Numerical Solution to Volterra-Type Integro-Differential Equations of the Second Kinds by Legendre Collocation Method

Abstract:

The purpose of this paper is to propose an efficient numerical method for solving Volterra-type integro-differential equation of the second kinds. This method based on Legendre-Gauss-Radau collocation, which is easy to be implemented especially for nonlinear and possesses high accuracy. Also, the method can be done by proceeding in time step by step. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

• Periodical: Applied Mechanics and Materials (Volumes 687-691)
• Pages: 1522-1527
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.687-691.1522
• Online since: November 2014
• Authors: Ting Jing Zhao*
• Keywords: Integro-Differential Equation, Legendre Collocation Method, Volterra-Type

References

[1] A. M. Wazwaz: Appl. Math. Comput. Vol. 77(1996), p.79.

[2] K. Maleknejad and N. Aghazadeh: Appl. Math. Comput. Vol. 161 (2005), p.915.

[3] H. Brunner, A. Makroglou and R. K. Miller: Appl. Numer. Math. Vol. 23 (1997) , p.381.

[4] W. Wang: Appl. Math. Comput. Vol. 172(2006), p.1323.

[5] C. T. H. Baker: J. Comput. Appl. Math. Vol. 125 (2000), p.217.

[6] H. Brunner: Collocation Methods for Volterra Integral and Related Functional Equations (Cambridge University Press, New York 2004).

[7] L. M. Delves and J. L. Mohamed: Computational Methods for Integral Equations(Cambridge University Press, Cambridge 1985).

[8] J. G. Blom and H. Brunner: ACM Trans. Math. Software Vol. 17(2)(1991) , p.167.

[9] T. Tang: IMA J. Numer. Anal. Vol. 13 (1)(1993) , p.93.

[10] M. Muhammad, A. Nurmuhammad, M. Mori and M. Sugihara: J. Comput. Appl. Math. Vol. 177(2005), p.269.

[11] P. W. Sharp and J. H. Verner: Appl. Numer. Math. Vol. 34 (2-3)(2000), p.261.

[12] P. W. Sharp and J. H. Verner: SIAM J. Numer. Anal. Vol. 38 (2)(2000), p.347.

[13] R. Hagen and B. Silbermann: Operator equations and numerical analysis, Seminar Analysis, Karl-Weierstras-Institutfur Mathematik, Berlin, 1987. Ch. On stability of the qualocation method, pp.43-52.

[14] I. H. Sloan: Numer. Math. Vol. 54(1988), p.41.

[15] I. H. Sloan: J. Comput. Appl. Math. Vol. 125(2000), p.461.

[16] I. H. Sloan and W. L. Wendland: Zeit. f. Anal. u. ihre Anw. Vol. 8(1989), p.361.

[17] X. H. Ma and C. M. Huang: Appl. Math. Modelling Vol. 38(2014), p.1434.

[18] Z. Q. Wang and B. Y. Guo: J. Sci. Comput. Vol. 52(2012), p.226.

[19] B. Y. Guo and Z. Q. Wang: Comput. Methods Appl. Mech. Eng. Vol. 196(2007), p.3726.

[20] B. Y. Guo and Z. Q. Wang: Adv. Comput. Math. Vol. 30(2009), p.249.

[21] B. Y. Guo and Z. Q. Wang: Discrete Contin. Dyn. Syst., Ser. B Vol. 14(2010), p.1029.