Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation deflned on the standard interval (i1;1), so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approx- imation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, poly- nomials approximation theory for orthogonal polynomials and the operator theory. The spectral rate of convergence for the proposed method is established in the L 1 -norm and weighted L 2 -norm. Numerical results are presented to demonstrate the efiectiveness of the proposed method.

[1]  Stefan Samko,et al.  Sonine Integral Equations of the First Kind in L p (0;b). , 2003 .

[2]  Sean McKee,et al.  Collocation methods for second-kind Volterra integral equations with weakly singular kernels , 1994 .

[3]  K. C.,et al.  SZEGŐ’S CONJECTURE ON LEBESGUE CONSTANTS FOR LEGENDRE SERIES , 2012 .

[4]  Giuseppe Mastroianni,et al.  Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey , 2001 .

[5]  Guo Ben-yu,et al.  Jacobi interpolation approximations and their applications to singular differential equations , 2001 .

[6]  David L. Ragozin,et al.  Polynomial approximation on compact manifolds and homogeneous spaces , 1970 .

[7]  C. Lubich,et al.  Fractional linear multistep methods for Abel-Volterra integral equations of the second kind , 1985 .

[8]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[9]  G. P. Névai,et al.  Mean convergence of Lagrange interpolation, II☆ , 1976 .

[10]  Tao Tang,et al.  A note on collocation methods for Volterra integro-differential equations with weakly singular kernels , 1993 .

[11]  Hermann Brunner,et al.  Polynomial Spline Collocation Methods for Volterra Integrodifferential Equations with Weakly Singular Kernels , 1986 .

[12]  Ben-yu Guo,et al.  Jacobi interpolation approximations and their applications to singular differential equations , 2001, Adv. Comput. Math..

[13]  Xiang Xu,et al.  Accuracy Enhancement Using Spectral Postprocessing for Differential Equations and Integral Equations , 2008 .

[14]  Wang,et al.  JACOBI PSEUDOSPECTRAL METHOD FOR FOURTH ORDER PROBLEMS , 2006 .

[15]  A. GOGATISHVILIa,et al.  The Generalized Hardy Operator with Kernel and Variable Integral Limits in Banach Function Spaces , .

[16]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .

[17]  Hermann Brunner,et al.  Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels , 1983 .

[18]  Tao Tang,et al.  Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations , 1992 .

[19]  Paul Nevai,et al.  Mean convergence of Lagrange interpolation. III , 1984 .

[20]  Christopher J. Holden,et al.  Graham and I , 1980 .

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

[22]  Q. Hu.  Stieltjes derivatives and b-polynomial spline collocation for Volterra integrodifferential equations with singularities , 1996 .

[23]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[24]  H. J. Riele,et al.  Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution , 1982 .

[25]  Jie Shen,et al.  Spectral and High-Order Methods with Applications , 2006 .

[26]  Jie Shen,et al.  Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials , 2006, J. Sci. Comput..

[27]  David L. Ragozin,et al.  Constructive polynomial approximation on spheres and projective spaces. , 1971 .

[28]  Lars-Erik Persson,et al.  Weighted Inequalities of Hardy Type , 2003 .

[29]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[30]  D. Willett,et al.  A linear generalization of Gronwall’s inequality , 1965 .

[31]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[32]  Ian H. Sloan,et al.  Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ${\mathbb R}^3$ , 2002, Numerische Mathematik.

[33]  Tang,et al.  ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS , 2008 .

[34]  Jan Lang,et al.  The generalized hardy operator with kernel and variable integral limits in banach function spaces , 1999 .