Review Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations K. Maleknejad * , H. Almasieh, M. Roodaki Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran article info Article history: Received 13 May 2008 Received in revised form 2 August 2008 Accepted 7 December 2009 Available online 4 January 2010 Keywords: Integral equations Nonlinear Triangular functions Volterra–Fredholm abstract A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra–Fredholm integral equations. The orthogonal triangular functions are utilized as a basis in collocation method to reduce the solution of nonlinear Volterra–Fredholm integral equations to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Some numerical examples illustrate the proposed method. Ó 2009 Elsevier B.V. All rights reserved. Contents 1. Introduction ........................................................................................... 3293 2. Brief review of orthogonal triangular functions ............................................................... 3294 3. Problem statement ...................................................................................... 3295 4. Convergence analysis .................................................................................... 3296 5. Numerical examples..................................................................................... 3296 6. Conclusion ............................................................................................ 3297 References ............................................................................................ 3298 1. Introduction Integral equations of various types appear in many fields of science and engineering. To obtain the solution of integral equations, many different basis functions are introduced. The commonly used methods are based on piecewise constant ba- sis functions (PCBF) [1], Chebyshev polynomial [2] and wavelets [3]. PCBFs has attracted much attention for nine decades. Then wavelets theory is started with the introduction of Haar function in 1910 [3]. Several numerical methods based on dif- ferent wavelets such as Legendre and Walsh were designed for analysis of control systems and various related applications [3,4], and approximating the solution of integral equations [4,5]. The techniques based on PCBF and wavelets are effective to obtain the solution of integral equations. But calculating constant coefficients requires the use of integration formula. Deb 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.12.015 * Corresponding author. Tel.: +98 261 340 74 81; fax: +98 261 341 02 78. E-mail addresses: maleknejad@iust.ac.ir (K. Maleknejad), halmasieh@yahoo.co.uk (H. Almasieh), Roodaki-1436@yahoo.com (M. Roodaki). Commun Nonlinear Sci Numer Simulat 15 (2010) 3293–3298 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns