1,720,973 research outputs found
Compounded Product Integration rules on (0, +∞)
Full text linkThe paper deals with the approximation of integrals of the type I(f , y) =: \int_{0}^{+\infty} f (x)k(x, y)\rho(x) dx, \rho(x) := e^{−x} x^\gamma , where f is a sufficiently smooth function and the kernel k collects criticisms of many different types (highly oscillating, weakly singular, "nearly" singular, etc.). We propose an extended product rule based on the approximation of f by an extended Lagrange process at Laguerre zeros. We prove that the rule is stable and convergent with order of the best polynomial approximation in suitable function spaces. Furthermore, by combining the stated rule with a related product formula, we define a pattern that allows a significant saving in number of function evaluations. We give details on the construction of the coefficients of the rule for some selected kernels. Finally, some numerical tests are proposed to show the efficiency of the compounded quadrature scheme
On the numerical solution of Volterra integral equations on equispaced nodes
Full text linkIn the present paper, a Nystrom-type method for second kind Volterra integral equations is introduced and studied. The method makes use of generalized Bernstein polynomials, defined for continuous functions and based on equally spaced points. Stability and convergence are studied in the space of continuous functions. Numerical tests illustrate the performance of the proposed approach
A product integration rule on equispaced nodes for highly oscillating integrals
Full text linkThis paper provides a product integration rule for highly oscillating integrands of the type
\int_{-a}^a e^{-\ii \omega (x-y)} f(x) dx, \quad a>0, \quad \ii=\sqrt{-1}, \quad y \in [-a,a], \quad \omega \in \RR^+,
based on the approximation of by means of the Generalized Bernstein polynomials . The rule involves the samples of at equally spaced points of and differently from the classical Bernstein polynomials, the suitable modulation of the parameter \ell\in \NN allows to increase the accuracy of the product rule, as the smoothness of increases. Stability and error estimates are proven for belonging to the space of continuous functions and their Sobolev-type subspaces. Finally, some numerical tests which confirm such theoretical estimates are shown
A Nyström method for Volterra-Fredholm integral equations with highly oscillatory kernel
Full text linkIn the present paper, we propose a Nyström method for a class of Volterra-Fredholm integral equations containing a fast oscillating kernel. The approximation tool consists of the l− Iterated Boolean sums of Bernstein operators, also known as Generalized Bernstein (GB) operators, based on equally spaced nodes of the interval [−1,1]. The corresponding GB polynomials associated with any continuous function depend on the additional parameter l, which can be suitably chosen in order to improve the rate of convergence, as the smoothness of the function increases. Hence, the low degree of approximation by the classical Bernstein polynomials or by piecewise polynomials functions, typically based on equispaced nodes, is overcome in some sense. The numerical method we propose here is stable and convergent in the space of the continuous functions equipped with the uniform norm. Error estimates are proved in Hölder-Zygmund type subspaces and some numerical tests confirm the theoretical error estimates
SIMULTANEOUS APPROXIMATION OF HILBERT AND HADAMARD TRANSFORMS ON BOUNDED INTERVALS
Full text linkn this paper, we propose a compound scheme of different product integration rules for the simultaneous approximation of both Hilbert and Hadamard transforms of a given function f. The advantages of such a scheme are multiple: a saving in the number of function evaluations and the avoidance of the derivatives of the density function f when approximating the Hadamard transform. Stability and convergence of the proposed method are proved in the space of locally continuous functions in (-1; 1) with possible algebraic singularities at the endpoints, equipped with weighted uniform norms. The theoretical estimates are confirmed by several numerical tests
A global method for solving second-kind Volterra–Fredholm integral equations
Full text linkThe paper presents a Nystr & ouml;m-type method to approximate the solution of second-kind Volterra-Fredholm integral equations. Two forms are considered, that is the disjoint form, in which the Volterra and Fredholm operators are additive integrals; and the mixed one, in which the two integrals appear in a single term through composition. In both situations, the right-hand side and the kernel functions may have algebraic singularities at +/- 1 and hence equations are treated in suitable weighted spaces equipped with the uniform norm. The proposed methods, based on product and Gauss rules, are stable and convergent. The error is of the order of the best polynomial approximation of the given functions. Numerical examples are presented to illustrate the accuracy of the method
Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind
Full text linkIn this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given
Numerical approximation of Fredholm integral equation by the constrained mock-Chebyshev least squares operator
Full text linkIn this paper, we propose two numerical approaches for approximating the solution of the following kind of integral equation f(y)−μ∫−11f(x)k(x,y)w(x)dx=g(y),y∈[−1,1],where f is the unknown solution, μ∈R∖{0}, k,g are given functions not necessarily known in the analytical form, and w is a Jacobi weight. The proposed projection methods are based on the constrained mock-Chebyshev least squares polynomials, and starting from data known at equally spaced points, provide a fine approximation of the solution. Such peculiarity can be helpful in all cases we deal with experimental data, typically measured at equispaced points. We prove the introduced methods are stable and convergent in some Sobolev subspace of C[−1,1]. Several numerical tests confirm the theoretical estimates and numerical effectiveness of the proposed method
A high order numerical scheme for a nonlinear nonlocal reaction–diffusion model arising in population theory
Full text linkThis paper provides a numerical method for nonlinear equation arising in mathematical biology. It is an extension of another one recently proposed for the linear, less realistic, situation. The main novel result is the proof that the convergence of the numerical method is of order four, as to our knowledge no similar high accuracy results exist yet in the current literature for usually employed simulation schemes for nonlocal equations
Lavorare tra le mura: un approfondimento sul ruolo e sul benessere professionale della polizia penitenziaria.
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