1,720,975 research outputs found
Error bounds for a gauss-type quadrature rule to evaluate hypersingular integrals
No full textIn the present paper we consider hypersingular integrals of the following type (Formula presented) where the integral is understood in the Hadamard finite part sense, p is a positive integer, wα(x) = e−xxαis a Laguerre weight of parameter α ≥ 0 and t > 0. In [6] we proposed an efficient numerical algorithm for approximating (1), focusing our attention on the computational aspects and on the efficient implementation of the method. Here, we introduce the method discussing the theoretical aspects, by proving the stability and the convergence of the procedure for density functions f s.t. f(p)satisfies a Dini-type condition. For the sake of completeness, we present some numerical tests which support the theoretical estimates
Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis
Full text linkThis paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function f by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point. We prove the stability and the convergence of the rule, giving error estimates for functions belonging to weighted Sobolev spaces equipped with uniform norm. We also show how the proposed rule can be used for the numerical solution of hypersingular integral equations. Numerical tests which confirm the theoretical estimates and comparisons with other existing quadrature rules are presented
Numerical method for boundary value problems on the real line
No full textThis paper deals with the global approximation of the solutions of Boundary Value Problems (BVPs) of second order on the real line. We first reduce the BVP to an equivalent Fredholm integral equation of the second kind and then approximate its solution by a Nyström type method based on a suitable product quadrature rule. Such quadrature formula is based on a truncated interpolation process at the Hermite zeros. The stability and the convergence of the method as well as the well conditioning of the involved linear systems are studied in weighted spaces of continuous functions. Numerical tests confirming the theoretical error estimates are shown
A numerical method for linear Volterra integral equations on infinite intervals and its application to the resolution of metastatic tumor growth models
Full text linkA Nyström method for linear second kind Volterra integral equations on unbounded intervals, with sufficiently smooth kernels, is described. The procedure is based on the use of a truncated Lagrange interpolation process and of a truncated Gaussian quadrature formula. The stability and the convergence of the method in suitable weighted spaces of functions are studied and some numerical examples showing its reliability are presented. In particular, the proposed method has been tested for the numerical resolution of some Volterra integral equations arising from the reformulation of differential models describing metastatic tumor growth whose unknown solutions represent biological observables as the metastatic mass or the number of metastases
Filtered interpolation for solving Prandtl’s integro-differential equations
No full textIn order to solve Prandtl-type equations we propose a collocation-quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case and cut off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular, we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit
A NUMERICAL METHOD FOR SOLVING SYSTEMS OF HYPERSINGULAR INTEGRO-DIFFERENTIAL EQUATIONS
Full text linkThis paper is concerned with a collocation-quadrature method for solving systems of Prandtl's integro-differential equations based on de la Vallee Poussin filtered interpolation at Chebyshev nodes. We prove stability and convergence in Holder-Zygmund spaces of locally continuous functions. Some numerical tests are presented to examine the method's efficacy
On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces
Full text linkThis paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nyström method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical tests which confirm the efficiency of the method are shown
Hermite-Fejér and Grünwald interpolation at generalized Laguerre zeros
Full text linkWe introduce special Hermite-Fejer and Grünwald operators at the zeros of the generalized Laguerre polynomials. We will prove that these interpolation processes are uniformly convergent in
suitable weighted function space
Modeling metastatic tumor evolution, numerical resolution and growth prediction
Full text linkIn this paper we consider a generalized metastatic tumor growth model that describes the primary tumor growth by means of an Ordinary Differential Equation (ODE) and the evolution of the metastatic density using a transport Partial Differential Equation (PDE). The numerical method is based on the resolution of a linear Volterra integral equation (VIE) of the second kind, which arises from the reformulation of the ODE–PDE model. The convergence of the method is proved and error estimates are given. The computation of the approximate solution leads to solving well conditioned linear systems. Here we focus our attention on two different case studies: lung and breast cancer. We assume five different tumor growth laws for each of them, different metastatic emission rates between primary and secondary tumors, and lastly that the newborn metastases can be formed by clusters of several cells
MatLab Toolbox for the numerical solution of linear Volterra integral equations arising in metastatic tumor growth models
Full text linkThis paper introduces VIE Toolbox composed by fourteen MatLab functions used for the numerical resolution of Volterra Integral Equations (VIEs) of the second kind on infinite intervals. An application to metastatic tumor growth models is also considered, assuming five different tumor growth laws, e.g. exponential, power-law, Gompertz, generalized logistic and von Bertalanffy-West laws, for lung and breast tumors data
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