Spectral methods for weakly singular Volterra integral equations with pantograph delays

Abstract

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L^∞-norm and the weighted L²-norm. Numerical examples are presented to complement the theoretical convergence results.

Keywords

Volterra integral equation / vanishing delay / weakly singular kernel / Jacobi-spectral collocation method / error analysis

References

[1] Ali I, Brunner H, Tang T. A spectral method for pantograph-type delay differential equations and its convergence analysis. J Comput Math, 2009, 27: 254-265

[2] Ali I, Brunner H, Tang T. Spectral method for pantograph-type differential and integral equations with multiple delays. Front Math China, 2009, 4: 49-61

[3] Brunner H. Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J Numer Anal, 1983, 20: 1106-1119

[4] Brunner H. The numerical solutions of weakly singular Volterra integral equations by collocation on graded mesh. Math Comp, 1985, 45: 417-437

[5] Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations Methods. Cambridge Monographs on Applied and Computational Mathematics, Vol 15. Cambridge: Cambridge University Press, 2004

[6] Canuto C, Hussaini M Y, Qarteroni A, Zang T A. Spectral Methods Fundamentals in Single Domains. New York: Springer-Verlag, 2006

[7] Chen Y, Tang T. Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math, 2009, 233: 938-950

[8] Chen Y, Tang T. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comp, 2010, 79: 147-167

[9] Diogo T, McKee S, Tang T. Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proc Roy Soc Edinburgh Sect A, 1994, 124: 199-210

[10] Gapobianco G, Cardone A. A parallel algorithm for large systems of Volterra integral equations of Abel type. J Comput Appl Math, 2008, 220: 749-758

[11] Gapobianco G, Conte D. An efficient and fast parallel methods for Volterra integral equations of Abel type. J Comput Appl Math, 2006, 189: 481-493

[12] Gapobianco G, Conte D, Prete I D. High performance parallel numerical methods for Volterra equations with weakly singular kernels. J Comput Appl Math, 2009, 228: 571-579

[13] Gapobianco G, Crisci M R, Russo E. Nonstationary waveform relaxation methods for Abel equations. J Integral Equations Appl, 2004, 16: 53-65

[14] Gogatishvill A, Lang J. The generalized hardy operator with kernel and variable integral limits in banach function spaces. J Inequal Appl, 1999, 4(1): 1-16

[15] Guo B, Wang L. Jacobi interpolation approximations and their applications to singular differential equations. Adv Comput Math, 2001, 14: 227-276

[16] Guo B, Wang L. Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J Approx Theory, 2004, 128: 1-41

[17] Henry D. Geometric Theory of Semilinear Parabolic Equations. New York: Springer-Verlag, 1989

[18] Kufner A, Persson L E. Weighted Inequalities of Hardy Type. New York: World Scientific, 2003

[19] Mastroianni G, Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey. J Comput Appl Math, 2001, 134: 325-341

[20] Nevai P. Mean convergence of Lagrange interpolation. III. Trans Amer Math Soc, 1984, 282: 669-698

[21] Ragozin D L. Polynomial approximation on compact manifolds and homogeneous spaces. Trans Amer Math Soc, 1970, 150: 41-53

[22] Ragozin D L. Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc, 1971, 162: 157-170

[23] Samko S G, Cardoso R P. Sonine integral equations of the first kind in Lp(0, b).Fract Calc Appl Anal, 2003, 6(3): 235-258

[24] Shen J, Tang T. Spectral and High-Order Methods with Applications. Beijing: Science Press, 2006

[25] Shen J, Wang L L. Some recent advances on spectral methods for unbounded domains. Commun Comput Phys, 2009, 5: 195-241

[26] Tang T, Xu X. Accuracy enhancement using spectral postprocessing for differential equations and integral equations. Commun Comput Phys, 2009, 5: 779-792

[27] Tang T, Xu X, Cheng J. On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math, 2008, 26: 825-837

[28] Welfert B D. A note on classical Gauss-Radau and Gauss-Lobatto quadratures. Appl Numer Math, 2010, 60: 637-644

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg