complete lagrange family for the cube in finite element interpolations abstract noting a vagueness in the serendipity definition the complete lagrange family for the cube and the mixed complete lagrange family are introduced the unisolvent lagrange interpolations over rectangular finite elements are then presented definitively in hierarchy ranking expressions so that the polynominal completeness is realized with appropriately necessary minimum nodes the corresponding interpolation bases in the conventional form can easily be derived through the basis transformation without matrix inversion

convergence and stability of numerical solutions to sddes with markovian switching a class of linear stochastic delay differential equations with markovian switching is considered the main aim of this paper is to investigate the convergence and stability of the euler method of the equations it is proved that the euler method is convergent with strong order p 1 2 the ms stable properties of the euler scheme are also studied

symplectic conditions and stochastic generating functions of stochastic runge kutta methods for stochastic hamiltonian systems with multiplicative noise abstract this paper develops numerical approximate algorithms for the solutions of stochastic hamiltonian systems with multiplicative noise some conditions are captured to guarantee that a given stochastic runge kutta method is symplectic this paper also shows that stochastic symplectic runge kutta methods can be written in terms of stochastic generating functions based on the order and symplectic conditions some low stage stochastic symplectic runge kutta methods with strong global order 1 0 are constructed some numerical results are presented to demonstrate the efficiency of stochastic symplectic runge kutta methods

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

mean square stability of analytic solution and euler maruyama method for impulsive stochastic differential equations from the view of algebra the mean square stability of analytic solutions and numerical solutions for impulsive stochastic differential equations are considered by the logarithmic norm the conditions under which the analytic and numerical solutions for a linear impulsive stochastic differential equation are mean square stable ms stable respectively are obtained the conditions are simple and easy to use some numerical experiments are given to illustrate the results

numerical treatment of stochastic delay differential equations a global error bound abstract motivated by the error analysis of numerical solution for stochastic delay differential equations a new result indicating the dependency of convergence rate of the scheme on two separate quantities p and q where the former is the global order of underlying scheme and the latter is that of interpolation approximating the history of system is presented as an objective instance the stochastic runge kutta srk schemes of global order 0 5 1 and 1 5 in the strong convergence sense are utilized to formulate three one step schemes to approximate the solution of the problem moreover the mean square stability of these schemes will be asymptotically developed finally the theoretical results will be confirmed by some test problems

a boundary value problem for nematic liquid crystals with a variable degree of orientation in this paper we represent the degree of microscopic order through a scalar valued function the degree of orientation which vanishes where the fluid becomes isotropic

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

two boundedness and monotonicity preserving methods for a generalized fisher kpp equation two finite difference methods are proposed for a generalized fisher kpp equation these methods are capable of preserving the skew symmetry of the equation these methods preserve the positivity boundedness and monotonicity of solutions these methods are solvable these methods are convergent a semi explicit finite difference method and an implicit finite difference method are proposed for a generalized fisher kpp equation with two space variables it is proved that these two methods could preserve the skew symmetry of this equation moreover under a condition on the step sizes the semi explicit method is capable of preserving the positivity the boundedness and the spatial and temporal monotonicity of initial approximations and the implicit method is able to preserve these properties with no restriction on the step sizes the stability and convergence of these two methods are also analyzed respectively finally some numerical simulations are provided to verify the validity of our analytical results

on the exponential stability in mean square of neutral stochastic functional differential equations in 1 2 some efforts have been devoted to the investigation of exponential stability in mean square of neutral stochastic functional differential equations however the results derived there are either difficult to demonstrate in a straightforward way for practical situations or somewhat too restricted to be applied to general neutral stochastic functional differential equations for instance nonautonomous cases in this paper we shall establish some results which are more effective and relatively easy to verify to obtain the required stability

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

semi radial singularity mapping theory for line singularities in fracture mechanics abstract fundamental theory of the semi radial singularity mapping dealing with line singularities in the stress and strain is presented explicitly the brick type semi radial singularity mapping element as well as the wedge type one are proposed element familization is then described based on the conventional polynomial interpolations for the cube and simplex with examinations of the elaborate trial function spaces

on nematic liquid crystals with variable degree of orientation energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals this mathematical model was recently posed by j l erickson in this paper we shall prove the existence local uniqueness and global regularity of minimizers we shall also estimate the size of possible defects

analytical solutions for the incompressible viscous flow within a rectangular domain abstract analytical solutions of the velocity pressure and stream function are developed for the slow incompressible viscous fluid in a simple rectangular domain all solutions can be applied to the boundary integral techniques as typical fundamental solutions

exponential stability in mean square of neutral stochastic differential functional equations abstract the aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d x t g xt f t x t g t xt dt t xt dw t where xt x t s s 0 with 0 is the past history of the solution several interesting examples are a given for illustration

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

exponential stability of euler maruyama solutions for impulsive stochastic differential equations with delay this paper establishes a method to study the exponential stability of euler maruyama em method for impulsive stochastic differential equations with delay by using the properties of m matrix and stochastic analysis technique some conditions under which the em solution is exponentially mean square stable are obtained some examples are provided to illustrate the results

singular element construction using a mapping technique conformal mapping regularizes in the new plane the singular stress field around the crack tip so conventional finite elements can be used in the transformed plane the inverse mapping of the above elements gives another class of mapped elements the crack tip mapped element is a singular element thus using various elements the parent elements in the mapped plane we obtain an infinity of singular elements the product elements these singular elements are introduced naturally and have no defect as many other singular elements have

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

stochastic symplectic partitioned runge kutta methods for stochastic hamiltonian systems with multiplicative noise a new class of sprk methods and their order conditions are given symplectic conditions of the sprk methods are obtained stochastic generating functions of the ssprk methods are obtained it is proved that the ssprk methods can preserve the quadratic invariants some low stage ssprk methods are constructed some new stochastic partitioned runge kutta sprk methods are proposed for the strong approximation of partitioned stochastic differential equations sdes the order conditions up to strong global order 1 0 are calculated the sprk methods are applied to solve stochastic hamiltonian systems with multiplicative noise some conditions are captured to guarantee that a given sprk method is symplectic it is shown that stochastic symplectic partitioned runge kutta ssprk methods can be written in terms of stochastic generating functions in addition this paper also proves that the ssprk methods can conserve the quadratic invariants of original stochastic systems based on the order and symplectic conditions some low stage ssprk methods with strong global order 1 0 are constructed finally some numerical results are presented to demonstrate the efficiency of the ssprk methods

discrete orthogonal polynomials inherent to sampling points at equal intervals abstract the discrete orthogonal polynomials in finite series are proposed for arbitrary number of sampling points at equal intervals recurrence rules are presented so as to realize recurrent computations in the spectral data smoothing applications

infinite domain elements a parametric element is formulated which enables the economic modelling of infinite domain type problmes a typical problem is an opening in a stress field in an infinite medium either in two or three dimensions the strategy is to model around the opening with two or three layers of conventional isoparametric finite elements and surround these with a single layer of infinite domain elements several sample problems has been analysed for circular square and spherical openings in infinite media and the results compared with either theoretical or boundary element solutions which include the infinite boundary in their solution technique copyright

introduction to the numerical analysis of stochastic delay differential equations abstract we consider the problem of the numerical solution of stochastic delay differential equations of ito form d x t f x t x t d t g x t x t d w t t 0 t and x t t for t 0 with given f g wiener noise w and given 0 with a prescribed initial function we indicate the nature of the equations of interest and give a convergence proof for explicit single step methods some illustrative numerical examples using a strong euler maruyama scheme are provided

a boundary integral approach in the geoelectrical cavity prospecting abstract the fredholm integral equation of the second kind with respect to electrical potential is derived associated with the cavity concealed within the earth in this approach the positioning constant to be attached to the governing equation for the fundamental solution is of great significance and we formulate the positioning constant at an unsmooth point on the boundary interface through the spherical geometry it is shown that the numerical solutions are in sufficient agreement with the experimental data

full explicit interpolation formulas for the argyris triangle abstract explicit interpolation formulas are presented in terms of area coordinates for the argyris triangle with 21 degrees of freedom so that c 1 continuity is clearly observed it is further shown that 5 unisolvency can be proved in a very direct manner

general complete lagrange family for the cube in finite element interpolations abstract in this paper we have first derived the interpolation polynomials for the general serendipity elements which allow arbitrarily placed nodes along the edges we have then presented a method to determine the interpolation functions for the general complete lagrange elements which allow arbitrarily placed nodes explicit expressions for interpolation functions of the serendipity and complete lagrange family elements which allow uniform spacing of nodes over the element domain are derived for elements of orders 4 10 we have also modified the shape functions of complete lagrange family so that they can correctly interpolate the complete polynomial in the global space for angular distortions

some general lagrange interpolations over simplex finite elements with reference to derivative singularities abstract for simplex finite elements the native lagrange family with arbitrarily placed nodes is presented in hierarchy ranking expressions it includes the well known complete lagrange family as well as the mid edge lagrange family to be proposed in this paper this new family enables us to utilize harmonious combinations of interpolations of different orders in finite element analysis as an application of developed simplex interpolations to fracture mechanics where some derivative singularities are needed we then describe the semi radial singularity mapping with examinations of peculiar trial function spaces

analytical integral formulae related to convex quadrilateral finite elements abstract analytical quadrature formulae for rational functions integrated over finite elements with quadrilateral geometry are presented definitively for second order differential equations these formulae produce algebraic finite element relations associated with the isoparametrical or subparametrical rectangular interpolations

the mapped elements for the solution of cracked bodies abstract using a conformal mapping the singular stress field around crack tips is transformed to a regular one with respect to the new variables thus the displacement field is approximated in the mapped plane the inverse mapping gives shape functions which are not polynomials the strain field obtained after a derivation of the displacement field has the appropriate square root singularity the elements constructed by using this procedure are called mapped elements this simple method was applied with excellent results to the solution of a series of problems

singularity and positioning constants in poisson s equation with a point source of unit intensity abstract the singularity constants and positioning constants are described in the potential field governed by poisson s equation with a point source of unit intensity it is then demonstrated that the singularity constants play an important role in the integral equation formulations utilizing rather complicated fundamental solutions

quadrilateral element with semi radial singularity mapping the singularity mapping elementology based on the constant and semi radial field conditions is presented in an originally quadrilateral finite element with the complete evaluations of singularities in the stress and strain as well as of the strain energy finiteness it is demonstrated that this singularity element family is highly versatile differences and some identities between the semi radial mapping and the so called quarter point mapping are also examined

infinite elements with 1 rn type decay this paper presents a method of developing a family of 1 rn type infinite elements for the analysis of problems definite in unbounded media the proposed method is a direct extension of the conventional finite element method the resulting improper integrals are integrated exactly over the infinite element domains two numerical examples in elastic half space static problems are investigated to illustrate the applicability and accuracy of the method the use of the proposed infinite elements yields excellent results and preserves all the advantages of the finite element method

mean square stability of semi implicit euler method for linear stochastic differential equations with multiple delays and markovian switching in this paper we study the mean square ms stability and general mean square gms stability of semi implicit euler method for linear stochastic differential equations with multiple delays and markovian switching it is proved that semi implicit euler method is ms stable and gms stable under suitable conditions a numerical example is provided to illustrate the theoretical results

exponential stability of the split step θ method for neutral stochastic delay differential equations with jumps abstract the exponential mean square stability of the split step method for neutral stochastic delay differential equations nsddes with jumps is considered new conditions for jumps are proposed to ensure the exponential mean square stability of the trivial solution if the drift coefficient satisfies the linear growth condition it is shown that the split step method can reproduce the exponential mean square stability of the trivial solution for the constrained stepsize then by applying the chebyshev inequality and the borel cantelli lemma the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained since split step method covers euler maruyama em method and split step backward euler ssbe method the conclusions are valid for these two methods moreover they can adapt to the nsddes and the sddes with jumps finally a numerical example illustrates the effectiveness of the theoretical results

on unified concepts of detectability and observability for continuous time stochastic systems this paper is concerned with detectability and observability of continuous time stochastic linear systems by adopting the idea used in defining these two concepts for time varying systems and markovian jump linear systems that have been studied in the literature corresponding definitions for continuous time stochastic linear systems are proposed these two definitions are not only able to unify some recent definitions on these two concept reported in the literature but also allow us to propose an efficient rank criterion to test observability of continuous time stochastic linear systems it seems that this rank criterion is quite analogous to the rank criterion for deterministic linear systems with the help of these two concepts and the new criteria the stochastic lyapunov equation is revisited and some recent published work on this equation are generalized numerical examples are given to illustrate the effectiveness of the proposed approach

landau de gennes theory of nematic liquid crystals the oseen frank limit and beyond we study global minimizers of a continuum landau de gennes energy functional for nematic liquid crystals in three dimensional domains subject to uniaxial boundary conditions we analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly in w1 2 to a global minimizer predicted by the oseen frank theory for uniaxial nematic liquid crystals with constant order parameter moreover the convergence is uniform in the interior of the domain away from the singularities of the limiting oseen frank global minimizer we obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the landau de gennes global minimizer we also study the interplay between biaxiality and uniaxiality in landau de gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

a finite element alternative to infinite elements abstract in this paper a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior and not in the far field by developing a finite element mesh that approximates the stiffness of an unbounded domain directly without approximating the far field displacement profile first the current formulation provides a superior alternative to infinite elements ies that have long been used to incorporate unbounded domains into the finite element method fem in contrast to most ies the present formulation a requires no new shape functions or special integration rules b is proved to be both accurate and efficient and c is versatile enough to handle a large variety of domains including those with anisotropic stratified media and convex polygonal corners in addition to this the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved this error expression can be used to calculate the accuracy of a given mesh a priori this in turn allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem we formulate such an optimization problem solve it and use the results to develop a practical mesh generation methodology this methodology does not require any additional computation on the part of the user and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method

explicit interpolation formulas for the bell triangle abstract explicit interpolation formulas are presented mainly using area coordinates for the bell triangle with 18 degrees of freedom so that c 1 continuity is clearly observed the bell triangle is derived by imposing the p 4 admissible constraints on the argyris triangle by which the 4 unisolvency is proved variable node expressions combining the argyris and bell triangles are further developed

s rock methods for stochastic delay differential equations with one fixed delay abstract we propose stabilized explicit stochastic runge kutta methods of strong order one half for ito stochastic delay differential equations with one fixed delay the family of the methods is constructed by embedding runge kutta chebyshev methods of order one for ordinary differential equations the values of a damping parameter of the methods are determined appropriately in order to obtain excellent mean square stability properties numerical experiments are carried out to confirm their order of convergence and stability properties

gauss jacobi quadrature rules for n simplex with applications to finite element methods abstract for numerical integrations over the n simplex the gauss jacobi quadrature operators are presented definitively it is demonstrated in finite element applications that the gauss jacobi quadratures could be highly effective under the isoparametric mapping as well as under the semi radial singularity mapping

some properties of minimal surfaces in singular spaces this paper involves the generalization of minimal surface theory to spaces with singularities let x be an npc space i e a metric space of non positive curvature we define a parametric minimal surface in x as a conformal energy minimizing map using this definition many propertlies of classical minimal surfaces can also be observed for minimal surfaces in this general setting in particular we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in x

convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay in this paper we study the convergence and stability of the stochastic theta method stm for a class of index 1 stochastic delay differential algebraic equations first in the case of constrained mesh i e the stepsize is a submultiple of the delay it is proved that the method is strongly consistent and convergent with order 1 2 in the mean square sense then the result is further extended to the case of non constrained mesh where we employ linear interpolation to approximate the delay argument later under a sufficient condition for mean square stability of the analytical solution it is proved that when the stepsizes are sufficiently small the stm approximations reproduce the stability of the analytical solution finally some numerical experiments are presented to illustrate the theoretical findings

split step theta method for stochastic delay integro differential equations with mean square exponential stability abstract in this paper we propose the split step theta method for stochastic delay integro differential equations by the lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme it is shown that the split step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper a numerical example is given at the end of the paper to illustrate the method and conclusion of the paper in addition the convergence of the split step theta method is proved in the appendix

the pricing of european options on two underlying assets with delays abstract in the paper the pricing of european options on two underlying assets with delays is discussed by using the approach of equivalent martingale measure transformation the market is proved to be complete with exchange option as a particular example we obtain the explicit pricing formula in a subinterval of option period the robust euler maruyama method is combined with the monte carlo simulation to compute exchange option prices within the whole option period numerical experiments indicate that there is an increasing possibility of the difference between the delayed and black scholes option prices with the increase of delay

rectifiability of line defects in liquid crystals with variable degree of orientation in 2 hardt lin and the author proved that the defect set of minimizers of the modified ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and holder continuous curves with finitely many crossings in this article we show that each holder continuous curve in the defect set is of finite length hence locally the defect set is rectifiable for the most part the proof closely follows the work of de lellis et al rectifiability and upper minkowski bounds for singularities of harmonic q valued maps arxiv 1612 01813 2016 on harmonic mathcal q valued maps the blow up analysis in alper et al calc var partial differ equ 56 5 128 2017 allows us to simplify the covering arguments in 11 and locally estimate the length of line defects in a geometric fashion

fundamental theory of the semi radial singularity mapping with applications to fracture mechanics abstract fundamental theory of the semi radial singularity mapping is presented which yields the necessary singularities in the stress and strain with bounded strain energy in particular the quadrilateral semi radial singularity mapping as well as the triangular one with collapse are described explicitly with some verification example in linear fracture mechanics the trial function spaces due to the conventional polynomial interpolations with these peculiar mappings are examined in detail

generalized lagrange family for the cube with special reference to infinite domain interpolations abstract as a general family allowing arbitrarily placed nodes the generalized lagrange family for the cube is presented in hierarchy ranking expressions this new family contains the well known regular mid edge and complete lagrange families as subfamilies the generalized lagrange interpolations are then applied to the infinite domain and the semi radial infinity mapping is proposed with examinations of elaborate trial function spaces

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we