics and acoustics of isotropic media the infinite families of three dimensional plane wave cauchy operators are found by a direct tensor method these families form involutive lie groups their generators n can be found by taking square roots from the unit tensors of the wavefront subspaces with outer normal n in optics the basic structural elements of n are complex involutive operators reflection isometries described by pairs of complex vectors s and c which satisfy the metric condition and also by a pair of projective operators of the two dimensional space of a plane orthogonal to n in the acoustics of isotropic media in view of the inequality of the longitudinal and transverse wave velocities the generators n are represented as a linear combination of an involutive operator and a diad it is shown that the projection of the average energy flux of the wave is conserved in the general case the families of vectors being a part of n are indicated for these families the global operators acting on initial field vectors give states described by the right hand and left hand elliptical helices the wave normal n characterizes the direction of the angular momentum of the field and for the case turns out to be equivalent to the darboux vector known in geometry

cubic spline for a class of singular two point boundary value problems in this paper we have presented a method based on cubic splines for solving a class of singular two point boundary value problems the original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation the tridiagonal system resulting from the spline approximation is efficiently solved by thomas algorithm some model problems are solved and the numerical results are compared with exact solution

cubic spline for a class of non linear singular boundary value problems arising in physiology abstract cubic spline method to analyze a class of non linear singular boundary value problems defined by y x m x y f x y y 0 0 y 1 y 1 is presented the quesilinearization technique is used to reduce the given non linear problem to a sequence of linear problems the resulting set of differential equations are modified at the singular point and are treated by using cubic spline for finding the numerical solution the numerical method is tested for its efficiency by considering two examples from physiology

parametric spline method for a class of singular two point boundary value problems we develop a three point formula based on cubic spline in compression for a class of singular two point boundary value problems for different values of parameters we obtain the classes of methods these methods are illustrated by three numerical examples two linear and one non linear

on the convergence of a finite difference method for a class of singular boundary value problems arising in physiology using chawla s identity bit 29 1989 566 a finite difference method based on uniform mesh is described for a class of singular boundary value problems p x y p x f x y 0 x 1 y 0 a y 1 y 1 or y 0 0 y 1 y 1 with p x xb0 g x b0 0 and it is shown that the method is of second order accuracy under quite general conditions on p x and f x y this work also extends the method developed by chawla et al bit 26 1986 326 for p x xb0 b0 1 to a general class of function p x xb0 g x b0 0 numerical examples for general function p x verify the order of the convergence of the method and two physiological problems have also been solved

cubic spline solution of two point boundary value problems with significant first derivatives a cubic spline function has been applied to the solution of the second order nonlinear two point boundary value problems with significant first derivatives the method obtained involves some parameters these parameters may be suitably chosen to obtain an oscillation free solution of the test equation y ky k 1 depending upon the choice of the parameters methods of order two three and four may be obtained the methods have been tested on both linear and nonlinear problems

tage iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations abstract in the present paper we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm the schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method computational results are presented comparing the two methods in terms of accuracy and execution times the results indicate the advantage of using parallel implementation of the new method

pointwise solution bounds for a class of singular diffusion problems in physiology a numerical technique for obtaining pointwise bounds for the solution of a class of nonlinear boundary value problems in physiology is presented simple analytic bounding functions are obtained using an integral representation for the solution the computations are performed in interval arithmetic thus obtaining lower and upper bounds simultaneously the oxygen diffusion problem in spherical cells and a nonlinear heat conduction model of the human head are presented as illustrative examples for these examples the present technique is computationally more efficient than the existing ones in that it yields sharper bounds with fewer integration steps

new cubic b spline approximation for solving third order emden flower type equations abstract in this article the typical cubic b spline collocation method equipped with new approximations for second and third order derivatives is employed to explore the numerical solution of a class of third order non linear singular boundary value problems the singularity is removed by means of l hospital s rule the taylor s series expansion of the error term reveals that our new scheme is fifth order accurate the proposed technique is tested on several third order emden flower type equations and the numerical results are compared with those found in the current literature it is found that our new approximation technique performs superior to the existing methods due to its simple implementation straight forward interpolation and very less computational cost

nonpolynomial spline approach to the solution of a system of second order boundary value problems we use a cubic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods convergence analysis of the method is discussed a numerical example is given to illustrate practical usefulness of our method

refractive index tensors in connection with problems of photon scattering in terms of the refractive index tensors n the operator solving a problem of reflection and refraction of electromagnetic waves on plane boundary between isotropic dielectrics is given separately the behaviour of beltrami fields on the boundary is considered they are described by infinite sets of branches of the traceless indefinite tensors n which are generators of involutive maxwell groups for photon antiphoton meeting pairs the connections of the traceless tensors n with fresnel reflection and transmission coefficients stokes relations beltrami fields and reflectional coxeter s groups are established

a linearisation method for non linear singular boundary value problems this paper presents a novel numerical approach for the solution of a class of nonlinear singular boundary value problems arising in physiology the approach is based on a new application of the successive linearisation method slm three illustrative examples are presented to demonstrate the effectiveness of the proposed method the new approach is found to give accurate results comparable to results in the literature found using existing numerical methods

frequency domain wave splitting techniques for plane stratified bianisotropic media a plane harmonic electromagnetic wave obliquely incident on a plane stratified bianisotropic medium with multiple discontinuities in the parameters is considered a covariant wave splitting approach based on the use of the formula of integration by parts for multiplicative integrals and the impedance concept is presented it encompasses various types of decomposition of the total internal field into two waves propagating in opposite directions including the physical and vacuum wave splittings treated earlier in the literature and provides a convenient means for both analytical investigation and numerical calculation of evolution operators green s functions and impedance tensors of split waves as well as characteristic matrices and reflection and transmission tensors of stratified bianisotropic media the potentialities of the approach are illustrated by its application to the problems of reflection transmission and guided propagation and by generalizing the method of multiple reflections to the case of stratified bianisotropic media

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

how to avoid a compact set abstract a first order expansion of the r vector space structure on r does not define every compact subset of every r n if and only if topological and hausdorff dimension coincide on all closed definable sets equivalently if a r k is closed and the hausdorff dimension of a exceeds the topological dimension of a then every compact subset of every r n can be constructed from a using finitely many boolean operations cartesian products and linear operations the same statement fails when hausdorff dimension is replaced by packing dimension

b spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems abstract this paper is concerned with the construction and convergence analysis of two b spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems ddsbvp the first method is based on uniform mesh while the second method is based on non uniform mesh for the second method we use a grading function to construct the non uniform grid we prove that the method based on uniform mesh is of second order accuracy and the method based on non uniform mesh is of fourth order accuracy three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods moreover we consider some special cases of the problem under consideration in order to compare our methods with other existing methods it is shown that our second method based on cubic b spline basis functions has the same order of convergence as quartic b spline collocation method 1 moreover our methods yield more accurate results and are computationally attractive than the methods developed in 1 2 3 4 5 6 7 8 the proposed methods are applied on three real life problems the first problem describes the distribution of radial stress on a rotationally shallow membrane cap the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy

wavelet galerkin method for solving nonlinear singular boundary value problems arising in physiology abstract in this study wavelet galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science the quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic b spline wavelets and galerkin method these wavelets are used as testing and weighting functions the method is computationally attractive and applications are demonstrated through illustrative examples

a fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations the method uses evaluation at two off steps three neighbouring knots and it reduces to simple five terms recurrence relations the formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method the root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method

on a class of regular singular two point boundary value problems abstract existence of a unique solution for a class of regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established wherep x satisfies i p x 0 on 0 b ii p x c1 0 r and for somer b iii xp x p x is analytic in z z xp x p x b 0 b 1 x b 0 1 and with quite general conditions onf x y these conditions onf x y are sharp which is seen through one example regions for multiple solutions have also been determined

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method

fresnel s reflection and transmission operators for stratified gyroanisotropic media for the general case of an inhomogeneous anisotropic and gyrotropic medium a differential tensor equation expressing the evolution of the tangential component of the field vectors of an electromagnetic wave is obtained a fundamental solution of this equation is given by a multiplicative integral a plane stratified system of anisotropic and gyrotropic layers is considered by means of the characteristic matrix of such a medium fresnel s reflection and transmission operators are derived these operators have wide utility because they describe exactly the interaction of light with any plane stratified gyroanisotropic structure the conservation of the normal component of the poynting vector in such a structure allows the authors to find a correlation between the operators of reflection and transmission the operator dispersion equation of the multilayer gyroanisotropic waveguide is presented all the calculations in this paper are based on the direct manipulation of tensors and their invariants eliminating the use of coordinate systems this facilitates solutions and provides results of great generality which are suitable for computer use

manifestly covariant coordinate free dyadic expression for planar homogeneous lorentz transformations parametrizing a planar homogeneous lorentz tranformation p by any timelike or spacelike vector b lying in the transformation plane and its transform a pb yields a dyadic expression for p with several advantages it provides an immediate solution to the problem of finding a homogeneous lorentz transformation converting a given timelike or spacelike vector into a second similar vector its manifestly covariant coordinate free form is valid in any lorentz frame and reduces easily to coordinate form it unifies timelike including boosts spacelike including pure spatial rotations and null planar transformations and also orthochronous and nonorthochronous planar transformations into a single form these classifications depend on the vectors a and b only if a b does the expression fail but then its limit as a b still exists and provides a valid expression for p

on a class of weakly regular singular two point boundary value problems ii abstract existence of a unique solution of a class of weakly regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established where p x satisfies i p x 0 on 0 b ii p x c 1 0 r and for some r b iii xp x p x is analytic in z z r with taylor expansion xp x p x b 0 b 1 x b 0 0 1 and with quite general conditions on f x y these conditions on f x y are sharp which is seen through one example regions for multiple solutions have also been determined

quasianalytic denjoy carleman classes and o minimality we show that the expansion of the real field generated by the functions of a quasianalytic denjoy carleman class is model complete and o minimal provided that the class satisfies certain closure conditions some of these structures do not admit analytic cell decomposition and they show that there is no largest o minimal expansion of the real field

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples

real analytic desingularization and subanalytic sets an elementary approach we give a proof of a theorem on desingularization of real analytic functions which is a weaker version of h hironaka s result but has the advantage of being completely self contained and elementary and not involving any machinery from algebraic geometry we show that the basic facts about subanalytic sets can be proved from this result