On the numerical scheme for solving non-linear two-dimensional Hammerstein integral equations

Document Type : Research Paper


1 Department of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran.

2 Department of mathematics, Zarandieh Branch, Islamic Azad University, Zarandieh, Iran.


In this work, solving non-linear two-dimensional Hammerstein integral equations is considered by an iterative method of successive approximation. This method is an efficient approach based on a combination of the quadrature formula and the successive approximations method. Also, the convergence analysis and the numerical stability of the suggested method are studied. Finally, to survey the accuracy of the present method, some numerical experiments are given. 


  • [1]          M. Abdou, A. Badr, and M. Soliman, On a method for solving a two-dimensional nonlinear integral equation of the second kind, J. comput. appl. math., 235 (2011), 3589–3598.
  • [2]          R. P. Agarwal, N. Hussain, and M. A. Taoudi, Fixed point theorems in ordered banach spaces and applications to nonlinear integral equations, Abst. Appl. Anal., volume 2012, Hindawi Publishing Corporation, 2012.
  • [3]          A.  Altu¨rk,  The  regularization-homotopy  method  for  the  two-dimensional  fredholm  integral  equations  of  the  first kind, Math. Comput. Appl., 21 (2016), P 9.
  • [4]          K. E. Atkinson, The numerical solution of a nonlinear boundary integral equation on smooth surfaces, IMA J. Numer. Anal., 14 (1994), 461–483.
  • [5]          K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.
  • [6]          K. E. Atkinson and G. Chandler, BIE methods for solving Laplace’s equation with nonlinear boundary conditions: The smooth boundary case, Math. Comp., 55 (1990), 455–472.
  • [7]          K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer Science+ Business Media LLC, NewYork, 2007.
  • [8]          K. Atkinson and F. A. Potra, Projection and iterated projection methods for nonlinear integral  equations, SIAM J. Numer. Anal., 24 (1987), 1352–1373.
  • [9]          I. Aziz, F. Khan, and et al., A new method based on haar wavelet for the numerical solution of two-dimensional nonlinear integral equations, J. Comput. Appl. Math., 272 (2014), 70–80.
  • [10]        E. Babolian, S. Bazm, and P. Lima, Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1164–1175.
  • [11]        S. Bazm and A. Hosseini, Bernoulli operational matrix method for the numerical solution of nonlinear two- dimensional VolterraFredholm integral equations of Hammerstein type, Comput. Appl. Math., 39 (2020), p 49.
  • [12]        A. M. Bica, M. Curila, and S. Curila, About a numerical method of successive interpolations for functional Hammerstein integral equations, J. Comput. Appl. Math., 236 (2012), 2005–2024.
  • [13]        H. Brunner, Collocation methods for Volterra integral and related functional differential equations, volume 15, Cambridge University Press, 2004.
  • [14]        L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.
  • [15]        S. Fazeli, G. Hojjati, and H. Kheiri, A piecewise approximation for linear two-dimensional volterra integral equa- tion by chebyshev polynomials, Int. J. Nonlinear Sci., 16 (2013), 255–261.
  • [16]        H.  Guoqiang  and  W.  Jiong,  Extrapolation  of  nystr¨om  solution  for  two-dimensional  nonlinear  fredholm  integral equations, J. Comput. Appl. Math., 134 (2001), 259–268.
  • [17]        G. Han and R. Wang, Richardson extrapolation of iterated discrete galerkin solution for two-dimensional fredholm  integral equations, J. Comput. Appl. Math., 139 (2002), 49–63.
  • [18]        R. J. Hanson and J. L. Phillips, Numerical solution of two-dimensional integral equations using linear elements, SIAM J. Numer. Anal., 15 (1978), 113–121.
  • [19]        S. A. Hosseini, S. Shahmorad, and A. Tari, Existence of an Lp-solution for  two  dimensional  integral  equations  of the Hammerstein type, Bull. Iran. Math. Soc., 40 (2014), 851–862.
  • [20]        A. Jafarian and S. M. Nia, Utilizing feed-back neural network approach for solving linear fredholm integral equations system, Appl. Math. Model., 37 (2013), 5027–5038.
  • [21]        A. Jerri, Introduction to integral equations with applications, John Wiley & Sons, 1999.
  • [22]        M. Kazemi and R. Ezzati, Existence of solution for some nonlinear two-dimensional volterra integral equations via measures of noncompactness, Appl. Math. Comput., 275 (2016), 165–171.
  • [23]        M. Kazemi and R. Ezzati, Existence of Solutions for some Nonlinear Volterra Integral Equations via Petryshyn’s Fixed Point Theorem, Int. J. Nonlinear Anal. Appl., 9 (2018), 1–12.
  • [24]        M. Kazemi, H. M. Golshan, R. Ezzati, and M. Sadatrasoul, New approach to solve two-dimensional Fredholm integral equations, J. Comput. Appl. Math., 354 (2019 ), 66–79.
  • [25]        M. Kazemi, V. Torkashvand, and R. Ezzati, On a method based on Bernstein operators for 2D nonlinear Fredholm- Hammerstein integral equations, U.P.B. Sci. Bull., Series A, 83 (2021), 178–198.
  • [26]        A. Khajehnasiri, Numerical solution of nonlinear 2D VolterraFredholm integro-differential equations by two- dimensional triangular function, Int. J. Appl. Comput. Math., 2 (2016), 575–591.
  • [27]        B. H. Lichae, J. Biazar, and Z. Ayati, A class of RungeKutta methods for nonlinear Volterra  integral  equations  of the second kind with singular kernels, Adv. Differ. Equ., 349 (2018).
  • [28]        K. Maleknejad, N. Aghazadeh, and M. Rabbani, Numerical solution of second kind fredholm integral equations system by using a taylor-series expansion method, Appl. Math. Comput., 175 (2006), 1229–1234.
  • [29]        K. Maleknejad, S. Sohrabi, and B. Baranji, Application of 2D-BPFs to nonlinear integral equations, Commun. Nonlinea.r Sci. Numer. Simul., 15 (2010), 527–535.
  • [30]        K. Maleknejad and P. Torabi, Application of fixed point  method  for  solving  nonlinear  Volterra-Hammerstein  integral equation, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 74 (2012), 45–56.
  • [31]        S. McKee, T. Tang, and T. Diogo, An  Euler-type  method  for  two-dimensional  volterra  integral  equations  of  the first kind, J. Inst. Math. Its Appl., 20 (2000), 423–440.
  • [32]        F. Mirzaee and S. Alipour, Solving two-dimensional non-linear quadratic integral equations of fractional order via Operational matrix method, Multidiscipl. Model. Mater. Struct., 15 (2019), 1136–1151.
  • [33]        S. Najafalizadeh and R. Ezzati, A block pulse operational matrix method for solving two-dimensional nonlinear integro-differential equations of fractional order, J. Comput. Appl. Math., 326 (2017), 159–170.
  • [34]        B. G. Pachpatte, Multidimensional integral equations and inequalities, Springer Science & Business Media, 2011.
  • [35]        A. G. Ramm, Dynamical systems method for solving operator equations, Commun. Nonlinear Sci. Numer. Simul., 9 (2004), 383–402.
  • [36]        Qi. Tang and D. Waxman, An integral equation describing an asexual population in a changing environment, Nonlinear Anal. Theory Methods Appl., 53 (2003), 683–699.
  • [37]        A. Tari Marzabad and S. M. Torabi, Numerical solution of two-dimensional integral equations of the first kind by multi-step methods, Comput. Methods Diff. Equations., 4(2) (2016), 128–138.
  • [38]        F. G. Tricomi, Integral equations, volume 5, Dover Publications, 1982.
  • [39]        A. M. Wazwaz, Linear and nonlinear integral equations: methods and applications, Springer Science & Business Media, 2011.