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In this paper, two families of quadrature formulas deduced from the so-called piecewise polynomial product integration rules (see, e.g., Rabinowitz and Sloan, 1984) are considered. Convergence properties and error estimations are ...
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In this paper, two families of quadrature formulas deduced from the so-called piecewise polynomial product integration rules (see, e.g., Rabinowitz and Sloan, 1984) are considered. Convergence properties and error estimations are given. Furthermore, numerical examples are performed and a comparison with rules of similar characteristics is also made.
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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 class...
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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. [References: 28]
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We mainly study numerical integration of real valued functions defined on the d-dimensional unit cube with all partial derivatives up to some finite order r >= 1 bounded by one. It is well known that optimal algorithms that use n ...
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We mainly study numerical integration of real valued functions defined on the d-dimensional unit cube with all partial derivatives up to some finite order r >= 1 bounded by one. It is well known that optimal algorithms that use n function values achieve the error rate n(-r/d), where the hidden constant depends on r and d. Here we prove explicit error bounds without hidden constants and, in particular, show that the optimal order of the error is min{1, d n(-r/d)}, where now the hidden constant only depends on r, not on d. For n = m(d), this optimal order can be achieved by (tensor) product rules.
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We propose replacing the classical Gauss-Laguerre quadrature formula by a truncated version of it, obtained by ignoring the last part of its nodes. This has the effect of obtaining optimal orders of convergence. Corresponding quad...
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We propose replacing the classical Gauss-Laguerre quadrature formula by a truncated version of it, obtained by ignoring the last part of its nodes. This has the effect of obtaining optimal orders of convergence. Corresponding quadrature rules with kernels are then considered and optimal error estimates are derived also for them. These rules are finally used to define stable Nystrom-type interpolants for a second kind of integral equation on the real semiaxis whose solutions decay exponentially at infinity. [References: 21]
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This paper is concerned with the numerical integration of functions by piecewise polynomial product integration rules followed by application of extrapolation procedures. The studied rules can be considered as generalizations of t...
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This paper is concerned with the numerical integration of functions by piecewise polynomial product integration rules followed by application of extrapolation procedures. The studied rules can be considered as generalizations of the conventional trapezoidal rule. Euler-MacLaurin type asymptotic expansions are obtained with only even powers. Furthermore, numerical examples are given in order to show the effectiveness of these methods and a comparison with rules of similar characteristics is also made.
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We study two product integration rules, one for the Chebyshev weight of the first-kind based on the Chebyshev abscissae of the second-kind, and another one constructed the other way around, i.e., relative to the Chebyshev weight o...
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We study two product integration rules, one for the Chebyshev weight of the first-kind based on the Chebyshev abscissae of the second-kind, and another one constructed the other way around, i.e., relative to the Chebyshev weight of the second-kind and based on the Chebyshev abscissae of the first-kind. The new rules are shown to have positive weights given by explicit formulae. Furthermore, we determine the precise degree of exactness and we compute the variance of the quadrature formulae, we examine their definiteness or nondefiniteness, and we obtain error bounds for these formulae either asymptotically optimal by Peano kernel methods or for analytic functions by Hilbert space techniques. In addition, the convergence of the quadrature formulae is shown not only for Riemann integrable functions on [-1,1], but also, by generalizing a result of Rabinowitz, for functions having a monotonic singularity at one or both endpoints of [-1,1].
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Double integrals with algebraic and/or logarithmic singularities are of interest in the application of boundary element method, e.g. linear theory of the aerodynamics of slender bodies of revolution and in many other fields, for e...
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Double integrals with algebraic and/or logarithmic singularities are of interest in the application of boundary element method, e.g. linear theory of the aerodynamics of slender bodies of revolution and in many other fields, for example computational electromagnetics. Therefore, the numerical evaluation of such type of integrals deserves attention. In this connection we propose here product interpolatory rules based on suitable Jacobi zeros, giving numerical tests to show the goodness of the proposed algorithm as well as from a point of view of convergence also for the simplicity of implementation.
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This paper discusses the problem of joining production rules obtained from knowledge sources of different typology (experts, problem-oriented texts, electronic carriers in the form of databases) with a view to building complete an...
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This paper discusses the problem of joining production rules obtained from knowledge sources of different typology (experts, problem-oriented texts, electronic carriers in the form of databases) with a view to building complete and consistent knowledge bases in integrated expert systems. It describes the properties of distributed knowledge acquisition within the target-oriented methodology for building integrated expert sys- tems and automatic technology created on its basis that includes new-generation instruments - an AT-TECH- NOLOGY software complex. The problems of using the theory of multisets to join productive rules obtained from different knowledge sources are considered.
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It is well known that calculations of the entries of the stiffness matrix in the finite element approximations of nonlocal diffusion and mechanics models are often very time-consuming due to the double integration process over the...
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It is well known that calculations of the entries of the stiffness matrix in the finite element approximations of nonlocal diffusion and mechanics models are often very time-consuming due to the double integration process over the domain and the singularities of the nonlocal kernel functions. In this paper, we propose some effective and accurate quadrature rules for computing these double integrals for one-dimensional nonlocal problems; in particular, for problems with highly singular kernels, the corresponding inner integrals can be first evaluated exactly in our method, and the outer one then will be approximated by some popular quadrature rules. With these quadrature rules, the assembly of the stiffness matrix in the finite element method for the nonlocal problems becomes similar to that for the classical partial differential equations and is thus quite efficient. (C) 2016 Elsevier Inc. All rights reserved.
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Various weakly singular integrals over triangular and quadrangular domains, arising in the mixed potential integral equation formulations, are computed with the help of novel generalized Cartesian product rules. The proposed integ...
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Various weakly singular integrals over triangular and quadrangular domains, arising in the mixed potential integral equation formulations, are computed with the help of novel generalized Cartesian product rules. The proposed integration schemes utilize the so-called double exponential quadrature rule, originally developed for the integration of functions with singularities at the endpoints of the associated integration interval. The final formulas can easily be incorporated in the context of singularity subtraction, singularity cancellation and fully-numerical methods, often used for the evaluation of multidimensional singular integrals. The performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods.
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