A Nyström method for integral equations with fixed singularities of Mellin type in weighted Lp spaces

We consider integral equations of the second kind with fixed singularities of Mellin type. According to the behavior of the Mellin kernel, we first determine suitable weighted Lp spaces where we look for the solution. Then, for its approximation, we propose a numerical method of Nystrm type based on a GaussJacobi quadratura formula. Actually, a slight modification of the classical procedure is introduced in order to prove convergence results in weighted Lp spaces. Moreover, a preconditioning technique allows us to solve well conditioned linear systems. We show the efficiency of the proposed method through some numerical tests.

[1]  J. Elschner,et al.  The $h$-$p$-Version of Spline Approximation Methods for Mellin Convolution Equations , 1993 .

[2]  R. Kress.  Linear Integral Equations , 1989 .

[3]  A. Timan.  Theory of Approximation of Functions of a Real Variable , 1994 .

[4]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[5]  G. Mastroianni,et al.  Condition Numbers in Numerical Methods for Fredholm Integral Equations of the Second Kind , 2002 .

[6]  I. G. Graham,et al.  Numerical methods for integral equations of Mellin type , 2000 .

[7]  Z. Ditzian.  On interpolation of $L_p[a,\,b]$ and weighted Sobolev spaces. , 1980 .

[8]  Andreas Rathsfeld,et al.  On polynomial collocation for second kind integral equations with fixed singularities of Mellin type , 2003, Numerische Mathematik.

[9]  B. Silbermann,et al.  Numerical Analysis for Integral and Related Operator Equations , 1991 .

[10]  L. Scuderi.  A Chebyshev polynomial collocation BIEM formixed boundary value problem on nonsmooth boundaries , 2002 .

[11]  Giovanni Monegato,et al.  A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non‐smooth boundaries , 2003 .

[12]  G. Milovanović,et al.  Interpolation Processes: Basic Theory and Applications , 2008 .

[13]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[14]  Rainer Kress,et al.  A Nyström method for boundary integral equations in domains with corners , 1990 .

[15]  Youngmok Jeon,et al.  The collocation method for mixed boundary value problems on domains with curved polygonal boundaries , 1997 .

[16]  J. Elschner.  On Spline Approximation for a Class of Non‐Compact Integral Equations , 1990 .

[17]  Siegfried Prössdorf,et al.  An algorithm for the approximate solution of integral equations of Mellin type , 1995 .

[18]  Maria Carmela De Bonis,et al.  A modified Nyström method for integral equations with Mellin type kernels , 2016, J. Comput. Appl. Math..

[19]  G. Mastroianni,et al.  Nyström interpolants based on the zeros of Legendre polynomials for a non-compact integral operator equation , 1994 .

[20]  Concetta Laurita,et al.  A Nyström method for a boundary integral equation related to the Dirichlet problem on domains with corners , 2013, Numerische Mathematik.

[21]  Giovanni Monegato,et al.  A stable Nyström interpolant for some Mellin convolution equations , 1996, Numerical Algorithms.