
Sparse spectral methods for partial differential equations on spherical caps
In recent years, sparse spectral methods for solving partial differentia...
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An iterative approximate method of solving boundary value problems using dual Bernstein polynomials
In this paper, we present a new iterative approximate method of solving ...
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Gram quadrature: numerical integration with Gram polynomials
The numerical integration of an analytical function f(x) using a finite ...
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A Generalization of Classical Formulas in Numerical Integration and Series Convergence Acceleration
Summation formulas, such as the EulerMaclaurin expansion or Gregory's q...
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The Lagrangian remainder of Taylor's series, distinguishes O(f(x)) time complexities to polynomials or not
The purpose of this letter is to investigate the time complexity consequ...
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AlmostOrthogonal Bases for Inner Product Polynomials
In this paper, we consider lowdegree polynomials of inner products betw...
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MonteCarlo cubature construction
In numerical integration, cubature methods are effective, in particular ...
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Recursion formulas for integrated products of Jacobi polynomials
From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method (FEM). With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect.
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