Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kind x(t) + ∫ab k(t, s)x(s)ds = y(t), whose kernel k(t,s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t,s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value decomposition is indicated. Application to integro-differential Schrodinger equations with nonlocal potentials is given.

[1]  Lothar Reichel,et al.  Fast solution methods for fredholm integral equations of the second kind , 1990 .

[2]  Herman Feshbach,et al.  A Unified Theory of Nuclear Reactions, II , 1962 .

[3]  John Strain,et al.  Locally Corrected Multidimensional Quadrature Rules for Singular Functions , 1995, SIAM J. Sci. Comput..

[4]  G. Micula,et al.  Numerical Treatment of the Integral Equations , 1999 .

[5]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

[6]  D. Kouri,et al.  NONITERATIVE SOLUTIONS OF INTEGRAL EQUATIONS FOR SCATTERING. I. SINGLE CHANNELS. , 1969 .

[7]  R. Machleidt,et al.  The Bonn Meson Exchange Model for the Nucleon Nucleon Interaction , 1987 .

[8]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[9]  Ronald J. W. Henry,et al.  Noniterative Integral-Equation Approach to Scattering Problems. , 1973 .

[10]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[11]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[12]  L. Delves,et al.  Computational methods for integral equations: Frontmatter , 1985 .

[13]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[14]  F. Perey,et al.  A non-local potential model for the scattering of neutrons by nuclei , 1962 .

[15]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[16]  R. Taylor,et al.  The Numerical Treatment of Integral Equations , 1978 .

[17]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

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

[19]  K. Atkinson,et al.  A survey of numerical methods for the solution of Fredholm integral equations of the second kind , 1977 .

[20]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[21]  M. A. Kaashoek,et al.  Classes of Linear Operators Vol. I , 1990 .

[22]  Vladimir Rokhlin,et al.  High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions , 1997 .

[23]  L. Rosenfeld,et al.  Theory of nuclear reactions: I. Resonant states and collision matrix , 1961 .

[24]  Israel Koltracht,et al.  Integral Equation Method for the Continuous Spectrum Radial Schrödinger Equation , 1997 .

[25]  Bradley K. Alpert,et al.  Hybrid Gauss-Trapezoidal Quadrature Rules , 1999, SIAM J. Sci. Comput..

[26]  Song,et al.  Nonlocal nature of the nuclear force and its impact on nuclear structure. , 1996, Physical review. C, Nuclear physics.

[27]  Harold Widom,et al.  Review: I. C. Gohberg and I. A. Fel′dman, Convolution equations and projection methods for their solution , 1975 .

[28]  R. Landau.  Quantum Mechanics II , 1990 .

[29]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .