Document Type: Research Paper


Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran


‎In this paper‎, ‎a matrix based method is considered for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form $s^{\beta}(t-s)^{-\alpha}G(y(s))$ based on the Tau method‎. ‎In this method‎, ‎a transformation of the independent variable is first introduced in order to obtain a new equation with smoother solution‎. ‎Error analysis of this method is also presented‎. ‎Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method‎.


Main Subjects

[1] S. S. Allaei, T. Diogo, M. Rebelo, The jacobi collocation method for a class of nonlinear Volterra integral equations with weakly singular kernel, J. Sci. Comput. DOI 10.1007/s10915-016-0213-x.

[2] P. Baratella, A Nystrom interpolant for some weakly singular nonlinear Volterra integral equations, J. Comput. Appl. Math. 237 (2013), 542-555.

[3] T. Diogo, J. Ma, M. Rebelo, Fully discretized collocation methods for nonlinear singular Volterra integral equations, J. Comput. Appl. Math. 247 (2013), 84-101.

[4] G. Ebadi, M. Y. Rahimi-Ardabili, S. Shahmorad, Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method, Appl. Math. Comput. 188 (2) (2007), 1580-1586.

[5] N. M. B. Franco, S. Mckee, A family of high order product integration methods for an integral equation of Lighthill, Int. J. Comput. Math. 18 (1985), 173-184.

[6] N. B. Franco, S. McKee, J. Dixon, A numerical solution of Lighthill's equation for the surface temperature distribution of a projectile, Mat. Aplic. Comp. 12 (1983), 257-271.

[7] F. Ghoreishi, M. Hadizadeh, Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations, Numer. Algor. 52 (2009), 541-559.

[8] S. M. Hossieni, S. Shahmorad, Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases, Appl. Math. Model. 27 (2003), 145-154.

[9] S. Karimi Vanani, F. Soleymani, Tau approximate solution of weakly singular Volterra integral equations, Math. Comput. Model. 57 (2013), 494-502.

[10] I. J. Kumar, On the asymptotic solution of a nonlinear Volterra integral equation, Proc. Roy. Soc. Lond. A. 324 (1971), 45-61.

[11] J. M. Lighthill, Contributions to the theory of the heat transfer trough a laminar boundary layer, Proc. Roy. Soc. London. 202 (A) (1950), 359-377.

[12] R. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin, California, 1971.

[13] M. Nili Ahmadabadi, H. Laeli Dastjerdi, Tau approximation method for the weakly singular Volterra-Hammerstein integral equations, Appl. Math. Comput. 285 (2016), 241-247.

[14] B. Noble, The numerical solution of nonlinear integral equations and related topics, Anselone, 1964.

[15] E. L. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing. 27 (1981), 15-25.

[16] M. Rebelo, T. Diogo, A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel, J. Comput. Appl. Math. 234 (2010), 2859-2869.

[17] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), 1075-1081.