numerical solution of singular regular boundary value problems by pole detection with qd algorithm in this paper we present a method using qd algorithm combined with finite difference method for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficients we can also use qd algorithm combined with other numerical methods these problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry some illustrative examples are given to demonstrate the efficiency and high accuracy of the proposed method and compare it to the numerical results made with other methods

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

on the efficient numerical solution of systems of second order boundary value problems an explicit reduction is given of the centered euler scheme for a first order two dimensional system producing an accurate difference approximation for a second order scalar equation all of the advantages of the centered euler formula are retained except for the condition number results of numerical experiments are presented

a compact finite difference method for a general class of nonlinear singular boundary value problems with neumann and robin boundary conditions abstract in this paper we develop and analyze a high order compact finite difference method cfdm for solving a general class of two point nonlinear singular boundary value problems with neumann and robin boundary conditions arising in various physical models convergence result of this method is established through matrix analysis approach to illustrate the applicability and accuracy of the method we consider nine numerical examples including heat conduction in the human head equilibrium of isothermal gas sphere oxygen diffusion in a spherical cell and reaction diffusion process in a spherical permeable catalyst it is shown that the computational order of convergence of the proposed cfdm is four the obtained results are compared with those obtained by other existing numerical methods

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

numerical solution of singular boundary value problems via chebyshev polynomial and b spline in this paper we present b spline method for numerically solving singular two point boundary value problems for certain ordinary differential equation having singular coefficients these problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry to remove the singularity we first use chebyshev economizition in the vicinity of the singular point and the boundary condition at a point email protected in the vicinity of the singularity is derived the resulting regular bvp is then efficiently treated by employing b spline for finding the numerical solution some examples have been included and comparison of the numerical results made with other methods

b spline solution of singular boundary value problems homogeneous and non homogenous singular boundary value problems special case are solved using b splines the original differential equation is modified at singular point then the boundary value problem is treated by using b spline approximation the method is tested on some model problems from the literature and the numerical results are compared with exact solution

the use of cubic splines in the numerical solution of a system of second order boundary value problems abstract we use uniform cubic polynomial splines to develop some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for 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

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

the construction of λµ b spline curves and its application to rotational surfaces in order to provide more flexible approaches for designers the b spline curves are constructed as a generalization of the traditional cubic uniform b spline curves possessing multiple local shape control parameters b spline curves not only inherit the properties of cubic uniform b spline curves but also exhibit better performance when adjusting its local shapes through two local shape control parameters particularly to adjust and control the shapes of rotational surfaces more elegantly the b spline rotational surfaces with two local shape parameters are presented and utilized a rotational surface is produced by combining b spline with a transfinite vector valued rational interpolation function further the properties of rotational surfaces as well as its applications in rotational surface designs are explored finally the modeling examples are supplied to illustrate the proposed method in admitting the easy control of the shape of a surface which suggest the much wider applications to the pattern design system of apparel cad cam

chaos game for ifss on topological spaces abstract we explore the chaos game on continuous ifss on topological spaces we prove that the existence of an attractor implies that the chaos game can be used to construct approximations of the attractor the essential role of basin of attraction is discussed

numerical study for a class of singular two point boundary value problems using green s functions in this paper we present a numerical method based on a non uniform mesh for a class of singular two point boundary value problems green s function of the singular boundary value problem is used to derive an equivalent integral equation which is then solved by interpolation with quadratic polynomials it is shown to be third order convergent numerical results are presented which are in agreement with the theoretical error bound

a novel method for nonlinear two point boundary value problems combination of adm and rkm abstract in this paper a novel method is proposed for solving nonlinear two point boundary value problems bvps this method is based on a combination of the adomian decomposition method adm and the reproducing kernel method rkm a major advantage of this method over standard adm is that it can avoid unnecessary computation in determining the unknown parameters the proposed method can be applied to singular and nonsingular bvps numerical results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for solving nonlinear two point boundary value problems

a new method for solving a class of singular two point boundary value problems abstract in this paper we present a new method for solving a class of singular two point boundary value problems for certain ordinary differential equations having singular coefficients the exact solution u x is represented in the form of series in the reproducing kernel space in the mean time the n term approximate solution u n x is obtained by solving a linear system of equations and is proved to converge to the exact solution this method can avoid the gram schmidt orthogonalization process so as to improve the precision and decrease consumedly the runtime when the number of knots is the same especially when the number of knots is large it is efficiently applied to solve some model problems with the comparison between the numerical solutions and the exact solutions the high precision of this method is demonstrated by the numerical examples

b spline method for a class of singular two point boundary value problems using optimal grid in this paper we present b spline method for solving singular two point boundary value problems for a certain differential equation having singular coefficients the given singular boundary value problem is modified at the point of singularity the modified problem is then solved by b spline in the case of non linear problems we first linearize the equation using quasilinearization technique and the resulting problem is solved by b spline error estimates are obtained which enable a deferred correction to be made optimal grid has been generated using the method some numerical examples have been included to demonstrate the efficiency of the method

numerical solution of singular boundary value problems with logarithmic singularities by pade approximation and collocation methods in this paper we present a method using quintic b spline and pade approximation for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficient we use the modified quintic b spline to approximate the first coefficients in the taylor expansion on which all the other coefficients depend moreover we introduce the superposed pade approximation the resulting perturbed initial value problem is then efficiently treated

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

relaxed notions of curvature and a lumped strain method for elastic plates the paper proposes an approximation method lumped strain method lsm for elastic plates based on generalized notions of the gaussian and mean curvatures and on a relaxation of the energy functional to the space of continuous piecewise linear functions the method adapts ideas from the theory of gamma convergence we restrict our attention to the case of the simply supported plate and state the proof of convergence of the method we give an application to the rhombic plate and compare the results with those obtained by standard finite element approximations

γ convergence of external approximations in boundary value problems involving the bi laplacian in addition to the various uses it was introduced for the theory of convergence offers a rather natural setting for discussing and developing nonorthodox approximation methods for variational problems for certain boundary value problems involving the bi laplacian sequences of discrete functionals are here defined and are shown to converge to the corresponding functionals of the continuous problems the minimizers of the discrete functionals provide converging approximations to the solution of the limit problem in question thus we obtain approximation schemes that are nonconforming but direct and that can be treated by current algorithms for symmetric and positive definite functionals the class of problems considered in this paper includes the stokes problem in fluid dynamics the loading problem of 2 d isotropic elastostatics and some boundary value problems of the kirchhoff love theory of plates also discussed is an extension of the discretization method that seems suitable for treating more general boundary value problems of elastic plates but whose convergence is conditional to a conjecture that remains to be proved a relevant application to the so called babuska paradox is presented

degree mixing in multilayer networks impedes the evolution of cooperation traditionally the evolution of cooperation has been studied on single isolated networks yet a player especially in human societies will typically be a member of many different networks and those networks will play different roles in the evolutionary process multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society with this motivation we here consider two layer scale free networks with all possible combinations of degree mixing wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating we find that breaking the symmetry through assortative mixing in one layer and or disassortative mixing in the other layer as well as preserving the symmetry by means of assortative mixing in both layers impedes the evolution of cooperation we use degree dependent distributions of strategies and cluster size analysis to explain these results which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas

a numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall abstract this paper is concerned with multiple solutions of a singular nonlinear boundary value problem bvp on the interval 0 1 which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall for the singular nonlinear bvp the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution then a numerical technique is proposed to find all possible multiple solutions for the suction driven pipe flow with the expanding wall e g 2 we find a new solution numerically and classify it as a type vi solution the computed results agree well with what can be obtained by the bifurcation package auto in addition we also construct asymptotic solutions for a few cases of parameters which agree well with numerical solutions these serve as validations of our numerical results thus we believe that the numerical technique designed in the paper is reliable and may be further applied to solve a variety of nonlinear equations that arise from other flow problems

two population three player prisoner s dilemma game we present a two population three player prisoner s dilemma finite state automata are used different types of tit for tat strategies all possible payoffs are computed due to the computational advantage in symmetric games most researches have focused on the symmetric games instead of the asymmetric ones which need more computations in this paper we present prisoner s dilemma game involving three players and suppose that two players among them agree against the third player by choosing either to cooperate together or to defect together at each round according to that assumption the game is transformed from the symmetric three player model to asymmetric two player model such that the identities of the players cannot be interchanged without interchanging the payoff of the strategies each strategy in the resulting model is expressed with two state automata we determine the payoff matrix corresponding to the all possible strategies we noticed that for some strategies it is better to be a player of the first type independent player than being of the second type allies

a quartic b spline for second order singular boundary value problems second order singular boundary value problems are treated using quartic fourth degree b spline approximations the values of the coefficients c i are chosen via optimization error analysis is discussed the efficiency of the method is illustrated with several problems from the literature

haar wavelet collocation method for lane emden equations with dirichlet neumann and neumann robin boundary conditions abstract in this paper we present a numerically stable algorithm based on the haar wavelet collocation method hwcm for numerical solution of a class of lane emden equation with dirichlet neumann and neumann robin type boundary conditions arising in various physical models the haar wavelet collocation method uses simple box functions and hence the formulation of the numerical algorithm is straightforward unlike other methods the hwcm can be applied easily and accurately to the neumann type boundary conditions the advantage of the haar wavelet method is its efficiency and simple applicability for a variety of boundary conditions the haar wavelet collocation method is utilized to reduce the lane emden equation with boundary conditions to a system of algebraic equations then the newton s method is employed for numerical solutions nine examples of the lane emden equation with various types of boundary conditions are provided to demonstrate the accuracy and efficiency of the hwcm the numerical results are compared with the results obtained by other techniques and the exact solution

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

runge kutta software with defect control four boundary value odes a popular approach to the numerical solution of boundary value ode problems involves the use of collocation methods such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval on the other hand several authors have suggested as an alternative certain subclasses of the implicit runge kutta formulas known as mono implicit runge kutta mirk formulas which can be implemented at a lower cost per step than the collocation methods these latter formulas do not have a natural implementation that provides a continuous approximation to the solution rather only a discrete approximation at certain points within the problem interval is obtained however recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit runge kutta formula these ideas have recently been extended to develop continuous extensions of the mirk formulas in this paper we describe our investigation of the use of continuous mirk formulas in the numerical solution of boundary value ode problems a primary thrust of this investigation is to consider defect control based on the continuous mirk formulas as an alternative to the standard use of global error control as the basis for termination and mesh redistribution criteria

corners give problems when decoupling fourth order equations into second order systems decoupling a fourth order elliptic boundary value problem into a system of second order equations may give an alternative approach to obtaining a solution for the original problem the system approach is usually preferred for numerical methods since one may use piecewise linear finite elements to approximate a solution in this note we compare both settings for the plate equation and prove that for planar domains with corners the system approach may fail to produce the correct solution

numerical solution of burgers fisher equation by cubic b spline quasi interpolation abstract in this paper numerical solution of the burgers equation is presented based on the cubic b spline quasi interpolation at first the cubic b spline quasi interpolation is introduced moreover the numerical scheme is presented by using the derivative of the quasi interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable the accuracy of the proposed method is demonstrated by some test problems the numerical results are found in good agreement with exact solutions the advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement

an unconstrained mixed method for the biharmonic problem in this work we present a finite element method for the biharmonic problem based on the primal mixed formulation of ciarlet and raviart a mixed finite element method for the biharmonic equation in symposium on mathematical aspects of finite elements in partial differential equations c de boor ed academic press new york 1974 pp 125 143 we introduce a dual mesh and a suitable approximation of the constraint that enables us to eliminate the auxiliary variable with no computational effort thus the discrete problem turns to be governed by a system of linear equations with symmetric and positive definite coefficients and can be solved by classical algorithms the construction of the stiffness matrix is obtained by using courant triangles and can be done with great efficiency

matrix games with payoffs of belief structures imprecise matrix games such as interval valued matrix games and fuzzy matrix games have attracted much interest for a long time most of the previous studies on imprecise matrix games mainly focus on the fuzzy uncertainty of payoffs however the uncertainties of nonspecificity and discord involved in payoffs are not well addressed so far the purpose of this paper is to study the matrix game with such types of uncertainties in order to achieve that purpose we present a matrix game model with payoffs of belief structures so as to integrate discord and nonspecificity the proposed model can be used to express more imprecise interactions between players in the reality besides as another main contribution of the study an effective method for solving matrix games with belief structures payoffs is developed to help us find the equilibrium points of the kind of games at last an example is given to illustrate the proposed model and method

numerical solution of singular boundary value problems by invariant imbedding abstract an invariant imbedding method is proposed for numerically solving singular boundary value problems with a regular singular point at one end of the interval the singular problem is first reduced to a regular boundary value problem by using series solution in the vicinity of the singular point to subtract out the singularity the regular boundary value problem is then solved by employing invariant imbedding method some numerical examples have been included to demonstrate the efficiency of the method

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

uncovering cooperative behaviors with sparse historical behavior data in the spatial games for past decades the main attention of the evolutionary games has been focused on cooperation mechanism with the assumption that the strategy information of all players are known however it is difficult for observers to obtain the global information of players strategies in the real world and some players even hide their strategy information to confuse their opponents here we try to solve the problem to predicate the hidden strategies with sparse historical behavior data in the evolutionary games to quantify the similarity of strategies among the players in our method the euclidean distance of players is defined from the strategies of the players in the few past rounds then the hidden strategy of a player will be determined from the tendency that players with minimum euclidean distance will adopt similar strategies the method has good performance on determining hidden strategy of human beings in both the prisoner s dilemma game and the public goods game where strategies of twenty five percent players are hidden and the success rate to determine hidden strategy reaches up to 0 9 it is also found that the success rate to determine hidden strategy depends on both length of historical behavior data and tempting payoff b the prisoner s dilemma game or multiple factor r the public goods game

a new high order numerical method for solving singular two point boundary value problems abstract recently goh et al 2012 28 proposed a numerical technique based on quartic b spline collocation for solving a class of singular boundary value problems sbvp with neumann and dirichlet boundary conditions bc this method is only fourth order accurate in this paper we propose an optimal numerical technique for solving a more general class of nonlinear sbvp subject to neumann and robin bc the method is based on high order perturbation of the problem under consideration the convergence of the proposed method is analyzed to demonstrate the applicability and efficiency of the method we consider four numerical examples three of which arise in various physical models in applied science and engineering a comparison with other available numerical solutions has been carried out to justify the advantage of the proposed technique numerical result reveals that the proposed method is sixth order convergent which in turn is two orders of magnitude larger than in goh et al 2012 28

numerical method using cubic b spline for the heat and wave equation in the present paper the cubic b splines method is considered for solving one dimensional heat and wave equations a typical finite difference approach had been used to discretize the time derivative while the cubic b spline is applied as an interpolation function in the space dimension the accuracy of the method for both equations is discussed the efficiency of the method is illustrated by some test problems the numerical results are found to be in good agreement with the exact solution

collocation for singular boundary value problems of second order the collocation methods for solving linear boundary value problems with a singularity of the first kind are investigated the stability and convergence results are established particular attention is paid to the superconvergence properties of the methods which are different as in the classical case when the singularity is not present the results are extended to the nonlinear case and illustrated by numerical examples

cubic spline polynomial for non linear singular two point boundary value problems we present a cubic spline polynomial for the solution of non linear singular two point boundary value problems the quesilinearization technique is used to reduce the non linear problem to a sequence of linear problems the resulting sets of differential equations are modified at the singular point and are treated by using cubic spline for finding the numerical solution the method is tested on three physical model problems from the literature

singular boundary value problems for odes this paper is concerned with the numerical solution of a system of ordinary differential equations odes y sy t f t y p on an interval 0 b subject to boundary conditions 0 g y 0 y b p the odes have a coefficient that is singular at t 0 but it is assumed that the boundary value problem bvp has a smooth solution some popular methods for bvps evaluate the odes at t 0 this paper deals with the practical issues of solving this class of singular bvps with such a method the bvp4c solver of matlab has been modified accordingly so that it can solve a class of singular bvps as effectively as it previously solved non singular bvps

a numerical method for singular two point boundary value problems via chebyshev economizition in this paper we present a numerical method for solving a two point boundary value problem in the interval 0 1 with regular singularity at x 0 by employing the chebyshev economizition on 0 d where d is near the singularity we first replace it by a regular problem on some interval d 1 the stable central difference method is then employed to solve the problem over the reduced interval some numerical results are presented to demonstrate the applicability of the method

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

stable high order methods for differential equations with small coefficients for the second order terms abstract stable high order methods are developed for solving differential equations with small coefficients for the second order terms in particular second fourth and sixth order methods are developed the methods are developed for general linear two point boundary value problems and proved to be stable and accurate results are presented for the one dimensional convection diffusion equation involving problems with and without boundary layers the methods converged for all values of parameters attempted and the results compared favorably with other methods the methods developed in this paper are accurate stable easy to use and applicable to other problems

higher order finite difference method for a class of singular boundary value problems in this paper a fourth order finite difference method for a class of singular boundary value problems is presented the original differential equation is modified at the singular point the fourth order finite difference method is then employed to solve the boundary value problem some model problems are solved and the numerical results are compared with exact solution

design of software for odes we discuss the design of software that is easy to use for simple problems but still capable of solving complicated problems nowadays a design should accommodate related but fundamentally different tasks such as solving daes and ddes in addition to ivps and bvps for odes major issues are illustrated with the matlab ode suite and selected solvers written in fortran 90 and maple

a difference method for singular two point boundary value problems a difference method based on uniform mesh for solving singular two point boundary value problems of the form x ay f x y 0

evolutionary mixed games in structured populations cooperation and the benefits of heterogeneity evolutionary games on networks traditionally involve the same game at each interaction here we depart from this assumption by considering mixed games where the game played at each interaction is drawn uniformly at random from a set of two different games while in well mixed populations the random mixture of the two games is always equivalent to the average single game in structured populations this is not always the case we show that the outcome is in fact strongly dependent on the distance of separation of the two games in the parameter space effectively this distance introduces payoff heterogeneity and the average game is returned only if the heterogeneity is small for higher levels of heterogeneity the distance to the average game grows which often involves the promotion of cooperation the presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin and they also emphasize the importance of the population structure in amplifying facilitators of cooperation

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

a lumped stress method for plane elastic problems and the discrete continuum approximation this paper proposes a rational method to approximate a plane elastic body through a latticed structure composed of truss elements the method is based on the introduction of a relaxed stress energy that allows an extension of the original problem to a larger space of admissible stress fields including stresses concentrated along lines use is made of polyhedral approximations of the airy stress function the truss analogy is employed to obtain a displacement formulation the paper includes several numerical applications of the method to sample problems a numerical convergence study and comparisons with exact solutions and standard finite element approximations

convergence analysis for derivative dependent fredholm hammerstein integral equations with green s kernel abstract in this article we consider a class of derivative dependent fredholm hammerstein integral equations i e the integral equation where the nonlinear function inside the integral sign is dependent on derivative and the kernel function is of green s type we propose the piecewise polynomial based galerkin and iterated galerkin methods to solve these type of derivative dependent fredholm hammerstein integral equations we discuss the convergence and error analysis of the proposed methods and also obtain the superconvergence results for iterated galerkin approximations some numerical results are given to illustrate this improvement

reformulation of boundary value problems into standard form boundary value problems in ode s arising in various applications are frequently not in the standard form required by the currently existing software however many problems can be converted to such a form thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software here various conversion devices are surveyed

efficient mesh selection for collocation methods applied to singular bvps we describe a mesh selection strategy for the numerical solution of boundary value problems for singular ordinary differential equations this mesh adaptation procedure is implemented in our matlab code sbvp which is based on polynomial collocation we prove that under realistic assumptions our mesh selection strategy serves to approximately equidistribute the global error of the collocation solution thus enabling to reach prescribed tolerances efficiently moreover we demonstrate that this strategy yields a favorable performance of the code and compare its computational effort with other implementations of polynomial collocation

a stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems abstract we propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition the mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a lagrange multiplier as new unknowns working with a pair of bases forming a biorthogonal system we can easily eliminate the gradient of the solution and the lagrange multiplier from the saddle point system leading to a positive definite formulation using a superconvergence property of a gradient recovery operator we prove an optimal a priori estimate for the finite element discretization for a class of meshes

strategy updating depending on local environment enhances cooperation in prisoners dilemma game a new method of strategy updating is proposed in the prisoners dilemma game some players apply the replicator dynamic rule for strategy updating while the others utilize the comprehensive method the intelligent players would consider not only the neighbors payoff but also its environment the results indicate that with the increase of intelligent players the fraction of cooperation increases an evolutionary prisoners dilemma game is studied on a square lattice when local environment is considered and a new strategy updating rule is utilized in this game first individuals are classified as two classes stochastic players and intelligent players the stochastic players are the ones who adopt the rule of replicator dynamics to update strategies and the intelligent players are those who make a decision through comprehensive considerations of the local environment namely the information of different strategy neighbors environment second the intelligent players evaluation is calculated from payoffs and local environment by dempstershafer theory the results indicate that the new strategy updating method promotes cooperation when intelligent individuals utilize the information of different strategy neighborhoods moreover the increasing number of intelligent player promotes cooperation level furthermore it is noteworthy that cooperators will finally coexist with defectors no matter what the proportion of intelligence is which is worthy to think about