Numerical Method for Inverse Laplace Transform with Haar Wavelet Operational Matrix

Wavelets have been applied successfully in signal and image processing. Many attempts have been made in mathematics to use orthogonal wavelet function as numerical computational tool. In this work, an orthogonal wavelet function namely Haar wavelet function is considered. We present a numerical method for inversion of Laplace transform using the method of Haar wavelet operational matrix for integration. We proved the method for the cases of the irrational transfer function using the extension of Riemenn-Liouville fractional integral. The proposed method extends the work of J.L.Wu et al. (2001) to cover the whole of time domain. Moreover, this work gives an alternative way to find the solution for inversion of Laplace transform in a faster way. The use of numerical Haar operational matrix method is much simpler than the conventional contour integration method and it can be easily coded. Additionally, few benefits come from its great features such as faster computation and attractiveness. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to analytical solution. Examples on solving differential equation by Laplace transform method are also given. | operational matrix | numerical inversion | Inversion of Laplace transform | Haar wavelet | ® 2012IbnuSina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v8n4.149 Equation Chapter (Next) Section 1


INTRODUCTION
Laplace transforms is known to be an important tool in solving mathematical equations that arise in engineering problem.Since its discovery by a French mathematician, it has been widely applied and continuously researched by scholars from various fields.Those scholars had put through enormous amount of efforts in finding its inverse function numerically and analytically.This is because finding the inverse of Laplace transform is considered to be a difficult task due to its limitation in the inversion table of inverse Laplace transform,in the sense that it couldn'tcater most of the engineering problems which always associated with complexity of mathematical equation.
The objective of this paper is to propose a numerical inversion of Laplace transform using Haar operation matrix.The proposed method in this paper is an extension work of J.L. Wu et al (2001) that covers the whole time domain in finding inversion Laplace transform numerically using Haar wavelet operational matrix for integration.J.L. Wu et al. has proposed a new unified method to derive the operational matrix of any orthogonal functions for integration within the interval of 0 1 t ≤ < .We derive the Haar operationalmatrix based on Wu et al. works but extending it using generalised block pulse function operational matrix for integration [2,7] series for 0 t τ ≤ < .Before Haar wavelet operational matrices were used to find inversion of Laplace transform, there are other literatures that used other orthogonal functions as well.In 1977, C. F. Chen et al have been using Walsh operational matrices for solving various distributed-parameters systems such as heat conduction and percolation problem [8].Later, a more rigorous approach has been taken by Wang Chi-Hsu to derive the generalised block pulse operational matrices [7].According to Wang, inversions of Laplace transform for rational and irrational transfer function illustrated by using generalized block pulse operational matrices is proven to be more accurate compare to previous work by Chen [8].

MATHEMATICAL REVIEW Equation Chapter 2 Section 1 2.1 Haar Wavelet Function Equation Section (Next)
An analytic function ( ) f t can be expandedin a series 0 ( ) ( ) where ( ) n t ϕ is the basis in the Hilbert space 2 ( ) L R and n a is coefficient of the series.The coefficients can be obtained as follows, ( ) ( ) which is convenient as it will fit the expansion of Haar For example, if we have a function ( ) n n t t ψ = , we could expand the function using power series expansion such as Taylor series expansion.Same goes to a function with sinusoidal basis, we could use Fourier series expansion.In this work an orthogonal function namely Haar wavelet function is considered.The set of this function is a group of square waves in intervals of [0, ) τ and defined as below where 1 23 ( 1 2 ) , (( 1 2) 2 ) , ( 2 ) and the resolution J is a positive integer.While j and k denoted the integer decomposition of the index i , for example 2 1 h x is defined as a co nstant and called scaling function, while 1 ( ) h x is called mother wavelet function or fundamental square wave.All the others following Haar wavelet functions are generated from mother wavelet function, 1 ( ) h t with translation and dilation process.( ) 2 (2 ) where 2 1, 0, 0 2 Haar wavelet function also is an orthogonal function, so that it holds the property as below 0 ( ( ), ( )) ( ) ( ) 0 The orthogonal set of the first four Haar function ( 4) m = in the interval of (0 1) t ≤ < can be shown in Figure 1

Haar Series Expansion
Haar wavelet function is not continuous.As for Haar series expansion, any function ( ) x t can be decomposed into Haar series and can be written as 0 ( ) ( ) If the function ( ) x t may be approximated as a piecewise constant then the sum in equation (2.8) may be truncated after m terms and defined within interval 0 t τ ≤ < , then it becomes, where the Haar coefficient i c are determined by 0 ( ) ( )  [ ] Taking the collocation points as following 2 1 , 1, 2, , 2 It is defined that the m square Haar wavelet matrix, m For instance, the fourth Haar wavelet matrix ( 4) m = , 4 H in the interval of 0 1 t ≤ < can be represented in matrix form as below.) ) ) ) ) ) ) Haar wavelet is an orthogonal functions and it can be shown that ) by this method it is convenient to find the coefficient without performing the integration as equation (2.10) (2.17)Where x is a vector of a function ( ) x t at the collocation point as equation (2.13).

Integration of Haar Wavelet Function and its Operational Matrix
Let consider the integration of a Haar wavelet function ( ) where m Q is the generalised Haar operational matrix for integration of Haar wavelet function, ( ) where ( ) m B t is the block pulse function [2]   1 2 1 ( ) 0 elsewhere which defined on the interval (0, ] τ thus equation (2.18) can be written as It is known that the integration of block pulse function can be calculated as below 0 ( ) ( ) where m α F is taken from generalize blockpulse operational matrix for integrationwith 1 and (0 Besides that the generalised Haar operational matrix for integration, m Q also can be obtained from recursive formula by Chen Hsiao et.al [5] aftersome modifications were made to cover the interval of [0, ) τ .The generalised Haar operational matrix from recursive formula can be calculated by equation as below.

Riemann-Liouville Fractional Integral and Haar Wavelet Function
It is known that for integer n , the iterated integration with ( ) H t with integral of order 0 α > .Some modification is necessary to accommodate with expression in finding inversion of Laplace transform later.Firstly, we consider the fractional integral of Haar wavelet scaling function, 0 ( ) h t of order 0 α > and equation (2.32) is then become, ( 1)

33) by cross multiplying the above equation, yields
[ ]

NUMERICAL ANALYSIS OF INVERSION LAPLACE TRANSFORM
The Laplace transform of a function ( ) x t , denoted by ( ) X s is defined by an integral function equation We know the Laplace transform of integral is as below The integration in equation (3.2) and equation ( 2.18) are corresponding to the multiplication of 1 s in s domain and Haar operational matrix for integration m Q in t domain respectively.Thus we could replace the 1 s factor to the generalised Haar operational matrix, m Q .Assuming that the irrational transfer function has a form of ( ) where 0 1 α ≤ < and truncated to ( ) n n∈  .By cross multiplying equation (3.3), we have ( ) Then perform inverse Laplace transform of equation(3.4), at both side yields Taking the collocation points as equation (2.13), factorize , Multiplying both sides with 1 2 ( ) 3) by replacing 1 s with the generalised Haar operational matrix, m Q .

Example 1
Consider the irrational transfer function as By using this method, firstly, find expression of ( ) X s in terms of 1 s and denoted as ( ) Then, replace each terms of 1 s in equation (3.12) by the In the case of 16 m = and 4 τ = , the result is shown in Figure 2.  ( ) ( ) In the case of, 1 a = , 16 m = and 1 τ = , from equation (3.10), we obtain ( ) The exact solution is and the result is shown in Figure 4 Fig. 4 Comparison between the exact solution and present numerical results for 32 m =

Example 4
Consider the irrational transfer function as

Fig. 1
Fig. 1 First four Haar function 13) Lastly, by equation (3.10) the inversion of Laplace transform can be calculated by the below equation.

1 s
with generalised Haar wavelet operational matrix, m Q .
is shown in Figure5.

Fig. 5
Fig. 5 Comparison between the exact solution and present numerical results for 32 m =