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A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

Published online by Cambridge University Press:  28 May 2015

Yunxia Wei  and
Yanping Chen
Yunxia Wei*
Affiliation:
College ofMathematic and Information Science, Shandong Institute of Business and Technology, Yantai 264005, Shandong, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in L∞ norm and weighted L2 norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

Keywords

65R2045J0565N12Neutral Volterra integro-differential equationweakly singular kernelJacobi collocation discretizationconvergence analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 2 , May 2013 , pp. 424 - 446
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Brunner, H., A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations, J. Comput. Appl. Math., 8 (1982), pp. 213–229.CrossRefGoogle Scholar
[2]Brunner, H., Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations, Math. Comput., 42 (1984), pp. 95—109.CrossRefGoogle Scholar
[3]Brunner, H., Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6 (1986), pp. 221–239.CrossRefGoogle Scholar
[4]Brunner, H., The numerical solution of initial-value problems for integro-differential equations, in Numerical Analysis 1987 (Ed. by Griffiths, D. F. and Watson, G. A.), pp. 18–38.Google Scholar
[5]Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press 2004.CrossRefGoogle Scholar
[6]Brunner, H., Pedas, A. and Vainikko, G., Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), pp. 957–982.CrossRefGoogle Scholar
[7]Brunner, H. and Tang, T., Polynomial spline collocation methods for the nonlinear Basset equation, Comput. Math. Appl., 18 (1989), pp. 449–457.CrossRefGoogle Scholar
[8]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods Fundamentals in Single Domains, Springer–Verlag, 2006.CrossRefGoogle Scholar
[9]Chen, Y. and Tang, T., Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math., 233 (2009), pp. 938–950.CrossRefGoogle Scholar
[10]Chen, Y. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.CrossRefGoogle Scholar
[11]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, Springer-Verlag, Heidelberg, 2nd Edition, 1998.Google Scholar
[12]Dixon, J. A., A nonlinear weakly singular Volterra integro-differential equation arising from a reaction-diffusion study in a small cell, J. Comput. Appl. Math., 18 (1987), pp. 289–305.CrossRefGoogle Scholar
[13]Goldfine, A., Taylor series methods for the solution of Volterra integral and integro-differential equations, Math. Comput., 31 (1977), pp. 691–707.CrossRefGoogle Scholar
[14]Henry, D., Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1989.Google Scholar
[15]Jiang, Y., On spectral methods for Volterra-type integro-differential equations, J. Comput. Appl. Math., 230 (2009), pp. 333–340.CrossRefGoogle Scholar
[16]Kufner, A. and Persson, L. E., Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.CrossRefGoogle Scholar
[17]Lubich, C., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45 (1985), pp. 463–469.CrossRefGoogle Scholar
[18]Makroglou, A., Convergence of a block-by-block method for nonlinear Volterra integro-differential equations, Math. Comput., 35 (1980), pp. 783–796.CrossRefGoogle Scholar
[19]Makroglou, A., A block-by-block method for Volterra integro-differential equations with weakly singular kernel, Math. Comput., 37 (1981), pp. 95–99.CrossRefGoogle Scholar
[20]Mastroianni, G. and Occorsio, D., Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey, J. Comput. Appl. Math., 134 (2001), pp. 325–341.CrossRefGoogle Scholar
[21]Mckee, S., Cyclic multistep methods for solving Volterra integro-differential equations, SIAM J. Numer. Anal., 16 (1979), pp. 106–114.CrossRefGoogle Scholar
[22]Nevai, P., Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc., 282 (1984), pp. 669–698.CrossRefGoogle Scholar
[23]Papatheodorou, T. S. and Jesanis, M. E., Collocation methods for Volterra integro-differential equations with singular kernels, J. Comput. Appl. Math., 6 (1980), pp. 3–8.CrossRefGoogle Scholar
[24]Ragozin, D. L., Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc., 150 (1970), pp. 41–53.CrossRefGoogle Scholar
[25]Ragozin, D. L., Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), pp. 157–170.Google Scholar
[26]Rawashdeh, E., Mcdowell, D. and Rakesh, L., Polynomial spline collocation methods for second-order Volterra integro-differential equations, IJMMS, 56 (2004), pp. 3011–3022.Google Scholar
[27]Rawashdeh, E., Mcdowell, D. and Rakesh, L., Polynomial spline collocation methods for second-order Volterra integro-differential equations, IJMMS, 56 (2004), pp. 3011–3022.Google Scholar
[28]Schneider, C., Product integration for weakly singular integral equations, Math. Comput., 36 (1981), pp. 207–213.CrossRefGoogle Scholar
[29]Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[30]Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373–382.CrossRefGoogle Scholar
[31]Tang, T., Xu, X. and Chen, J., On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math., 26 (2008), pp. 825–837.Google Scholar
[32]Tang, T. and Yuan, W.The further study of a certain nonlinear integro-differential equation, J. Comput. Phys., 72 (1987), pp. 486–497.CrossRefGoogle Scholar
[33]Tarang, M., Stability of the spline collocation method for second order Volterra integro-differential equations, Math. Model. Anal., 9 (2004), pp. 79–90.CrossRefGoogle Scholar
[34]Vainikko, G., On the smoothness of the solution of multidimensional weakly singular integral equations, Math. USSR-Sb., 68 (1991), pp. 585–600.CrossRefGoogle Scholar
[35]van der|Houwen, P. J. and Riele, H. J. J. Te, Linear multistep methods for Volterra integral and integro-differential equations, Math. Comput., 45 (1985), pp. 439–461.Google Scholar
[36]Yuan, W. and Tang, T.The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation, Math. Comput., 54 (1990), pp. 155–168.CrossRefGoogle Scholar