A collocation method for solving nonlinear second kind Volterra integral equations through single-term Walsh series

Document Type : Original Paper


Department of Mathematics, Arak University, Arak, Iran. P. O. Box 38156-8943.


In this paper, a numerical method for solving second kind nonlinear Volterra integral equations

is presented. The method is based upon the extension of unknown function by single term Walsh series.

Indeed, the interval [0, 1) is divided to m equal subinterval and in each interval, the unknown function is extended by the first term of Walsh series functions. By using a collocation method the coefficients of these extensions are computed and a block-pulse approximation of the unknown function is obtained. By the block-pulse approximation both continuous and pointwise approximations can be obtained. A convergence analysis for continuous approximations are investigated. The numerical examples confirm the ability and

accuracy of the method. The method is computationally attractive and can easily be generalized for the systems of nonlinear volterra equations.


Main Subjects

[1] E. Babolian, M. Mordad, A numerical method for solving system of linear and nonlinear integral equations of the second kind by hat basis functions, Comput. Math. Appl. 62 (2011) 187-198.
[2] K. Balachandran, K. Murugesan, Analysis of nonlinear singular systems via STWS method, Int. J. Comp. Math. 36 (1990) 9-12.
[3] K. Balachandran, K. Murugesan, Numerical solution of a singular non-linear system from fluid dynamics, Int. J. Comp. Math. 38 (1991) 211-218.
[4] V. Balakumar, M. Murugesan, Single-Term Walsh Series method for systems of linear Volterra integral equations of the second kind, Appl. Math. Comp. 228 (2014) 371-376.
[5] H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal. 27(4) (1990) 987-1000.
[6] C. Canuto and M.Y. Hussaini and A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamic, Springer-Verlag, 1987.
[7] R. Chandra Guru Sekar, V. Balakumar, K. Murugesan, Method of Solving Linear System of Volterra Integro-Differential Equations Using the Single Term Walsh Series, International Journal of Applied and Computational Mathematics, 3(2) (2017) 549-559.
[8] R. Chandra Guru Sekar, K. Murugesan, STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind, International Journal of Computer Mathematics, 94(9) (2017) 1867–1878.
[9] R. Chandra Guru Sekar, K. Murugesan, System of linear second order Volterra integro-differential equations using Single Term Walsh Series technique, Applied Mathematics and Computation, 273(C) (2016) 484-492.
[10] K.B. Datta and M.M. Bosukonda, Orthogonal functions in systems and control, World scientific publishing Co. Pte. Ltd., 1995.
[11] C.H. Hsiao and C.F. Chen, Solving integral equations via Walsh functions, Comput. Elec. Engng. 6 (1979) 279-292.
[12] F. Mirzaee, Numerical solution of nonlinear Fredholm-Volterra integral equations via Bell polynomials, Computational Methods for Differential Equations, 5(2) (2017) 88-102.
[13] A. Pushpam, P. Anandhan, Numerical Solution of Non-linear Fuzzy Differential Equations using Single Term Walsh Series Technique, International Journal of Mathematics Trends and Technology, 45(1) (2017) 35-39.
[14] A. Pushpam, P. Anandhan, Solving Higher Order Linear System of Time-Varying Fuzzy Differential Equations Using Generalized STWS Technique, International Journal of Science and Research, 5(4) (2016) 57-61.
[15] G.P. Rao, K.R. Palanisamy, T. Srinivasan, Extension of computation beyond the limit of normal interval in Walsh series analysis of dinamical systems, IEEE. Trans. Autom. control, 25 (1980) 317-319.
[16] M. Razzaghi, J. Nazarzadeh, Walsh Functions, Wiley Encyclopedia of Electrical and Electronics Engineering, 23(2) (1999) 429-440.
[17] M. Razzaghi, B. Sepehrian, Single-Term Walsh Series Direct Method for the Solution of Nonlinear problems in the Calculus of Variations, Journal of Vibration and Control, 10 (2004) 1071-1081.
[18] A. Saadatmandi, M. Dehghan, A collocation method for solving Able’s integral equations of first and second kinds, Z. Naturforsch. 63a (2008) 752-756.
[19] B. Salehi, L. Torkzadeh, K. Nouri, Chebyshev cardinal wavelets for nonlinear Volterra integral equations of the second kind, Mathematics interdiciplinary Research, 7 (2022) 281-299.
[20] B. Sepehrian, Single-term Walsh series method for solving Volterra’s population model, Int. J. Appl. Math. Reaserch, 3(4) (2014) 458-463.
[21] B. Sepehrian, M. Razzaghi, A new method for Solving nonlinear Volterra-Hammerstein integral equations via single-term Walsh series, Mathematical Analysis and Convex Optimization, 1(2) (2020) 59-70.
[22] B. Sepehrian, M. Razzaghi, Single-term Walsh series method for the Volterra integro-differential equations, Engineering Analysis with Boundary Elements, 28 (2004) 1315-1319.
[23] B. Sepehrian, M. Razzaghi, Solution of nonlinear Volterra-Hammerstein integral equations via single-term Walsh series method, Math. prob. Eng. 5 (2005) 547-554.
[24] H. Zhang, Y. Chen, C. Guo, Z. Fu, Application of radial basis function method for solving nonlinear integral equations, Journal of applied mathematics, J. Appl. Math. (2014). DOI: 10.1155/2014/381908.
[25] H. Zhang, Y. Chen, X. Nie, Solving the linear integral equations based on radial basis function interpolation, J. Appl. Math. (2014), DOI: 10.1155/2014/793582.