1,720,974 research outputs found
A stable BIE method for Laplace’s equation with Neumann boundary conditions in domains with piecewise smooth boundaries
Full text linkThis paper deals with a new boundary integral equation method for the numerical solution of the exterior Neumann problem for the Laplace equation in planar domains with corners. Using the single layer representation of the potential, the differential problem is reformulated in terms of a boundary integral equation (BIE) whose solution has singularities at the corners. A “modified” Nyström-type method based on a Gauss–Jacobi–Lobatto quadrature formula is proposed for its approximation. Convergence and stability results are proved in proper weighted spaces of continuous functions. Moreover, the use of a smoothing transformation allows one to increase the regularity of the solution and, consequently, the order of convergence of the method. The efficiency of the proposed method is illustrated by some numerical tests
A Nyström method for a boundary integral equation related to the Dirichlet problem on domains with corners
No full textThe authors consider the interior Dirichlet problem for Laplace’s equation on planar domains with corners. They provide a complete analysis of a natural method of Nyström type based on the global Gauss–Lobatto quadrature rule, in order to approximate the solution of the corresponding double layer boundary integral equation. Mellin-type integral operators are involved and, as usual, a modification of the method close to the corners is needed. A new modification is proposed and the convergence and stability of the “modified” quadrature method are proved. Some numerical tests are also included
A Nyström method for mixed boundary value problems in domains with corners
Full text linkIn this paper we propose a new approach to the numerical solution of the mixed Dirichlet-Neumann boundary value problem for the Laplace equation in planar domains with piecewise smooth boundaries. We consider a perturbed BIE system associated to the problem and present a Nyström method for its numerical solution. As Mellin type integral operators are involved, we need to modify the method close to the corners in order to prove its stability and convergence. Some numerical tests are also given to show the efficiency of the method here described
On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners
Full text linkThe authors propose a "modified" Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests
Confronto di panel addestrati e inesperti nell'analisi descrittiva del formaggio pecorino
No full textA global approximation method for second-kind nonlinear integral equations
Full text linkA global approximation method of Nyström type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first occurrence, the method uses a Gauss-Legendre rule whereas in the second one resorts to a product rule based on Legendre nodes. Stability and convergence are proved in functional spaces equipped with the uniform norm and several numerical tests are given to show the good performance of the proposed method. An application to the interior Neumann problem for the Laplace equation with nonlinear boundary conditions is also considered
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
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
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