Comparison between Sinc approximation and differential transform methods for nonlinear Hammerstein integral equations

Document Type : Research Paper


1 Department of Mathematics, Shirvan Branch, Islamic Azad University, Shirvan, Iran

2 Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Zip code 14676-86831, Iran


Here, the comparison between Sinc method in combination with double exponential transformations (DE) and approximation by means of differential transform method (DTM) for nonlinear Hammerstein integral equations is considered. Convergence analysis is presented. Detection of effectiveness from various aspects such as run time, different norms, condition number are highlighted and plotted graphically. Results of two schemes are practically well, but in manner of separable kernel, DTM solution is more accurate and so fast.


[1] I.H. Abdel-Halim Hassan, Differential transformation technique for solving higher-order initial value problems, Appl. Math. Comput. 154 (2004) 299–311.
[2] A. Arikoglu and I. Ozkol, Solution of boundary value problems for integro-differential equations by using differential transform method Appl. Math. Comput. 168 (2005) 1145–1158.
[3] N. Bildik, A. Konuralp, F. Bek and S. Kucukarslan, Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. Math. Comput. 127 (2006) 551–567.
[4] C.K. Chen and S.H. Ho, Application of differential transformation to eigenvalue problems, Appl. Math. Comput. 79 (1996) 173–188.
[5] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962.
[6] F. Ayaz, Application of differential transform method to differential-algebraic equations, Appl. Math. Comput. 152 (2004) 649–657.
[7] M.A. Fariborzi Araghi and Gh. Kazemi Gelian, Numerical solution of integro differential equations based on double exponential transformation in the sinc-collocation method, App. Math. and Comp. Intel. 1 (2012) 48–55.
[8] M.A. Fariborzi Araghi and Gh. Kazemi Gelian, Numerical solution of nonlinear Hammerstien integral equations via Sinc collocation method based on double exponential transformation, Math. Sci. 30 (2013).
[9] M.A. Fariborzi Araghi and Gh. Kazemi Gelian, Solving fuzzy Fredholm linear integral equations using Sinc method and double exponential transformation, Soft Comput. 19(4) (2015) 1063–1070.
[10] R. Ghoochani-Shirvan, J. Saberi-Nadjafi and M. Gachpazan, An analytical and approximate solution for nonlinear Volterra partial integro-differential equations with a weakly singular kernel using the fractional differential transform method, Int. J. Diff. Equ. 2018 (2018).
[11] S. Haber, Two formulas for numerical indefinite integration, Math. Comp. 60 (1993) 279–296.
[12] M. Hadizadeh and Gh. Kazemi Gelian,Error estimate in the Sinc collocation method for Volterra-Feredholm integral equations based on DE transformations, ETNA 30 (2008) 75–87.
[13] I.H. Hassan, Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems Chaos Solitons Fract. 36(1) (2008) 53–65.
[14] Gh. Kazemi Gelian and M. A. Fariborzi Araghi, Numerical solution of the Burgers’ equation based on Sinc method, Theory Approx. Appl. 13(1) (2019) 27–42.
[15] H. Liu and Y. Song, Differential transform method applied to high index differential-algebraic equations, Appl. Math. Comput. 184 (2) (2007) 748–753.
[16] J. Lund and k. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992.[17] M. Mori and M. Sugihara,The double exponential transformation in numerical analysis, J. Comput. Appl. Math.127 (2001) 287–296.
[18] M. Muhammad, A. Nurmuhammad, M. Mori and M. Sugihara, Numerical solution of integral equations by means of the Sinc colocation based on the double exponential transformation, J. Comput. Appl. Math. 177(2) (2005) 269–286.
[19] M. Muhammad and M. Mori,Double exponetial formulas for numerical indefinite integration, J. Comput. Appl. Math. 161 (2003) 431–448.
[20] A. Nurmuhammad, M. Muhammad and M. Mori,Double exponential transformation in the Sinc collocation method for a boundry value problem, J. Comput. Appl. Math. Appl. 38 (1999) 1–8.
[21] F. Stenger,Numerical Methods Based on Sinc and Analytic Functions, Springer, 1993.
[22] M. Sugihara, Optimality of the double exponential formula - functional analysis approach, Numer. Math. 75 (1997) 379–395.
[23] M. Sugihara and T. Matsuo,Recent development of the Sinc numerical methods, J. Comput. Appl. Math. 164 (2004) 673–689.
[24] H. Takahasi and M. Mori, Double exponetial formulas for numerical integration, Publ. Res. Inst. Math. Sci. 9 (1974) 721–741.
[25] J. K. Zhou, Differential Transformation and its Application for Electrical Circuits, Huarjung University Press, Wuhan, China, 1986.
Volume 13, Issue 1
March 2022
Pages 1291-1301
  • Receive Date: 27 February 2020
  • Revise Date: 17 April 2020
  • Accept Date: 20 April 2020