Solving two-dimensional nonlinear Volterra integral equations using Rationalized Haar functions

Document Type : Research Paper


1 Department of Science, School of Mathematical Sciences, University of Zabol, Zabol, Iran

2 Faculty of Engineering, Sabzevar University of New Technology, Sabzevar, Iran


In this paper, we have introduced a computational method for a class of two-dimensional nonlinear Volterra integral equations, based on the expansion of the solution as a series of Haar functions. To achieve this aim it is necessary to define the integral operator. The Banach fixed point theorem guarantees that under certain assumptions this operator has a unique fixed point, we have introduced an orthogonal projection and by interpolation property, we have achieved an operational matrix of integration. Also, by using the Banach fixed point theorem, we get an upper bound for the error of our method. Since our examples in this article are selected from different references, so should be the numerical results obtained here can be compared with other numerical methods.


[1] I. Aziz and R. Amin, Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet, Appl. Math. Model. 40 (2016), 10286–10299.
[2] I. Aziz, S. Islam and M. Asif, Haar wavelet collocation method for three-dimensional elliptic partial differential equations, Comput. Math. Appl. 73 (2017), 2023–2034.
[3] I. Aziz and I. Khan, Numerical solution of diffusion and reaction–diffusion partial integro-differential equations. Int. J. Comput. Meth. 15 (2018), 1850047.
[4] P. Assari, On the numerical solution of two-dimensional integral equations using a meshless local discrete Galerkin scheme with error analysis, Eng. Comput. 35 (2019), no. 3, 893–916.
[5] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.
[6] A. Babaaghaie and K. Maleknejad, Numerical solutions of nonlinear two-dimensional partial Volterra integrodifferential equations by Haar wavelet, J. Comput. Appl. Math. 317 (2017), 643–651.
[7] E. Babolian, S. Bazm and P. Lima, Numerical solution of nonlinear two-dimensional integral equations using Rationalized Haar functions, Commun. Nonl. Sci. Numer. Simul. 16 (2011), no. 3, 1164–1175. [8] H. Brunner and J.-P. Kauthen, The numerical solution of two-dimensional Volterra integral equations by collocation and iterated collocation, IMA J. Numer. Anal. 9 (1989), no. 1, 47–59.
[9] H. Brunner, On the numerical solution of nonlinear Volterra–Fredholm integral equations by collocation methods, SIAM J. Numer. Anal. 27 (1990), no. 4, 987–1000
[10] C.F. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEE Proc. Contr. Theor. Appl. 144 (1997), 87–94.
[11] H.J. Dobner, Bounds for the solution of hyperbolic problems, Comput. 38 (1987), 209–218
[12] M. Erfanian, M. Parsamanesh and A. Akrami, Solving two-dimensional nonlinear Fredholm integral equations using rationalized Haar functions in the complex plane, Int. J. Appl. Comput. Math. 5 (2019), 47.
[13] M. Erfanian and M. Gachpazan, A new method for solving of telegraph equation with Haar wavelet, Int. J. Comput. Sci. 3 (2016), 6–10 .
[14] M. Erfanian, M. Gachpazan and S. Kosari, A new method for solving of Darboux problem with Haar Wavelet, SeMA J. 74 (2017), 475–487.
[15] M. Erfanian and A. Mansoori, Rationalized Haar wavelet bases to approximate the solution of the first Painleve equations, J. Math. Model. 7 (2019), 107–116
[16] M. Erfanian, The approximate solution of nonlinear integral equations with the RH wavelet bases in a complex plane, Int. J. Appl. Comput. Math. 4 (2018), 31.
[17] M. Erfanian,The approximate solution of nonlinear mixed Volterra-Fredholm Hammerstein integral equations with RH wavelet bases in a complex plane, Math. Method Appl. Sci. 41 (2018), 8942–8952.
[18] G.Q. Han, K. Hayami, K. Sugihara and J. Wang, Extrapolation method of iterated collocation solution for twodimensional non-linear Volterra integral equation, Appl. Math. Comput. 112 (2000), 49–61.
[19] Z. Kamont and H. Leszczynski, Numerical solutions to the Darboux problem with functional dependence, Georgian Math. 5 (1998), no. 1, 71–90
[20] R.T. Lynch and J.J. Reis, Haar transform image conding, Proc. Nat. Telecommun. Conf., Dallas, TX, 1976, pp. 441–443.
[21] U. Lepik, Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comput. 176 (2006), 324–333.
[22] U, Lepik, Solving fractional integral equations by the Haar wavelet method, Appl. Math. Comput. 214 (2009), 468–478.
[23] F. Mirzaee and N. Samadyar, Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order, Appl. Math. Comput. 344 (2019), 191–203.
[24] K. Maleknejad, S. Sohrabi and B. Baranji, Two-dimensional PCBFs: application to nonlinear Volterra integral equations, Proc. Worldcong. Engin. (WCE), vol II. July 1–3, London, UK. 2009.
[25] S. Mckee, T. Tang and T. Diogo, An Euler-type method for two-dimensional Volterra integral equations of the first kind, IMA J. Numer. Anal. 20 (2000), 423–440.
[26] S. Nemati, P.M. Lima and Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math. 242 (2013), 53–69.
[27] M. Rastegar, A. Bazrafshan Moghaddam, M. Erfanian and B.B. Moghaddam, Using matrix-based rationalized Haar wavelet method for solving consolidation equation, Asian-Eur. J. Math. 12 (2019), 1950086.
[28] M. Razzaghi and J. Nazarzadeh, Walsh functions, Wiley Encycl. Electric. Electron. Engin. 23 (1999), 429–440.
[29] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge, 1997.
Volume 14, Issue 8
August 2023
Pages 95-105
  • Receive Date: 14 March 2022
  • Revise Date: 08 August 2022
  • Accept Date: 03 September 2022