an implicit explicit domain decomposition algorithm for a singularly perturbed parabolic problem abstract this paper deals with an iterative algorithm for domain decomposition applied to solving of a singularly perturbed parabolic problem this algorithm is based on an implicit scheme inside boundary layers and on an explicit scheme outside them and suitable for parallel computing convergence properties of the algorithm are established and numerical experiments are presented

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

on stability in fuzzy linear programming problems abstract this study focuses on the problem of stability with respect to changes of centres of fuzzy parameters of the solution in fuzzy linear programming flp problems with symmetrical triangular fuzzy numbers and extended operations and inequalities

numerical methods for systems of nonlinear parabolic equations with time delays abstract the purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays the system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations the mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system a theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained it is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero

smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems abstract we construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form d y t f t y t with caputo type fractional derivatives d y of order 0 using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed algorithms two numerical examples are given

fourth order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems in this paper we study numerical methods for a class of two dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains a compact finite difference method with an anisotropic mesh is proposed for the problems the existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions and two sufficient conditions for the uniqueness of the solution are also given the optimal error estimate in the discrete l norm is obtained under certain conditions the error estimate shows the fourth order accuracy of the proposed method when two spatial mesh sizes are proportional by using an upper solution or a lower solution as the initial iteration an almost optimal picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently numerical results are presented to confirm our theoretical analysis

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

monotone schwarz iterates for a semilinear parabolic convection diffusion problem this paper deals with a discrete monotone iterative algorithm for solving a nonlinear singularly perturbed convection diffusion problem of parabolic type on each time level the monotone method known as the method of lower and upper solutions is applied to computing a nonlinear upwind difference scheme obtained after discretisation of the continuous problem a monotone domain decomposition algorithm based on a modification of the schwarz alternating method is constructed the rate of convergence of the monotone schwarz method is estimated uniform convergence properties of the monotone domain decomposition algorithm are studied numerical experiments are presented

some boundary value problems of fractional differential equations and inclusions in this paper we study the existence of solutions for nonlinear fractional differential equations and inclusions of order q 1 2 with families of mixed and closed boundary conditions in case of inclusion problems the existence results are established for convex as well as nonconvex multivalued maps our results are based on leray schauder degree theory nonlinear alternative of leray schauder type and some fixed point theorems for multivalued maps some interesting special cases are also discussed

monotone iterative methods for finite difference system of reaction diffusion equations this paper presents an existence comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary value problems the basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system using upper and lower solutions as two distinct initial iterations two monotone sequences from a suitable linear system are constructed it is shown that these two sequences converge monotonically from above and below respectively to a unique solution of the nonlinear discrete equations this formulation leads to a well posed problem for the nonlinear discrete system applications are given to several models arising from physical chemical and biological systems numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences

on stability analysis of multiple objective dynamic input output model the popularity of multiple objective dynamic input output modio models in real application was much restricted by their instabilities in this paper we investigate the mechanism of the instability of the original modio models theoretically and develop a modified modio model in virtue of the idea of goal programming it is shown that the proposed model is stable while remaining the characteristics of the original modio models

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

fuzzy programming in multiobjective linear programming a parametric approach abstract zimmermann s fuzzy approach to the compromise solution concept in the multi objective linear programming problem is considered it is shown that given a class of membership functions of fuzzy goals assigned to objective functions in the problem wider than primarily proposed by zimmermann the use of classical linear programming methods to solve and analyse the problem is also possible as a result of application of the method based on the parametric programming technique one can obtain a fuzzy solution of the problem this solution is a certain fuzzy subset of the set of weakly efficient solutions of the problem the solution which belongs to this set to the highest degree is a maximizing one in bellman and zadeh s terminology it may also be obtained by use of the usual non parametric simplex method

a numerical approach to the solution of abel integral equations of the second kind with nonsmooth solution abstract in this paper we study the feasibility of numerical solutions analogous to those methods introduced by bownds 1979 1982 and bownds and wood 1976 1979 for volterra integral equations of the second kind with weakly singular kernels

on stability analysis of multicriteria lp problems with fuzzy parameters abstract in this paper we investigate the stability of solution in fuzzy multiobjective linear programming with symmetrical triangular fuzzy numbers with respect to change of fuzzy parameters

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

monotone iterates for solving nonlinear integro parabolic equations of volterra type the paper deals with numerical solving of nonlinear integro parabolic problems based on the method of upper and lower solutions a monotone iterative method is constructed existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given construction of initial upper and lower solutions is discussed numerical experiments are presented

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

a research on the deterministic grey dynamic model with multiple inputs dgdmmi 1 1 1 accumulated generating operation ago is one of the most important characteristics of grey theory and the purpose of ago is to reduce the randomness of the data the incompleteness of messages is the primary characteristic of grey theory and it is necessary to whiten a system by way of inserting more messages in the system the new model presented in this article is called the deterministic grey dynamic model with multiple inputs abbreviated as dgdmmi 1 1 1 additional several series can be introduced to act as the leading indicators input series except the predicted series output series by dgdmmi 1 1 1 model the messages for a system can be inserted sufficiently a transfer function model would be used to combine the ago series of input series with that of output series and then the deterministic convergence scheme is used to evaluate the unbiased estimates of the system parameters the relationship between input series and output series is erected sufficiently by dgdmmi 1 1 1 and high prediction accuracy of a dynamic system can be obtained then high prediction accuracy of economy and indirect measurement of engineering quantities can be broadly anticipated

an inexact monotone method for solving semilinear parabolic problems abstract the paper deals with numerical solving semilinear parabolic problems based on the method of upper and lower solutions an inexact monotone iterative method is constructed where monotone linear systems are solved by the jacobi or gauss seidel methods only approximately an analysis of convergence rates of the inexact monotone iterative method based on different stopping tests is given results of numerical experiments implemented in the framework of an inexact newton method are presented where iteration counts of the inexact monotone method are compared with a monotone iterative method whose convergence rate is linear

periodic solutions of parabolic systems with time delays existence of maximal and minimal periodic solutions of a coupled system of parabolic equations with time delays and with nonlinear boundary conditions is discussed the proof of the existence theorem is based on the method of upper and lower solutions and its associated monotone iterations this method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic parabolic system an application is given to a competitor competitor mutualist model which consists of a coupled system of three reaction diffusion equations with time delays

a hybrid collocation method for volterra integral equations with weakly singular kernels the commonly used graded piecewise polynomial collocation method for weakly singular volterra integral equations may cause serious round off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time dependent equations to round off errors the singularity preserving nonpolynomial collocation method is known to have only local convergence to overcome the shortcoming of these well known methods we introduce a hybrid collocation method for solving volterra integral equations of the second kind with weakly singular kernels in this hybrid method we combine a singularity preserving nonpolynomial collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain we prove the optimal order of global convergence for this method the convergence analysis of this method is based on a singularity expansion of the exact solution of the equations we prove that the solutions of such equations can be decomposed into two parts with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function a numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem this paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic reaction diffusion problem firstly the monotone method known as the method of lower and upper solutions is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem secondly a monotone domain decomposition algorithm based on a modification of the schwarz alternating method is constructed this monotone algorithm solves only linear discrete systems at each iterative step of the iterative process the rate of convergence of the monotone domain decomposition algorithm is estimated numerical experiments are presented

uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given numerical experiments are presented

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

a uniform monotone alternating direction scheme for nonlinear singularly perturbed parabolic problems this paper deals with a monotone alternating direction adi scheme for solving nonlinear singularly perturbed parabolic problems monotone sequences based on the method of upper and lower solutions are constructed for a nonlinear difference scheme which approximates the nonlinear parabolic problem the monotone sequences possess quadratic convergence rate an analysis of uniform convergence of the monotone adi scheme to the solutions of the nonlinear difference scheme and to the continuous problem is given numerical experiments are presented

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

fuzzy optimization an appraisal abstract this paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community some technical hurdles exist to hinder a unanimity reasons for this as well as possible ways for improvement are also discussed

numerical methods for nonlinear integro parabolic equations of fredholm type abstract this paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction diffusion convection equations under either linear or nonlinear boundary conditions the discrete approximation of the problem is based on the finite difference method and the computation of the finite difference solution is by the method of upper and lower solutions three types of quasi monotone reaction functions are considered and for each type a monotone iterative scheme is obtained each of these iterative schemes yields two sequences which converge monotonically from above and below respectively to a unique solution of the finite difference system this monotone convergence leads to an existence uniqueness theorem as well as a computational algorithm for the computation of the solution an error estimate between the computed approximations and the true finite difference solution is obtained for each iterative scheme these error estimates are given in terms of the strength of the reaction function and the effect of diffusion convection and are independent of the true solution applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given numerical experiments are presented

asymptotic behavior of solutions for a cooperation diffusion model with a saturating interaction this paper is concerned with a lotka volterra cooperation diffusion model with a saturating interaction term for one species the goal of the paper is to investigate the asymptotic behavior of the time dependent solution in relation to the corresponding steady state solutions under homogeneous neumann boundary condition some simple and easily verifiable conditions are given to the rate constants so that for every nontrivial nonnegative initial function the corresponding time dependent solution converges to one of the nonnegative constant steady state solutions as time tends to infinity this convergence result leads to the existence and uniqueness of a positive or nonnegative steady state solution and the global asymptotic stability of a given nonnegative constant steady state solution in terms of ecological dynamics it also gives some coexistence permanence and extinction results for the model

grey dynamic input output analysis abstract a system whose information is partially known and partially unknown is named as a grey system and it is general situation in reality on the basis of grey number matrix and their covered operation some definitions of grey functions are enumerated combining grey system theory with traditional dynamic input output analysis we propose the grey dynamic input output analysis it has two types i e the continuous and the disperse the most important results are the computational formulas of the matrix covered set of inverse grey investment coefficient matrix by using them we can forecast and control the economic system under the uncertain situation for the calculation of the matrix covered set of inverse grey matrices we obtain the desired formulas which can be applied to general grey matrices the formulas can greatly promote the development of grey theory and enrich the applied fields of uncertain mathematics the modified case illustrates our method

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

interval analysis and fuzzy set theory an overview of interval analysis its development and its relationship to fuzzy set theory is given possible areas of further fruitful research are highlighted

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

optimization of primal and dual network models of distribution abstract a direct one pass method of solution of the distribution problem is developed basis transformation is then applied to the original constraint equations of this to obtain a quadratic problem which can also be obtained by summation of finite element matrices in which the element constitutive parameter is respectively the route cost or inverse cost for the dual and primal problems using the method of steepest descent extrema for both these are obtained which are equal or close to the exact solutions

comparison of static response of structures using convex models and interval analysis method in this paper by combining the finite element analysis and non probabilistic convex models we present the numerical algorithm of non probabilistic convex models and interval analysis method for the static displacement of structures with uncertain but bounded parameters under the condition of the box or interval vector determined from the ellipsoid of the uncertain but bounded structural parameter vector by comparing the numerical algorithm of non probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non probabilistic convex models is tighter than those produced by the interval analysis method copyright 2003 john wiley sons ltd

finite difference reaction diffusion systems with coupled boundary conditions and time delays abstract this paper is concerned with finite difference solutions of a coupled system of reaction diffusion equations with nonlinear boundary conditions and time delays the system is coupled through the reaction functions as well as the boundary conditions and the time delays may appear in both the reaction functions and the boundary functions the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference equations for both the time dependent problem and its corresponding steady state problem this investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions monotone convergence of the time dependent solution to a maximal or a minimal steady state solution for quasimonotone functions and local and global attractors of the time dependent system including the convergence of the time dependent solution to a unique steady state solution also discussed are some computational algorithms for numerical solutions of the steady state problem when the reaction function and the boundary function are quasimonotone all the results for the coupled reaction diffusion equations are directly applicable to systems of parabolic ordinary equations and to reaction diffusion systems without time delays

an accelerated monotone iterative method for the quantum corrected energy transport model a non stationary monotone iterative method is proposed and analyzed for the quantum corrected energy transport model in nanoscale semiconductor device simulation for the density gradient equations it is analytically and numerically shown that the convergence rate of the method is optimal in the sense of gummel s decoupling iteration this is a globally convergent method in the sense that the initial guess can be taken as a lower or an upper solution which is independent of applied voltages the method integrates the monotone parameters grid sizes and scharfetter gummel fitting in an adaptive and automatic way to treat the singularly perturbed nature of the model that incurs boundary junction and quantum potential layers in the device

a variant of tikhonov regularization for parabolic pde with space derivative multiplied by a small parameter in this paper we examine the applicability of a variant of tikhonov regularization for parabolic pde with its highest order space derivative multiplied by a small parameter the solution of the operator equation u t 2 u x 2 a x t f x t is not uniformly convergent to the solution of the operator equation u t a x t f x t when 0 this violates one of the conditions of hadamard well posedness and hence the perturbed parabolic operator equation is ill posed since the operator is unbounded we first discuss the general theory for unbounded operators and propose an a posteriori parameter choice rule for choosing a regularization parameter we then apply these techniques in the context of perturbed parabolic problems finally we implement our regularized scheme and compare with other basic existing schemes to assert the adaptability of the scheme as an alternate approach for solving the problem

monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem in this paper we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u bu 2 based on piecewise linear polynomials and the lumping operator these iterations construct the monotonically decreasing and increasing sequences and convergence proofs are given finally we present some numerical examples verifying the effectiveness of the theory

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

grey enterprise input output analysis a system whose information is partially known and partially unknown to the investigator is named a grey system due to various inevitable noises contained in the process of data collection managers of enterprise usually have to deal with poor information and make decisions under the influence of uncertainty in this article we propose grey enterprise input output analysis the value and the physical types of grey models are established in particular we present a detailed way of calculating the matrix covered set of the inverse of grey triangular matrix the proposed method makes the management of enterprises more practically possible when the available data contains uncertainty at the end we look at a case study to show the practical feasibility of our work

on a domain decomposition algorithm for a singularly perturbed reaction diffusion problem this paper deals with an iterative algorithm for domain decomposition applied to the solution of a singularly perturbed reaction diffusion problem this algorithm is based on finite difference domain decomposition approach and suitable for parallel computing convergence properties of the algorithm are established numerical results for a test singularly perturbed problem are presented

a value of information in flp problems via sensitivity analysis abstract a fuzzy linear programming flp problem is formulated and the value of information is discussed if the fuzziness of coefficients is decreased by using information about the coefficients we can expect to obtain a more satisfactory solution than without information with this view the prior value of information is discussed using the sensitivity analysis technique a method of allocating the given investigation cost to each fuzzy coefficient is proposed

finite difference domain decomposition algorithms for a parabolic problem with boundary layers abstract parallel algorithms combining a time discretization and finite difference domain decomposition methods are applied to solution of a singularly perturbed parabolic problem different time discretizations are applied in different subdomains convergence results for the algorithms are established

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

numerical procedures for input output analysis input output analysis is one of the most important tools of management science one ideally suited to today s ubiquitous pc the present paper introduces the method with a miniscule solution routine with this lagrange multiplier constraints are easily included and the resulting formulation is then condensed to an equivalent penalty factor result simple cases of this being especially easy to use in practice finally a general purpose band solution routine of the type much used in the finite element method is given it being useful to structure ioa models in an element fashion and indeed to understand the continuum character that large ioa models may possess

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a high order hodie finite difference scheme for 1d parabolic singularly perturbed reaction diffusion problems this paper deals with the numerical approximation of the solution of 1d parabolic singularly perturbed problems of reaction diffusion type the numerical method combines the standard implicit euler method on a uniform mesh to discretize in time and a hodie compact fourth order finite difference scheme to discretize in space which is defined on a priori special meshes condensing the grid points in the boundary layer regions the method is uniformly convergent having first order in time and almost fourth order in space the analysis of the uniform convergence is made in two steps splitting the contribution to the error from the time and the space discretization although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the euler method some numerical results are given corroborating in practice the theoretical results

accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems abstract an accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented the mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite difference or finite element method it is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions this result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions an application is given to a tabular reactor model from chemical engineering for numerical solutions and the number of iterations are compared with that by the regular monotone iterative scheme

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

spline collocation methods for linear multi term fractional differential equations abstract some regularity properties of the solution of linear multi term fractional differential equations are derived based on these properties the numerical solution of such equations by piecewise polynomial collocation methods is discussed the results obtained in this paper extend the results of pedas and tamme 2011 15 where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer in the present paper we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by some numerical examples

time delayed finite difference reaction diffusion systems with nonquasimonotone functions this paper is concerned with finite difference solutions of a system of reaction diffusion equations with coupled nonlinear boundary conditions and time delays the reaction functions and the boundary functions are not necessarily quasimonotone and the time delays may appear in the reaction functions as well as in the boundary functions the investigation is devoted to the finite difference system for both the time dependent problem and its corresponding steady state problem some monotone iteration processes for the finite difference systems are given and the asymptotic behavior of the time dependent solution in relation to the steady state solution is discussed the asymptotic behavior result leads to some local and global attractors of the time dependent problem including the convergence of the time dependent solution to a unique steady state solution an application and some numerical results to an enzyme substrate reaction diffusion problem are given all the results are directly applicable to parabolic ordinary systems and to reaction diffusion systems without time delays

monotone iterative methods for nonlinear equations involving a noninvertible linear part we study monotone iterative methods for the numerical solution of a class of nonlinear elliptic boundary value problems by applying a theorem of ortega and rheinboldt this generalizes the work of earlier authors to the case of nonlinear perturbations of linear problems with a nontrivial kernel

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

central difference approximation of convection in caputo fractional derivative two point boundary value problems a rigorous numerical analysis is given for a fractional derivative two point boundary value problem the highest order term of the differential operator is a caputo fractional derivative of order 1 2 the second and higher order derivatives of the solution of the problem are in general unbounded at one end of the interval which creates difficulties for the analysis the problem is discretized on an equidistant mesh of diameter h using a standard central difference approximation of the convection term and nodal convergence of order o h 1 is proved provided that h satisfies a stability condition that depends on and the convective coefficient this condition may be restrictive when is near 1 when it is violated the solution may exhibit small oscillations but these can be removed by a simple and inexpensive postprocessing technique numerical results are given to display the performance of the method

finite difference reaction diffusion equations with nonlinear diffusion coefficients a monotone iterative method for numerical solutions of a class of finite difference reaction diffusion equations with nonlinear diffusion coefficient is presented it is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system it is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero

singularly perturbed problems in partial differential equations a survey this survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980 2000 in this review we have covered singularly perturbed partial differential equations about odes the survey has already been done by us see m k kadalbajoo k c patidar appl math comput 130 2002 457 510

uniform convergence of a weighted average scheme for a nonlinear reaction diffusion problem this paper deals with a weighted average scheme or q scheme for solving a nonlinear singularly perturbed parabolic reaction diffusion problem the uniform convergence of the weighted average scheme on piecewise uniform and log meshes is established numerical experiments are presented

collocation methods for second kind volterra integral equations with weakly singular kernels in this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of volterra integral equations vies are smooth if the exact solutions are not smooth however suitable transformations can be made so that the new vies possess smooth solutions spline collocation methods with uniform meshes applied to these new vies are then shown to be able to yield optimal global convergence rates the general theory is applied to a typical case i e the integral kernels consisting of the singular term t s

iterative domain decomposition algorithms for the solution of singularly perturbed parabolic problems abstract a combination of a time discretization and domain decomposition methods for the solution of singularly perturbed semilinear parabolic problems is considered on this basis iterative algorithms suitable for parallelization are constructed convergence properties of these algorithms are established the implementation of the algorithms on a shared memory multiprocessor is examined numerical results for a test singularly perturbed problem are presented

numerical solving nonlinear integro parabolic equations by the monotone weighted average method the paper deals with numerical solving nonlinear integro parabolic problems of volterra type based on the weighted average method a monotone iterative method is presented construction of initial upper and lower solutions is given existence and uniqueness of a solution to the nonlinear difference scheme are established an analysis of convergence rates of the monotone iterative method is given numerical experiments are presented

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

a higher order uniformly convergent method with richardson extrapolation in time for singularly perturbed reaction diffusion parabolic problems a b s t r a c t in this paper we are interested in solving efficiently an initial boundary value singularly perturbed time dependent problem of reaction diffusion type on a priori special mesh we construct a high order uniformly convergent finite difference scheme which combines the implicit euler method to discretize in time together with the richardson extrapolation technique and a hodie scheme to discretize in space the analysis of the uniform convergence splits completely the contribution to the global error of both the time and the space discretizations we show numerical results for different test problems confirming in practice the order of uniform convergence proved

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence