Legendre wavelet collocation method for the numerical solution of singular initial value problems

Introduction

Recently, the studies of singular initial value problems in the second order ordinary differential equations (ODEs) have fascinated the attention of many mathematicians and physicists. Many methods including numerical and perturbation methods have been used to solve such type of problems. The approximate solutions for these problems were presented by many researchers for examples Wazwaz [1,2] were using the adomain decomposition method (ADM) and Yildirim and Ozis [3] were using the variational iteration method (VIM). In numerical analysis, classical discritization methods, such as finite differences, finite elements, spectral elements, are powerful tools for solving differential equations. However, singularities and step changes often emerge in many phenomena, like stress concentration, elastoplasticity, shock wave and crack. Since small-scale features only exist in a small percentage of the solution domain, if one chooses a uniform numerical grid fine enough to resolve the small-scale characteristics, then the solution to the equations will be over-resolved in the majority of the domain. One would like ideally, to have a dense grid where small-scale structure exists and a sparse grid where the solution is only composed of large-scale features [4,5,6] . It demands for the usage of non-uniform grids and adaptive grids or moving elements to dynamically adapt to the changes in the solution [7] . That is where wavelets play a role. Among the different wavelet families, most simple are the Haar wavelets. Haar wavelets have been used by many researchers because of their simplicity and better convergence. Some of the relevant work regarding Haar and Chebyshev wavelets can be found in [8][9][10][11][12][13][14][15][16][17][18] . The weaker side of using the Haar and Chebyshev basis functions for approximating functions is that they are lower in accuracy. To cover this aspect, Legendre wavelets are considered to get more accurate approximation in the absence of jumps or sharp transitions. Legendre wavelets use Legendre polynomials as their basis functions. Because of their improved smoothness and good interpolating properties, accuracy of Legendre wavelets is better than Haar and Chebyshev wavelets in case of approximating smooth functions even on a coarse grid. Applications of Legendre wavelets for numerical approximations can be found in [11, 19, 20 & 21] . In this paper, an attempt is made to propose the Legendre wavelet collocation method (LWCM) for the numerical solution of singular initial value problems. The paper is organized in the following way. Section 2 is devoted to the preliminaries of wavelets. Method of solution is presented in section 3. Section 4 deals with the numerical examples. Concluding remarks of the paper is discussed in Section 5.

Wavelet Preliminaries 2.1 Wavelets

In the recent years, wavelets have been applied extensively for signal processing in communications and physics research, and have proved to be a wonderful mathematical tool. Wavelets can be used for algebraic manipulations in the system of equations obtained which leads to better resulting system. Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets; , 1 ( ) , , , 0

The best way to understand wavelets is through a multi-resolution analysis. Given a function 2 () f L R  a multi-resolution analysis (MRA) of 2 ( ) L R produces a sequence of subspaces 1 , , . . .,

such that the projections of f onto these spaces give finer and finer approximations of the function f as j  .

A multi-resolution analysis of 2 () LR is defined as a sequence of closed subspaces

e. all the j V are invariant under translation.

V is used to approximate general functions by defining appropriate projection of these functions onto these spaces.

Since the union of all the j V is dense in 2 () LR , so it guarantees that any function in 2 () LR can be approximated arbitrarily close by such projections. As an example the space   , j V j Z  can be defined like

For each j the space j W serves as the orthogonal complement of j V in 1 j V  . The space j W include all the functions in 1 j V  that are orthogonal to all those in j V under some chosen inner product. The set of functions which form basis for the space j W are called wavelets.

Legendre wavelets and operational matrix of integration

For any positive integer k , the Legendre wavelets family is defined [19] as follows; Equivalently, for any positive integer k , the Legendre wavelets family is defined [11] as fallows;

. By varying the values of i with respect to the collocation points 0.5 , 1, 2, ,

, we get the Legendre matrix of order N N  , where

and Legendre polynomials used in the approximation are of degree less than M .

The integration of Legendre wavelets is given [19] as where P is the NN  operational matrix for integration and is given in [19] as

Where U and D are MM  matrices given by 2 0 0 0 0 0 0 0 0 0 0 0

Method of solution

Consider the singular initial value problem of the form

Subject to the initial condition (0) , (0)

Substituting the values of (3.3)-(3.5) in (3.1) then we have,

Proof. The error at th k level may be defined as,

From the above lemma, the error bound is inversely proportional to the level of the resolution of the Legendre wavelets. This ensures that the convergence of the Legendre wavelets approximation by increasing the level of resolution.

Rate of convergence ( )

c RN: The rate of convergence is defined as

Numerical examples

In this section, we consider some of the singular initial value problems to demonstrate the applicability of the proposed Legendre wavelet collocation method.

Problem 1. First consider the equation of the type [3] ,

with the initial conditions (0) 1, '(0) 0 uu  .

Using the method explained in section 3, we obtained the LWCM solution and is presented in comparison with exact solution 2 ()

 , Haar wavelet collocation method (HWCM) (as the method explained in [15] ), Chebyshev wavelet collocation method (CWCM) [18] solutions and VIM solution in Table 1 for N=16 (M=8 & k=2) & Fig. 1 for N=32 (M=8 & k=3). The error analysis for higher values of N is given in Table 2. 3 for N=32 (M=8 & k=3). The error analysis for higher values of N is given in Table 6.

Conclusion

A new efficient method that is, Legendre wavelet collocation method has been proposed for the numerical solution of singular initial value problems. The accurate implementation of the classical numerical methods for the numerical solution of the singular initial value problems has been found to involve some difficulty. It has been shown here that the present method can be easily implemented and the results obtained are most accurate. The performance of LWCM is superior to the HWCM, CWCM and other classical methods for example VIM & ADM, which is justified through the illustrative problems. Hence the present method has a clear advantage over the classical methods. A distinct feature of the proposed method is its simple applicability for a variety of singular initial value problems.

Acknowledgement

AcknowledgementThe authors thankful to the UGC, New Delhi, India, for the financial support of UGC's Research Fellowship in Science for Meritorious Students vide sanction letter No. F.4-1/2006(BSR)/7-101/2007(BSR), dated-02/01/2013 during Ph. D time and also thanks to KLE CET, Chikodi for their support to research.

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