Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations

A class of numerical methods is developed for second order Volterra integro- differential equations by using a Legendre spectral approach. We provide a rigorous error analysis for the proposed methods, which shows that the numerical errors decay exponentially in the L ∞ -norm and L 2 -norm. Numerical examples illustrate the conver- gence and effectiveness of the numerical methods. AMS subject classifications: 45L10, 65R20, 65D15

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

[2]  Tao Tang,et al.  Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems , 2001 .

[3]  N. SIAMJ..  STABLE AND EFFICIENT SPECTRAL METHODS IN UNBOUNDED DOMAINS USING LAGUERRE FUNCTIONS , 2000 .

[4]  Algorithms for the solution of second order Volterra integro-differential equations , 1991 .

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

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

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

[8]  S. Anthoni,et al.  CONTROLLABILITY OF SECOND ORDER SEMILINEAR VOLTERRA INTEGRODIFFERENTIAL SYSTEMS IN BANACH SPACES , 1999 .

[9]  Edris Rawashdeh,et al.  Polynomial spline collocation methods for second-order Volterra integrodifferential equations , 2004, Int. J. Math. Math. Sci..

[10]  M. Tarang.  Stability of the spline collocation method for second order Volterra integro‐differential equations 1 , 2004 .

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

[12]  Solution of linear Volterra integro-differential equations via Sinc functions , 2009 .

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

[14]  Mauro Bologna,et al.  Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels , 2010 .

[15]  A. Makroglou.  A block-by-block method for Volterra integro-differential equations with weakly-singular kernel , 1981 .

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

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

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

[19]  A parallel shooting method for second order volterra integro-differential equations with two point boundary conditions , 1993 .

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

[21]  Hermann Brunner,et al.  Stability of numerical methods for volterra integro-differential equations , 2005, Computing.

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

[23]  Giovanni Monegato,et al.  Nyström interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations , 1997 .

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

[25]  M. Aguilar,et al.  Collocation methods for second-order Volterra integro-differential equations , 1988 .

[26]  Jie Shen,et al.  Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval , 2000, Numerische Mathematik.

[27]  Hermann Brunner,et al.  Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution , 1997 .

[28]  B. Guo,et al.  Hermite pseudospectral method for nonlinear partial differential equations , 2000 .

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