Showing papers in "Computers & Mathematics With Applications in 2001"
TL;DR: A new approach in a posteriori error estimation is studied, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm.
Abstract: In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.
370 citations
TL;DR: In this article, the fixed-point theorem of Leggett-Williams was generalized to prove the existence of three positive solutions to a second-order discrete boundary value problem, which is a generalization of the fixed point theorem of the Banach space.
Abstract: We generalize the fixed-point theorem of Leggett-Williams, which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a Banach space. We then show how to apply our theorem to prove the existence of three positive solutions to a second-order discrete boundary value problem.
353 citations
TL;DR: In this paper, the authors established criteria on uniform asymptotic stability for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques, and showed that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behavior.
Abstract: In this paper, criteria on uniform asymptotic stability are established for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques. It is shown that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behavior. Some examples are also discussed to illustrate the theorems.
240 citations
TL;DR: In this paper, the authors proposed an approximate approach for ranking fuzzy numbers based on the left and right dominance of one fuzzy number over the other, which is useful in ranking the fuzzy numbers when membership functions cannot be acquired.
Abstract: This study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance. The proposed approach only requires a few left and right spreads at some α-levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other. The total dominance is then determined by combining the left and right dominance based on a decision maker's optimistic perspectives. Such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired. The approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers. By using a few left and right spreads, two groups of examples demonstrate the accuracy and applicability of the proposed approach.
169 citations
TL;DR: In this paper, the convergence of projection methods is based on a new iterative algorithm for the approximation-solvability of the following system of nonlinear variational inequalities (SNVI): determine elements x*, y* ϵ K such that 〈ϱT(y*) + x* − y*, x − x*
Abstract: The convergence of projection methods is based on a new iterative algorithm for the approximation-solvability of the following system of nonlinear variational inequalities (SNVI): determine elements x*, y* ϵ K such that 〈ϱT(y*) + x* − y*, x − x*〉 ≥ 0, for all x ϵ K and for ϱ > 0, and 〈γT(x*) + y* − x*, x − y*〉 ≥ 0, for all x ∈ K and for γ > 0, where T : K → H is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. This new class of generalized nonlinear variational inequalities reduces to standard class of nonlinear variational inequalities, which are widely studied and applied to various problems arising from mathematical sciences, optimization and control theory, and other related fields.
157 citations
TL;DR: This paper extends the alternating Turing machine (A-TM) to model multiplayer games of incomplete information, and defines TEAM-PRIVATE-PEEK and TEAM-BLIND-PEE, extending the previous models of PEEK, which can be shown to be complete for their respective classes.
Abstract: This paper (see also [1]) extends the alternating Turing machine (A-TM) of Chandra, Kozen and Stockmeyer [2], the private and the blind alternating machines of Reif [3,4] to model multiplayer games of incomplete information. We use these machines to provide matching lower bounds for our decision algorithms described in our companion paper [5]. We also apply multiple person alternation to other machine types. We show that multiplayer games of incomplete information can be undecidable general. However, one form of incomplete information games that is decidable we term as hierarchical games (defined later in this paper). In hierarchical multiplayer games, each additional clique (subset of players with same information) increases the complexity of the outcome problem by a further exponential. Consequently, if a multiplayer game of incomplete information with k cliques has a space bound of S ( n ), then its outcome can be k repeated exponentials harder than games of complete information with the same space bound S ( n ). This paper proves that this exponential blow-up must occur in the worst case. We define TEAM-PRIVATE-PEEK and TEAM-BLIND-PEEK, extending the previous models of PEEK. These new games can be shown to be complete for their respective classes. We use these games to establish lower bounds on complexity of multiplayer games of incomplete information and blindfold multiplayer games. We analyze the time bounded alternating machines, and conclude that time is not a very critical resource for multiplayer alternation. We also show DQBF (a variant of QBF) to be complete in NEXPTIME .
151 citations
TL;DR: In this article, a nonlinear shooting method for solving two-point boundary value problems was proposed and numerical experiments with various initial velocity conditions were conducted. And the authors discussed and analyzed the numerical solutions which were obtained by the shooting method.
Abstract: We study a new nonlinear shooting method for solving two-point boundary value problems and show numerical experiments with various initial velocity conditions. We discuss and analyze the numerical solutions which are obtained by the shooting method.
139 citations
TL;DR: In this paper, a new family of multipoint methods to approximate a solution of a nonlinear operator equation in Banach spaces is introduced, and error estimates are provided for these iterations using a technique based on a new system of recurrence relations.
Abstract: We introduce a new family of multipoint methods to approximate a solution of a nonlinear operator equation in Banach spaces. An existence-uniqueness theorem and error estimates are provided for these iterations using a technique based on a new system of recurrence relations. To finish, we apply the results obtained to some nonlinear integral equations of the Fredholm type.
127 citations
TL;DR: In this article, a model describing dynamics of Hopfield neural networks involving variable delays is considered and the existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established.
Abstract: In this article, a model describing dynamics of Hopfield neural networks involving variable delays is considered. Existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established. Further, we derive sufficient criteria of global asymptotic stability (GAS) of the equilibrium point.
110 citations
TL;DR: In this article, a twin fixed-point theorem was applied to obtain the existence of at least two positive solutions for the right focal boundary value problem with respect to the two-dimensional discrete right focal value problem.
Abstract: A new twin fixed-point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y″ + f(ity) = 0, 0 <- t <- 1, y(0) = y′(1) = 0. It is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem Δ2u(k) + g(u(k)) = 0, k ϵ {0, … ,N}, u(0) = Δu(N + 1) = 0.
108 citations
TL;DR: In this paper, the authors improve and further generalize some Ostrowski-Gruss type inequalities involving bounded once and twice differentiable mappings, and propose a new type inference algorithm.
Abstract: In this paper, we improve and further generalize some Ostrowski-Gruss type inequalities involving bounded once and twice differentiable mappings.
TL;DR: In this article, the authors give necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2 and derive a new representation of the general common solution.
Abstract: We give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2 and derive a new representation of the general common solution to these two equations. We apply this result to determine new necessary and sufficient conditions for the existence of an Hermitian solution and a representation of the general Hermitian solution to the matrix equation AXB = C.
TL;DR: In this paper, the authors studied the existence of positive solutions to the boundary value problem where a ϵ C([0, 1], [0, ∞]), and f ϵC([0 ∞), [0 √ ∞)).
Abstract: Let ai ≥ 0 for i = 1,…, m − 3 and am−2 > 0. Let ξi satisfy 0 < ξ1 < ξ2 < … < ξm−2 < 1 and Σm−2i=1 aiξi < 1. We study the existence of positive solutions to the boundary-value problem where a ϵ C([0, 1], [0, ∞)), and f ϵ C([0, ∞), [0, ∞)). We show the existence of at least one positive solution if f is either superlinear or sublinear by applying a fixed-point theorem in cones.
TL;DR: In this article, the authors present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions.
Abstract: We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.
TL;DR: In this article, the authors proposed an algorithm for solving the nonlinear two-point boundary value problem that has at least one positive solution for λ in a compatible interval, which stems mainly from combining the decomposition series solution obtained by Adomian decomposition method with Pade approximations.
Abstract: In this paper, we propose an algorithm for solving the nonlinear two-point boundary value problem that has at least one positive solution [1–6] for λ in a compatible interval. Our method stems mainly from combining the decomposition series solution obtained by Adomian decomposition method with Pade approximates. The validity of the approach is verified through illustrative numerical examples
TL;DR: In this paper, the boundedness character, the periodic nature, and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions were investigated.
Abstract: We investigate the boundedness character, the periodic nature, and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions.
TL;DR: In this paper, it was proved that both the Mann and Ishikawa type iteration processes converge strongly to x * ϵ in a real normed linear space and that the strong convergence of these iteration processes with errors is also proved.
Abstract: Let E be a real normed linear space and let A : E ↦ 2 E be a uniformly continuous and uniformly quasi-accretive multivalued map with nonempty closed values such that the range of ( I – A ) is bounded and the inclusion f ϵ Ax has a solution x * ϵ E . It is proved that Ishikawa and Mann type iteration processes converge strongly to x *. Further, if T : E ↦ 2 E is a uniformly continuous and uniformly hemicontractive set-valued map with bounded range and a fixed point x * ϵ E , it is proved that both the Mann and Ishikawa type iteration processes converge strongly to x *. The strong convergence of these iteration processes with errors is also proved.
TL;DR: In this paper, the authors give sufficient conditions under which every solution of the nonlinear difference equation with variable delay x (n + 1) − x ( n ) + p n f ( x ( g ( n ))) tends to zero as n → ∞.
Abstract: In this paper, we give sufficient conditions under which every solution of the nonlinear difference equation with variable delay x ( n + 1) − x ( n ) + p n f ( x ( g ( n ))) = 0, n = 0, 1, 2, … tends to zero as n → ∞. Here, p n is a nonnegative sequence, f : R → R is a continuous function with xf ( x ) > 0 if x ≠ 0, and g : N → Z is nondecreasing and satisfies g ( n ) ≤ n for n ≥ 0 and lim n →∞ g ( n ) = ∞.
TL;DR: In this article, the authors used uniform cubic polynomial splines to develop some consistency relations which were 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.
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.
TL;DR: In this paper, the hyperbolic trigonometric law of cosines and sines in the Poincare ball model of n-dimensional Hyperbolic geometry is presented and illustrated along lines parallel to Euclidean trigonometry.
Abstract: Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincare ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincare ball model of three-dimensional hyperbolic geometry is becoming increasingly important in the construction of hyperbolic browsers in computer graphics. These allow in computer graphics the exploitation of hyperbolic geometry in the development of visualization techniques. It is, therefore, clear that hyperbolic trigonometry in the Poincare ball model of hyperbolic geometry, as presented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics. Hyperbolic trigonometry is governed by gyrovector spaces in the same way that Euclidean trigonometry is governed by vector spaces. The capability of gyrovector space theory to capture analogies and its powerful elegance is thus demonstrated once more.
TL;DR: In this article, a method for computing the zeros of a quaternion polynomial with all terms of the form q k X k was developed. But this method is based essentially on Niven's algorithm, which consists of dividing the polynomials by a characteristic poynomial associated to a zero, and the information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the quaternions.
Abstract: A method is developed to compute the zeros of a quaternion polynomial with all terms of the form q k X k . This method is based essentially in Niven's algorithm [1], which consists of dividing the polynomial by a characteristic polynomial associated to a zero. The information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the polynomial. The companion matrix is represented by a matrix with complex entries. Three numerical examples using Mathematica 2.2 version are given.
TL;DR: In this article, a new version of extragradient method for the variational inequality problem is proposed, which uses a new searching direction which differs from any one in existing projection-type methods, and is of a better step-size rule.
Abstract: In this paper, we propose a new version of extragradient method for the variational inequality problem. The method uses a new searching direction which differs from any one in existing projection-type methods, and is of a better step-size rule. Under a certain generalized monotonicity condition, it is proved to be globally convergent.
TL;DR: In this paper, it was shown that the Ishikawa (and Mann) iteration process with errors converges strongly to some fixed point of the nonempty closed convex subset of a uniformly convex Banach space.
Abstract: Let C be a nonempty closed convex subset of a uniformly convex Banach space and let T : C → C be completely continuous asymptotically nonexpansive in the intermediate sense. In this paper, we prove that the Ishikawa (and Mann) iteration process with errors converges strongly to some fixed point of T, which generalizes the recent results due to Huang [1].
TL;DR: In this article, the authors considered the discrete focal boundary value problem and proved the existence of three positive solutions under various assumptions on f and the integers a, t 2, and b. To prove their results, they applied a generalization of the Leggett-Williams fixed-point theorem.
Abstract: We are concerned with the discrete focal boundary value problem Δ 3 x ( t − k ) = f ( x ( t )), x ( a ) = Δ x ( t 2 ) = Δ 2 x ( b + 1 = 0. Under various assumptions on f and the integers a , t 2 , and b we prove the existence of three positive solutions of this boundary value problem. To prove our results, we will apply a generalization of the Leggett-Williams fixed-point theorem.
TL;DR: In this article, the economic lot scheduling problem (ELSP) without capacity constraints under power-of-two (PoT) policy is investigated and the optimal objective value is piece-wise convex.
Abstract: We present further analysis on the economic lot scheduling problem (ELSP) without capacity constraints under power-of-two (PoT) policy. We explore its optimality structure and discover that the optimal objective value is piece-wise convex. By making use of the junction points of this function, we derive an effective (polynomial-time) search algorithm to secure a global optimal solution. The conclusions of this research lay the foundation for deriving an efficient heuristic, and also creates a benchmark for evaluating the quality of the heuristics for the conventional ELSP under PoT policy.
TL;DR: In this article, a weakly coupled system of quasi-variational inequalities for finite element approximation of Hamilton-Jacobi-Bellman equations is presented, and a convergence and a quasi-optimal L∞-error estimate are established.
Abstract: This paper deals with the finite element approximation of Hamilton-Jacobi-Bellman equations. We establish a convergence and a quasi-optimal L∞-error estimate, involving a weakly coupled systems of quasi-variational inequalities for the solution of which an interative scheme of monotone kind is introduced and analyzed.
TL;DR: In this paper, a review of the special solutions of discrete (difference) and q-discrete Painleve equations in terms of discrete special functions is presented. But the authors do not consider the special solution of the discrete painleve equation with affine Weyl groups.
Abstract: In this paper, we present a review of the special solutions of discrete (difference) and q-discrete Painleve equations in terms of discrete special functions These solutions exist whenever the parameters of the Painleve equation satisfy a particular constraint The discrete special functions belong to the hypergeometric family, although in some cases they seem to go beyond the known discrete and q-discrete hypergeometric functions The equations studied in this paper are chosen on the basis of our recent classification of discrete Painleve equation with the help of affine Weyl groups
TL;DR: In this paper, a family of new iteration methods without employing derivatives is proposed and proved that these new methods are quadratic convergence by numerical experiments, and they are comparable to well-known methods of Newton and Steffensen.
Abstract: A family of new iteration methods without employing derivatives is proposed in this paper. We have proved that these new methods are quadratic convergence. Their efficiency is demonstrated by numerical experiments. The numerical experiments show that our algorithms are comparable to well-known methods of Newton and Steffensen. Furthermore, combining the new method with bisection method we construct another new high-order iteration method with nice asymptotic convergence properties of the diameters ( b n − a n ).
TL;DR: In this paper, the multicommodity reliability of a capacitated-flow network with a unique source mode satisfies a demand (d 1, d 2,…, d p ) at the unique sink node, where p is the demand of commodity k.
Abstract: Traditionally, many researchers solved the multicommodity maximum flow problem by assuming that the arcs of the flow network are deterministic. When the arcs are stochastic (i.e., the capacity of each arc has several values), this article studies how to calculate the probability that a capacitated-flow network with a unique source mode satisfies a demand ( d 1 , d 2 ,…, d p ) at the unique sink node, where d k is the demand of commodity k . Such a probability is named the multicommodity reliability and is dependent on capacities of arcs. One solution procedure is proposed to evaluate the multicommodity reliability, which includes two parts: an algorithm to generate all ( d 1 , d 2 ,…, d p )-MPs and a method to calculate the multicommodity reliability in terms of ( d 1 , d 2 ,…, d p )-MPs. Two illustrative examples are given.
TL;DR: In this paper, a class of operator-integral equations of Volterra-Stieltjes type is investigated and the solvability of those equations in the space of continuous functions is studied.
Abstract: We investigate a class of operator-integral equations of Volterra-Stieltjes type and we study the solvability of those equations in the space of continuous functions. Equations in question create a generalization of numerous integral equations considered in nonlinear analysis. The main tool used in our considerations is the technique associated with measures of noncompactness. We show the applicability of our existence result in the study of a few integral equations of Volterra type.