Haar wavelet–quasilinearization technique for fractional nonlinear differential equations

Abstract

In this article, numerical solutions of nonlinear ordinary differential equations of fractional order by the Haar wavelet and quasilinearization are discussed. Quasilinearization technique is used to linearize the nonlinear fractional ordinary differential equation and then the Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Haar wavelet method. The results are compared with the results obtained by the other technique and with exact solution. Several initial and boundary value problems are solved to show the applicability of the Haar wavelet method with quasilinearization technique.

Introduction

In recent years, numerous applications of fractional order ordinary and partial differential equations have appeared in many areas of physics and engineering. There have found a number of works, especially in hereditary solid mechanics and in viscoelasticity theory, where fractional order derivatives are used for a better description of material properties. For most of fractional order differential equations, exact solutions are not known. Therefore different numerical methods have been applied for providing approximate solutions. Some of these techniques include, the Adomian decomposition method (ADM) [1], [2], the homotopy perturbation method (HPM) [3], [4], [5], the variational iteration method (VIM) [6], [7], [8], and the generalized differential transform method (DTM) [9], [10], [11].

Haar wavelet is the lowest member of Daubechies family of wavelets and is convenient for computer implementations due to availability of explicit expression for the Haar scaling and wavelet functions [12]. Operational approach is pioneered by Chen et al. [13] for uniform grids. The basic idea of Haar wavelet technique is to convert differential equations into a system of algebraic equations of finite variables.

The Haar wavelet technique for solving linear homogeneous/inhomogeneous, constant and variable coefficients has been discussed in [14], [15], [16] for uniform grids. The function having smooth behavior i.e., there is no abrupt behavior are well approximated by the Haar wavelets over uniform grids. The Haar wavelets with non-uniform grids are suitable for function having abrupt changes.

The differential equation arising from locally disturbed vibrations is solved by Lepik [17] by using non-uniform grids because solution has abrupt behavior. The non-uniform Haar wavelets are used by Fazal-i-Haq et al. [18] for the solution of singularly perturbed boundary value problem with different value of epsilon. An exact upper bound for the error analysis of the solution of fractional differential equations with variable coefficient by using Haar wavelets was presented by Chen et al. [19]. Li et al. [20] extend the Haar wavelet operational matrix to solve the fractional order differential equations. Gao et al. [21] presented an algorithm for fractional order integral by the Haar wavelet.

Boundary value problems are considerably more difficult to deal with than initial value problems. The Haar wavelet method for boundary value problems is more complicated than for initial value problems. Bratu-type boundary value problems are considered by Venkatesh et al. [22]. Second-order boundary value problems are solved in [23] by the Haar wavelets, they considered the six sets of different boundary conditions for the solution. Boundary value problems for fractional differential equations are solved in [24] in which considered the numerical solution by the Haar wavelet for different boundary value problems of fractional order.

The quasilinearization approach was introduced by Bellman and Kalaba [25], [26] as a generalization of the Newton–Raphson method [27] to solve the individual or systems of nonlinear ordinary and partial differential equations. The quasilinearization approach is suitable to a general nonlinear ordinary or partial differential equations of any order.

Jiwari [28] used a uniform Haar wavelet method with quasilinearization technique for the approximate solution of Burgers’ equation and compared the results with the solutions obtained by the other numerical methods and exact solution. The same approach used by Kaur et al. [29] for the solutions of nonlinear boundary value problems in which they treated the quadratic nonlinearity of unknown function. They solved the homogeneous and non-homogeneous Lane Emden equations [30], and compared the results with exact solution.

The Haar wavelets with quasilinearization technique [28], [29], [30] are applied for the approximate solution of integer order nonlinear differential equations over uniform grids. We utilize this approach for fractional order nonlinear differential equations over uniform as well as non-uniform grids. We used the non-uniform grids only when the solution of fractional nonlinear differential equation have abrupt behavior in a given domain.

In this paper, we consider the case of fractional order nonlinear ordinary differential equations which contain not only quadratic nonlinearity but also various other forms of nonlinearity. The main aim of the present paper is to get the numerical solutions of nonlinear fractional order initial and boundary value problems over a uniform grids with a simple method based on the Haar wavelets and quasilinearization technique. We use the non-uniform grids for solution of differential equations having abrupt behavior. Illustrative problems show the advantage of the method.

We use the cubic spline interpolation [31] to get the solution at grid points for the sake of comparison. For this purpose we use the MATLAB built-in function $y_i=\mathit{interp}1(x\text{,}y\text{,}{x}_i\text{,}‘\mathit{spline}’)$, for one-dimensional data interpolation by cubic spline interpolation.

The paper is arranged as follows: in Section 2 we review basic definition of fractional differentiation and integration, while in Section 3 and Section 4 we describe the Haar wavelets. In Section 5 we present the main features of the quasilinearization approach. In Section 6 we apply the Haar wavelet method with quasilinearization technique to some fractional nonlinear ordinary differential equations. Finally in Section 7 we conclude our work.