Spectral methods for weakly singular Volterra integral equations with smooth solutions

We propose and analyze a spectral Jacobi-collocation approximation for the linear Volterra integral equations (VIEs) of the second kind with weakly singular kernels. In this work, we consider the case when the underlying solutions of the VIEs are sufficiently smooth. In this case, we provide a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in the infinity norm and weighted Sobolev space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

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

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

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

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

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

[6]  A parallel algorithm for large systems of Volterra integral equations of Abel type , 2008 .

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

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

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

[10]  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.

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

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

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

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

[15]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

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

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

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

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

[20]  G. Capobianco,et al.  High performance parallel numerical methods for Volterra equations with weakly singular kernels , 2009 .

[21]  Ishtiaq Ali,et al.  Spectral methods for pantograph-type differential and integral equations with multiple delays , 2009 .

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

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

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

[25]  Tao Tang,et al.  Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel , 2010, Math. Comput..

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

[27]  Hermann Brunner,et al.  Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels , 2001, SIAM J. Numer. Anal..

[28]  Jie Shen,et al.  Some Recent Advances on Spectral Methods for Unbounded Domains , 2008 .

[29]  Dajana Conte,et al.  An efficient and fast parallel method for Volterra integral equations of Abel type , 2006 .

[30]  H. Brunner,et al.  A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS * , 2009 .

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

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

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