Rationalized Haar Wavelet Bases to Approximate Solution of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations with Error Analysis
Majid Erfanian, Morteza Gachpazan, Hossain Beiglo · 2015
Analytical solutions of integral equations, either do not exist or are hard to find. Due to this, many numerical methods have been developed for finding the solutions of integral equations. The use of wavelets has come to prominence during the last two decades. Wavelets can be used as analytical tools for signal processing, numerical analysis and mathematical modeling. The early works concerning wavelets were in the 1980s by Morlet, Grossmann, Meyer, Mallat and others. But in fact, it was the paper of Daubechies [1] in 1988 that caught the attention of the applied mathematics communities in signal processing, and numerical analysis. Most of the early works are discussed in [2,3] and [4-6]. The goal of the most modern wavelet researches is to create a set of basis functions and transform them, which yields an informative and useful description of a function or signal. Various types of wavelets have been applied for numerical solution of different kinds of integral equations. These include Haar, Legendre, trigonometric, CAS, Chebyshev, and Coifman wavelets. Lepik and Tamme in [7] have applied Haar wavelets to nonlinear Fredholm integral equations, but their method involves approximation of certain integrals. The orthogonal set of Haar functions is a group of square waves with magnitude of +2i=2, -2i=2 and 0, for any i=0; 1; : : :[8,9]. Lynch and Reis [10] have rationalized the Haar transform by deleting the irrational numbers and introducing the integral powers of two. This modification results in what is called the rationalized Haar (RH) transform. The RH transform preserves all the properties of the original Haar transform and can be efficiently implemented using digital pipeline architecture [11]. The corresponding functions are known as RH functions. The RH functions are composition of only three amplitude +1, -1 and 0. The aim of this work is to present a numerical method for approximating the solution of nonlinear Fredholm-Hammerstein integral equations of the second kind as follows: 1 11 0 () ( ) ( , ) ( , ( )) u x f x K x t W t u t dt α = + ∫