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.


Articles in Press, Corrected Proof
Available Online from 14 December 2022
  • Receive Date: 14 March 2022
  • Revise Date: 08 August 2022
  • Accept Date: 03 September 2022
  • First Publish Date: 14 December 2022