Numerical Solution to Differential Equations via Hybrid of Block-pulse and Rationalized Haar Functions

Many different bases functions have been used to estimate the solution to differential equations, such as orthogonal bases [3, 4, 14, 15], wavelets [7–8] and hybrid [2, 13, 16–17]. The various systems of orthogonal functions may be classified into two categories. The first is piecewise continuous function (PCBF) to which the orthogonal systems of Walsh functions [5], Block-pulse functions [4, 10] and Haar functions [1, 11, 12, 19] belong. The second group consists in continuous orthogonal functions such as orthogonal polynomials and sin-cos basis [3, 9]. We notice that, by approximating a discontinuous function by a continuous basis we can not properly model the discontinuities and therefore we must approximate such a function by PCBFs. One of the main characteristics of the orthogonal function techniques for solving different problems is to reduce these problems to those of solving a system of algebraic equations. These techniques have been presented, among others by Hwang and Shih [9] and Lepik [11]. Refs. [14, 16–17] introduced the hybrid of Block-pulse and orthogonal polynomials to approximate different problems. Refs. [2, 13] used the hybrid of Block-pulse and rationalized Haar functions to approximate time-varying differential equations and nonlinear Volterra-Hammerstein equations. The aim of this paper is to reduce the variables and get higher accuracy for approximation