boundary value methods the third way between linear multistep and runge kutta methods it is well known that the approximation of the solutions of odes by means of k step methods transforms a first order continuous problem in a kth order discrete one such transformation has the undesired effect of introducing spurious or parasitic solutions to be kept under control it is such control which is responsible of the main drawbacks e g the two dahlquist barriers of the classical lmf with respect to runge kutta methods it is however less known that the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem starting from such an idea a new class of multistep methods called boundary value methods bvms has been proposed and analyzed in the last few years of course they are free of barriers moreover a block version of such methods presents some similarity with runge kutta schemes although still maintaining the advantages of being linear methods in this paper the recent results on the subject are reviewed

are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

some numerical algorithms for solving the highly oscillatory second order initial value problems abstract in this paper some numerical algorithms spectral collocation method block spectral collocation method boundary value method block boundary value method implicit runge kutta method diagonally implicit runge kutta method and total variation diminishing runge kutta method are used to solve the highly oscillatory second order initial value problems we first derive these methods for the first order initial value problems and then extend these methods to the highly oscillatory nonlinear systems by matrix analysis methods these new methods preserve the accuracy of the original methods and the main advantages of these new methods are low storage requirements and high efficiency extensive numerical results are presented to demonstrate the convergence properties of these methods

third order boundary value problems with nonlocal boundary conditions we treat the existence of multiple positive solutions for a third order boundary value problem with nonlocal boundary conditions in our work each of the three boundary conditions is a linear functional on c 0 1 that is given by a stieltjes integral but is not necessarily a positive functional

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

positive solutions to a multi point higher order boundary value problem the authors consider a higher order multi point boundary value problem some existence and nonexistence results for positive solutions of the problem are obtained by using krasnosel skii s fixed point theorem examples are included to illustrate the results

symmetric positive solutions to singular system with multi point coupled boundary conditions in this paper we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi point coupled boundary conditions the arguments are based upon a specially constructed cone and the fixed point index theorem in cones an example is then given to demonstrate the applicability of our results

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i case of type 1 fuzzy sets abstract the grade of membership is assumed to be determined as the relative size of evidence grounds supporting a given element s membership compared to all the necessary reference grounds relevant to the judgement of membership grade and is interpreted as conditional probability conditioned by the given element by assigning the dependence relations of the evidence grounds to those of the fuzzy subsets the set operators are defined as probabilistic operators with dependence relations the properties of the operators are studied and it is found that with type 1 fuzzy subsets they satisfy all the basic properties satisfied by the crisp set operators illustrative examples are given describing the construction of membership functions and the selection of set operators in relation to the situations to be modelled the proposed operators a kind of parametrized operators with an index value of the dependence relations as a parameter cover the entire operators ranging from the so called max min operators to the bold union intersection operators an extension of the above operator range is also discussed

on multiple fixed sign solutions of a discrete system with hermite boundary conditions abstract we consider the discrete system m u i k p i k u 1 k u 2 k u n k k 0 1 n 1 i n together with hermite boundary conditions j u i k 0 j 0 m 1 1 r 1 i n where r 2 m 1 for 1 r 1rm m and k s are integers such that 0 k1

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

a meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit adi approach and interpolating element free galerkin efg method a new numerical algorithm has been investigated for solving time fractional reaction subdiffusion equation the fractional derivative of the considered equation is described in the riemann liouville sense firstly we discrete the temporal dimension of the considered model using a finite difference scheme a central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order o 1 moreover to achieve a full discretization scheme a type of meshless method has been improved that is known as element free galerkin efg method the efg method for integration uses a background mesh this method is based on the galerkin weak form in which the test and trial functions are shape functions of moving least squares mls approximation since the shape functions of traditional mls lack the kronecker property essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue to this end a new class of mls shape functions has been applied that is called shape functions of interpolating mls the new shape function has the mentioned property in the efg method calculating the appeared two dimensional integrals is a basic issue in this research work the alternating direction implicit approach is combined with the element free galerkin method then using the new proposed method the two dimensional integrals on rectangular domain will be changed to simple one dimensional integrals we prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

multiple solutions and eigenvalues for third order right focal boundary value problems abstract we are concerned with the right focal boundary value problem x t f t x t t 1 t t 3 x t 1 x t 2 x t 3 0 and the associated eigenvalue problem x t a t f x t with the same boundary conditions under various assumptions on f a and we establish intervals of the parameter which yield the existence of a positive solution of the eigenvalue problem by placing certain restrictions on the nonlinearity we prove the existence of at least one at least two at least three and infinitely many positive solutions of the boundary value problem by applying some known fixed point theorems as well as some recent generalizations of these fixed point theorems

block boundary value methods for the solution of ordinary differential equations block boundary value methods are considered and applied to solve initial value problems results on their convergence and stability properties are presented and their intermediate position between the class of multistep and runge kutta methods is established

compact finite difference scheme and rbf meshless approach for solving 2d rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives abstract this article is devoted to the comparison between a high order difference scheme and radial basis functions rbfs meshless approach for the numerical solution of two dimensional fractional rayleigh stokes problem the fractional derivative of problem is described in the riemann liouville sense in the high order difference scheme we discretize the spaces derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme also in the rbf meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the kansa s approach to approximate the spatial derivatives we prove the stability and convergence for high order difference scheme using fourier analysis and for time discrete scheme in rbf method via energy approach we compare the results of compact finite difference method and rbf meshless approach in terms of accuracy and cpu time also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method

numerical solution to the unsteady two dimensional schrodinger equation using meshless local boundary integral equation method a meshless local boundary integral equation lbie method is proposed to solve the unsteady two dimensional schrodinger equation the method is based on the lbie with moving least squares mls approximation for the mls approximation nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables a time stepping method is employed to deal with the time derivative an efficient method for dealing with singular domain integrations that appear in the discretized equations is presented finally numerical results are considered for some examples to demonstrate the accuracy efficiency and high rate of convergence of this method copyright 2008 john wiley sons ltd

element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi type equation with robin boundary condition the traditional element free galerkin efg approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares mls approximation in the current paper we propose a new version of the efg method based on the shape functions of reproducing kernel particle method rkpm in other words based on the developed approach in han and meng 2001 the fractional tricomi type equation will be solved using the new technique the fractional derivative has been introduced in the caputos sense and is approximated by a finite difference plan of order o 3 1 2 we use the efg rkpm to discrete the spatial direction we illustrate some numerical results on non rectangular domains the unconditional stability and convergence of the new technique have been proved numerical examples display the theoretical results and the efficiency of the proposed approach also the numerical results are compared with the finite element method fem and efg mls procedure

fuzzy control of model car this paper discusses fuzzy control of a model car fuzzy control rules are derived by modelling an expert s driving actions experiments are performed using a model car with a sensing unit and a micro computer

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

jacobi gauss lobatto collocation method for the numerical solution of 1 1 nonlinear schrodinger equations a jacobi gauss lobatto collocation j gl c method used in combination with the implicit runge kutta method of fourth order is proposed as a numerical algorithm for the approximation of solutions to nonlinear schrodinger equations nlse with initial boundary data in 1 1 dimensions our procedure is implemented in two successive steps in the first one the j gl c is employed for approximating the functional dependence on the spatial variable using n 1 nodes of the jacobi gauss lobatto interpolation which depends upon two general jacobi parameters the resulting equations together with the two point boundary conditions induce a system of 2 n 1 first order ordinary differential equations odes in time in the second step the implicit runge kutta method of fourth order is applied to solve this temporal system the proposed j gl c method used in combination with the implicit runge kutta method of fourth order is employed to obtain highly accurate numerical approximations to four types of nlse including the attractive and repulsive nlse and a gross pitaevskii equation with space periodic potential the numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method indeed for relatively few nodes used the absolute error in our numerical solutions is sufficiently small

a two step high order compact scheme for the laplacian operator and its implementation in an explicit method for integrating the nonlinear schrodinger equation we describe and test an easy to implement two step high order compact 2shoc scheme for the laplacian operator and its implementation into an explicit finite difference scheme for simulating the nonlinear schrodinger equation nlse our method relies on a compact double differencing which is shown to be computationally equivalent to standard fourth order non compact schemes through numerical simulations of the nlse using fourth order runge kutta we confirm that our scheme shows the desired fourth order accuracy a computation and storage requirement comparison is made between the 2shoc scheme and the non compact equivalent scheme for both the laplacian operator alone as well as when implemented in the nlse simulations stability bounds are also shown in order to get maximum efficiency out of the method we conclude that the modest increase in storage and computation of the 2shoc schemes is well worth the advantages of having the schemes compact and their ease of implementation makes their use very useful for practical implementations

a fourth order compact solution of the two dimensional modified anomalous fractional sub diffusion equation with a nonlinear source term this work is concerned to the study of high order difference scheme for the solution of a two dimensional modified anomalous sub diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses the space fractional derivatives are described in the riemann liouville sense in the proposed scheme we discretize the space derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme we prove the stability and convergence of proposed scheme using the fourier analysis the convergence order of the proposed method is o t h x 4 h y 4 comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme

error estimate for the numerical solution of fractional reaction subdiffusion process based on a meshless method in this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction subdiffusion equation firstly we obtain a time discrete scheme based on a finite difference scheme then we use the meshless galerkin method to approximate the spatial derivatives and obtain a full discrete scheme in the proposed scheme some integrals appear over the boundary and the domain of problem which will be approximated using gauss legendre quadrature rule then we prove that the time discrete scheme is unconditionally stable and convergent using the energy method we show convergence order of the time discrete scheme is o the aim of this paper is to obtain an error estimate and to show convergence for the meshless galerkin method based on the radial basis functions numerical examples confirm the efficiency and accuracy of the proposed scheme

numerical solution of a biological population model using he s variational iteration method this paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations the variational iteration method and adomian decomposition method are used for solving this equation and then numerical results are compared with each other showing that the variational iteration method leads to more accurate results furthermore variational iteration method overcomes the difficulty arising in calculating the adomian s polynomials which is an important advantage over the adomian decomposition method

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

two grid method for two dimensional nonlinear schrodinger equation by mixed finite element method abstract a conservative two grid mixed finite element scheme is presented for two dimensional nonlinear schrodinger equation one newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse grid solution as the initial guess moreover error estimates are conducted for the two grid method it is shown that the coarse space can be extremely coarse with no loss in the order of accuracy and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in the two grid method the numerical results show that this method is very effective

numerical solution of two dimensional nonlinear schrodinger equation using a new two grid finite element method abstract a new two grid finite element scheme is presented for two dimensional nonlinear schrodinger equation one newton iteration is applied on the fine grid to linearize the nonlinear system using the coarse grid solution as the initial guess and furthermore one more linear system on the coarse space is solved the error estimations of the two grid solution in the l 2 and h 1 norm are given it is shown both theoretically and numerically that the coarse space can be extremely coarse and two grid algorithm still achieves optimal approximation as long as the mesh sizes satisfy h o h 1 3

an element free based solution for nonlinear schrodinger equations using the icvmls ritz method an improved complex variable moving least squares ritz icvmls ritz method is proposed for predicting numerical solution of the two dimensional nonlinear schrodinger equation in this element free solution procedure the icvmls approximation is employed to reduce the number of unknown coefficients in the trial function it follows by the ritz procedure to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear schrodinger equation the central differencing scheme and newton s algorithm are adopted to solve the nonlinear equation system numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element free icvmls ritz method by comparing the computed results with the available analytical solutions

existence of three positive solutions to a second order boundary value problem on a measure chain growth conditions are imposed on f such that the boundary value problem x t f x t t t1 t2 x t1 x t1 0 and x t2 x t2 0 where t1 t2 from a measure chain t has at least three positive solutions by way of the five functionals fixed point theorem

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

the use of compact boundary value method for the solution of two dimensional schrodinger equation in this paper a high order and accurate method is proposed for solving the unsteady two dimensional schrodinger equation we apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulting linear system of ordinary differential equations the proposed method has fourth order accuracy in both space and time variables moreover this method is unconditionally stable due to the favorable stability property of boundary value methods the results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature these results show that the combination of a compact finite difference approximation of fourth order and a fourth order boundary value method gives an efficient algorithm for solving the two dimensional schrodinger equation

multi point boundary value problems of second order differential equations i abstract we study the nonlinear boundary value problem with nonhomogeneous multi point boundary condition u f t u u 0 t 0 1 u 0 i 1 m a i u t i 1 u 1 i 1 m b i u t i 2 sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relations using one of the results under some assumptions we obtain explicit ranges of values of 1 and 2 with which the problem has a solution has a positive solution and has no solution respectively furthermore we prove that the whole plane for 1 and 2 can be divided into two disjoint connected regions e and n such that the problem has a solution for 1 2 e and has no solution for 1 2 n

a semi discrete higher order compact scheme for the unsteady two dimensional schrodinger equation in this study an implicit semi discrete higher order compact hoc scheme with an averaged time discretization has been presented for the numerical solution of unsteady two dimensional 2d schrodinger equation the scheme is second order accurate in time and fourth order accurate in space the results of numerical experiments are presented and are compared with analytical solutions and well established numerical results of some other finite difference schemes in all cases the present scheme produces highly accurate results with much better computational efficiency

an unconditionally stable linearized ccdadi method for generalized nonlinear schrdinger equations with variable coefficients in two and three dimensions in this paper we propose a three level linearly implicit combined compact difference method ccd together with alternating direction implicit method adi for solving the generalized nonlinear schrdinger equation nlse with variable coefficients in two and three dimensions the method is sixth order accurate in space variable and second order accurate in time variable fourier analysis shows that the method is unconditionally stable comparing to the nonlinear ccdpradi scheme for solving the 2d cubic nlse with constant coefficients li etal 2015 current method is a linear scheme which generally requires much less computational cost moreover current method can handle 3d problems with variable coefficients naturally finally numerical results for both 2d and 3d cases are presented to illustrate the advantages of the proposed method

high order implicit galerkin legendre spectral method for the two dimensional schrodinger equation abstract in this paper we propose galerkin legendre spectral method with implicit runge kutta method for solving the unsteady two dimensional schrodinger equation with nonhomogeneous dirichlet boundary conditions and initial condition we apply a galerkin legendre spectral method for discretizing spatial derivatives and then employ the implicit runge kutta method for the time integration of the resulting linear first order system of ordinary differential equations in complex domain we derive the spectral rate of convergence for the proposed method in the l 2 norm for the semidiscrete formulation numerical experiments show our formulation have high order accuracy

three solutions of an nth order three point focal type boundary value problem abstract we consider the following differential equation u n t f t u t t 0 1 together with the three point focal type boundary conditions u j 0 0 0 j n 3 u n 2 p 0 u n 1 1 0 where 1 2 p 1 by using two different fixed point theorems we offer criteria for the existence of three positive solutions of this problem examples are also included to illustrate the results obtained

application of he s variational iteration method for solving the cauchy reaction diffusion problem in this paper the solution of cauchy reaction diffusion problem is presented by means of variational iteration method reaction diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields application of variational iteration technique to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution moreover this technique does not require any discretization linearization or small perturbations and therefore it reduces significantly the numerical computations

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

three positive solutions for second order m point boundary value problems for the second order m point boundary value problemu t f t u 0 0 0 i 1 2 m 2 0 x 1 x 2 x m 2 1 growth conditions are imposed on f which yield the existence of at least three positive solutions by using the leggett williams fixed point theorem the associated green s function for the m point boundary value problem is also given

a three point boundary value problem for nonlinear fourth order differential equations abstract in this paper the authors consider the nonlinear fourth order ordinary differential equation e u t g t f u 0 with the boundary conditions b u 0 u 1 u 0 u p u 1 0 some results on the existence and nonexistence of positive solutions to problem e b are obtained results on the existence of infinitely many positive solutions are also presented examples are included to demonstrate that the results are sharp