Numerical Solution of Nonlinear Fredholm-Volterra Integtral Equations via Piecewise Constant Function by Collocation Method

A. Shahsavaran

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DOI: 10.4236/ajcm.2011.12014                

Abstract

In this work, we present a computational method for solving nonlinear Fredholm-Volterra integral equations of the second kind which is based on replacement of the unknown function by truncated series of well known Block-Pulse functions (BPfs) expansion. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.

Keywords

Nonlinear Fredholm-Volterra Integral Equation, Block-Pulse Function, Error Analysis, Collocation Points

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Shahsavaran, E. Babolian, “Numerical Implementation of an Expansion Method for Linear Volterra Integral Equations of the Second Kind with Weakly Singular Kernels,” International Journal of Applied Mathematics, Vol. 3, No. 1, 2011, pp. 1-8.
[2]	E. Babolian, F. Fattahzadeh and E. G. Raboky, “A Chebyshev Approximation for Solving Nonlinear Integral Equations of Hammerstein Type,” Applied Mathematics and Computation Vol. 189, No. 1, 2007, pp. 641-646. doi:10.1016/j.amc.2006.11.181
[3]	F. G. Tricomi, “Integral Equations,” Dover, 1982.
[4]	H. Brunner, “Implicity Linear Collocation Method for Nonlinear Volterra Equations,” Applied Numerical Ma- thematics, Vol. 9, No. 3-5, 1982, pp. 235-247. doi:10.1016/0168-9274(92)90018-9
[5]	L. J. Lardy, “A Variation of Nysrtom’s Method for Hammerstein Integral Equations,” Journal of Integral Equations, Vol. 3, No. 1, 1982, pp. 123-129.
[6]	S. Kumar, I. H. Sloan, “A New Colloca-tion—Type Method for Hammerstein Integral Equations,” Journal of Computational Mathematics, Vol. 48, No. 178, 1987, 585-593.
[7]	H. Guoqiang, “Asymptotic Error Expansion Variation of A Collocation Method for Volterra—Hammerstein equations,” Applied Numerical Mathematics, Vol. 13, No. 5, 1993, pp. 357-369. doi:10.1016/0168-9274(93)90094-8
[8]	Y. Ordokhani, “Solution of Nonlinear Volterra-Fred-holm-Hammerstein Integral Equations Via Rationalized Haar Functions,” Applied Mathematics and Computation, Vol. 180, No. 2, 2006, pp. 436-443. doi:10.1016/j.amc.2005.12.034
[9]	S. Yousefi and M. Raz-zaghi, “Legendre Wavelet Method for the Nonlinear Volterra-Fredholm Integral Equations,” Mathematics and Computers in Simulation, Vol. 70, No. 1, 2005, pp. 1-8. doi:10.1016/j.matcom.2005.02.035
[10]	S. Yashilbas, “Taylor Polynomial Solution of Non- linear Volterra-Fredholm Integral Equations,” Applied Mathe-matics and Computation, Vol. 127 No. 2-3, 2002, pp. 195-200. doi:10.1016/S0096-3003(00)00165-X