Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels

In this paper, we introduce efficient methods for the approximation of solutions to weakly singular Volterra integral equations of the second kind with highly oscillatory Bessel kernels. Based on the asymptotic analysis of the solution, we derive corresponding convergence rates in terms of the frequency for the Filon method, and for piecewise constant and linear collocation methods. We also present asymptotic schemes for large values of the frequency. These schemes possess the property that the numerical solutions become more accurate as the frequency increases.

[1]  K. S. Kölbig,et al.  Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints , 1972 .

[2]  Shuhuang Xiang,et al.  Fast integration of highly oscillatory integrals with exotic oscillators , 2009, Math. Comput..

[3]  J. Gillis,et al.  Integrals of Bessel Functions , 1963 .

[4]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[5]  Christopher T. H. Baker,et al.  A perspective on the numerical treatment of Volterra equations , 2000 .

[6]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[7]  Virginia Kiryakova,et al.  Explicit solutions to hyper-Bessel integral equations of second kind , 1999 .

[8]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[9]  L. G. McAlevey.  Product integration rules for volterra integral equations of the first kind , 1987 .

[10]  Peter Linz.  Product integration methods for Volterra integral equations of the first kind , 1971 .

[11]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[12]  Shuhuang Xiang,et al.  Asymptotic expansion and Filon-type methods for a Volterra integral equation with a highly oscillatory kernel , 2011 .

[13]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[14]  Robert J Krueger,et al.  An electromagnetic inverse problem for dispersive media , 1985 .

[15]  Richard Weiss,et al.  On the solution of volterra integral equations of the first kind , 1973 .

[16]  Dugald B. Duncan,et al.  Stability and Convergence of Collocation Schemes for Retarded Potential Integral Equations , 2004, SIAM J. Numer. Anal..

[17]  Direct and inverse scattering problems in dispersive media-Green's functions and invariant imbedding techniques , 1990 .

[18]  Hermann Brunner,et al.  Discontinuous Galerkin approximations for Volterra integral equations of the first kind , 2009 .

[19]  Daan Huybrechs,et al.  Norges Teknisk-naturvitenskapelige Universitet Local Solutions to High Frequency 2d Scattering Problems Local Solutions to High Frequency 2d Scattering Problems , 2022 .

[20]  Stephen Langdon,et al.  A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem , 2006, SIAM J. Numer. Anal..

[21]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[22]  L. Filon.  III.—On a Quadrature Formula for Trigonometric Integrals. , 1930 .

[23]  J. Nédélec.  Acoustic and electromagnetic equations , 2001 .

[24]  A. Iserles,et al.  Efficient quadrature of highly oscillatory integrals using derivatives , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  Shuhuang Xiang,et al.  Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications , 2011 .

[26]  David Levin,et al.  Fast integration of rapidly oscillatory functions , 1996 .

[27]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[28]  A. Iserles,et al.  On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation , 2004 .