1,720,994 research outputs found
A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains
No full textThe author proposes a numerical procedure in order to approximate the solution of
a class of Fredholm integral equations of the third kind on unbounded domains. The
given equation is transformed in a Fredholm integral equation of the second kind. Hence,
according to the integration interval, the equation is regularized by means of a suitable
one-to-one map or is transformed in a system of two Fredholm integral equations that
are subsequently regularized. In both cases a Nyström method is applied, the convergence
and the stability of which are proved in spaces of weighted continuous functions. Error
estimates and numerical tests are also included
Weakly singular linear Volterra integral equations: A Nyström method in weighted spaces of continuous functions
Full text linkThis paper provides a Nyström method for the numerical solution of Volterra integral
equations whose kernels contain singularities of algebraic type. It is proved that the
method is stable and convergent in suitable weighted spaces. An error estimate is also
given as well as several numerical tests are presented
A projection method with smoothing transformation for second kind volterra integral equations
Full text linkIn this paper we present a projection method for linear second kind Volterra integral equations with kernels having weak diagonal and/or boundary singularities of algebraic type. The proposed approach is based on a specific optimal interpolation process and a smoothing transformation. The convergence of the method is proved in suitable spaces of functions, equipped with the uniform norm. Several tests show the accuracy of the presented method
Numerical Methods for Cauchy Bisingular Integral Equations of the First Kind on the Square
Full text linkIn this paper we investigate the numerical solution of Cauchy bisingular integral equations of the first kind on the square. We propose two different methods based on a global polynomial approximation of the unknown solution. The first one is a discrete collocation method applied to the original equation and then is a “direct” method. The second one is an “indirect” procedure of discrete collocation-type since we act on the so-called regularized Fredholm equation. In both cases, the convergence and the stability of the method is proved in suitable weighted spaces of functions, and the well conditioning of the linear system is showed. In order to illustrate the efficiency of the proposed procedures, some numerical tests are given
Volterra integral equations with highly oscillatory kernels: A new numerical method with applications
No full textThe aim of this paper is to present a Nyström-type method for the numerical approximation of the solution of Volterra integral equations of the second kind having highly oscillatory kernels. The method is based on a mixed quadrature scheme which combines the classical product rule with a dilation quadrature formula. The convergence and the stability of the method are investigated and the accuracy of the presented approach is assessed by some numerical tests. The proposed procedure is also applied to the computation of initial scattering data related to the initial value problem associated to the Korteweg-de Vries equation
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 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
A projection method for Volterra integral equations in weighted spaces of continuous functions
No full textThis paper is concerned with the numerical treatment of second kind Volterra integral equations whose integrands present diagonal and/or endpoint algebraic singularities. A projection method based on an optimal interpolating operator is developed in the spaces of weighted continuous functions endowed with the supremum norm. In such spaces, the uniqueness of the solution is discussed and suitable conditions are determined to assure the stability and the convergence of the method. Several numerical tests are presented to show the efficiency of the method and the agreement with the theoretical estimates
On the mathematical theory of living systems II: The interplay between mathematics and system biology
No full text- …
