International Journal of Computer Applications (0975 – 8887)
Volume 97– No.18, July 2014
Haar Wavelet Matrices for the Numerical Solutions of
Differential Equations
Sangeeta Arora Yadwinder Singh Brar Sheo Kumar
Research Scholar Department of ECE Department of Mathematics
Punjab Technical University Guru Nanak Dev Engineering Dr. B.R. Ambedkar NIT
Jalandhar College Ludhiana Jalandhar
ABSTRACT A review of the basic properties of the wavelets and the
Haar Wavelets has become important tool for solving number decomposition and the reconstruction of functions in terms of
of problems of science and engineering. In this paper a the wavelet bases is given by Strang [6]. Many families of
computational scheme is implemented using Haar matrices to wavelets have been proposed in the literature. If one wants to
find the numerical solution of differential equations with known use wavelets for the solution of differential equations, one
initial and boundary conditions. We also presented exact therefore has to choose one specific family which is most
solution, numerical solution and absolute error. Numerical advantageous for the intended application. Within one family
experiments presented here are comparable with the available there are also members of different degree. All these wavelet
data. The algorithm used in this is very simple and easy to families can be classified as either being an orthogonal or
implement. biorthogonal family. Each orthogonal wavelet family is
characterized by two functions- the mother scaling function and
Keywords the mother wavelet. With a solid historical as well as practical
Haar wavelets, Haar functions, Operational matrix, Differential background, Among the wavelet families, which are defined by
equation. an analytical expression, special attention deserves the Haar
wavelets.
In 1910, Alfred Haar introduced the notion of wavelets. The
1. INTRODUCTION Haar wavelet transform is one the earliest examples of what is
It has been observed from the literature that many researchers known now as a compact, dyadic, orthonormal wavelet
are developing fast and accurate numerical schemes to handle transform. Haar wavelets are made up of pairs of piecewise
the different problems arising in various fields of science and constant functions and are mathematically the simplest among
engineering. In the past finite element methods and finite all the wavelet families. A good feature of the Haar wavelets is
difference methods were commonly used for solving such the possibility to integrate them analytically arbitrary times. The
problems. Nowadays wavelet methods are extensively applied Haar wavelets are very effective for treating singularities, since
to the problems for numerical solutions as wavelets methods they can be interpreted as intermediate boundary conditions.
have several advantages over FEM and FDM. . FEM is one of Haar wavelets are easy to handle from the mathematical aspect.
the successful and dominant numerical methods in last century. Haar wavelets are very effective for solving ordinary
Wavelet analysis is a new technique that can be performed in differential and partial differential equations. Therefore the
several ways, a continuous wavelet transform, a discretized idea, to apply Haar wavelet technique was quite popular [7-10].
continuous wavelet transform, and a true discrete wavelet One property of the Haar wavelet is that it has compact support,
transform. With the rapid development of computer technology which means that it vanishes outside of a finite interval.
in the past few decades a broad range of numerical methods In this paper, we apply Haar wavelet matrices to solve ordinary
have been developed for different types of problems and differential equations with initial or boundary condition known
achieved a great success like Haar wavelets methods. Haar and we compare the results of numerical and exact solutions.
wavelet method is simple and possesses less computational This has been designed to promote the study of wavelets to
cost. In comparison with existing numerical schemes used to beginners. Thus, simplified calculations are presented with
solve the PDE’s, the Haar wavelet methods is an improvement necessary basic knowledge of haar functions and their
over other methods in terms of accuracyIt is extensively used in generation.
modeling and simulation of engineering and science due to its
versatility and flexibility.
A wavelet is a mathematical function used to divide a given 2. HAAR WAVELET MATRICES
function or continuous-time signal into different scale The Haar wavelet transform is the first known wavelet and was
components. The word wavelet is due to Morlet and proposed in 1909 by Alfred Haar. A haar wavelet is a system of
Grossmann. In the early 1980s they used the French word square wave having the first curve known as scaling function
ondelette, meaning “small wave”. Soon it was transferred to
English by translating “onde” into “wave”, giving “wavelet”.
The study of wavelets has attained the present growth due to
mathematical analysis of wavelets by Stromberg [1],
Grossmann and Morlet and Meyer [2,3]. The concept of and second curve is is given by
Multiresolution Analysis (MRA) was introduced by Mallat and
Meyer [4]. Daubechies in 1988 presented a method to construct
wavelets with compact support and scale functions [5].
33
International Journal of Computer Applications (0975 – 8887)
Volume 97– No.18, July 2014
This is also known as mother wavelet. To perform the wavelet
transform, haar wavelets uses translation and dilation using.
Moreover, for any square integrals function
approximation can be made using the haar functions as
3. NUMERICAL METHOD
In this section, we present some numerical examples that are
Where To express the considered to apply the solution procedure presented in the
haar matrix, we used the general notation as previous section. Basis functions considered here are family of
Haar wavelets. In the first example we solve the equation
Thus we have
With exact solution for this case is given
by
To start the solution procedure, we have
Integrate both the sides and put the values of and we
found the values of , again integrate the side and put the
value and we get the
after putting the values of in
Integration over the vector is given by
)
Operational matrix obtains the values as We found after simplification of this equation we get the
result
,
then find and put the values of then we have after
putting the values of we obtain numerical
solution then we compare with the exact solution and we have
the absolute error.
Table 1. Comparison of Exact and Haar Solution
x Exact Solution Haar Solution Absolute Error
0.125 0.0081 0.0088 0.0006
0.375 0.0791 0.0810 0.0019
0.625 0.2361 0.2399 0.0031
0.875 0.4952 0.4992 0.0041
34
International Journal of Computer Applications (0975 – 8887)
Volume 97– No.18, July 2014
If we take more collocation points and we will get more Another Example we will take
accurate results. Graphical representation of exact numerical
solution is shown below. With and exact solution for this
equation is
Graphical Representation of first example
For numerical solution of above example we will take
after solving this we get
substitute the values in the given equation, we obtained
numerical solution and compared with exact Solution.
Table 2. Comparison of Exact and Haar Solution
x Exact Solution Haar Solution Absolute Error
0.125 -0.0119 -0.0121 0.2159
0.375 -0.0334 -0.0340 0.5210
0.625 -0.0435 -0.0440 0.5022
0.875 -0.0259 -0.0261 0.2023
Graphical Representation of Second example
Figure 1(A): Exact Solution
Figure 2(A): Exact Solution
Figure 1(B) : Haar Solution
35
International Journal of Computer Applications (0975 – 8887)
Volume 97– No.18, July 2014
[5] Daubechies, I., 1988. “Orthonormal bases of compactly
supported wavelets”, Comm. Pure Appl. Math., 909-996.
[6] Strang, G., 1989. SIAM Rev. 31, 614.
[7] Hsiao, C.H., 2004. “Haar wavelet approach to linear stiff
systems”, Mathematics and Computers in Simulation, 561-
Figure 2(B): Haar Solution 567.
[8] Daubechies, I., 1992. Ten Lectures on Wavelets (SIAM,
Philadelphia,)
[9] Chui, C.K., 1992. “An introduction to wavelets”
(Academic Press, Boston, MA).
[10] Lepik, ULO., 2005. Numerical solution of differential
equations using Haar wavelets, Mathematics and
Computers in Simulation, 127-143.
[11] Daubechies, I., 1992. “Ten Lectures on Wavelets, SIAM,
Philadelphia, PA”.
[12] Kaiser, G., 1994. “A friendly guide to wavelets,” Boston:
Birkhauser.
[13] Lepik, U., 2007. “Numerical Solution of Evolution
Equations” by the Haar Wavelet Method. Appl. Math. and
Comput., 185, 695-704.
[14] Chen, C.F., and Hsiao, C.H., 1997. “Haar Wavelet Method
for Solving Lumped and Distributed Parameter Systems,”
IEE Proc, Number 144 in Control Theory Appl., 87-94.
[15] Hsiao, C.H., 1997. “State Analysis of Linear Time
4. CONCLUSION & FUTURE SCOPE Delayed Systems via Haar Wavelets,” Math. Comp.
This paper represented simple and straight forward numerical
Simulat., 44: 457-470.
technique based on Haar wavelet is proposed for solving the
differential equation. This method is simple and has small [16] Cattani, C., 2004. "Haar Wavelets Based Technique in
computation cost and also very convenient for solving variety Evolution Problems," Proc. Estonian Acad. Set Phys.
of boundary value problems. This paper presented numerical Math., 1: 45-63.
solution very close to the exact solution. So Haar wavelet
method is very simple, fast and reliable. It is observed that Haar [17] Siddiqui, Abul Hasan., 2003. "Wavelet Method for Partial
Wavelet method can be extended for more collocation points. Differential Equations and Image Processing, Numerical
Thus it will be interesting to study for what class of differential Methods," Wavelet Methods and Image Processing ISBN:
equations; the Haar wavelet will give better results. 978-0-8247-4097-9, ISBN: 978-0-203-91301-7, doi:
10.1201/97802039 13017. Ch 11.
5. REFERENCES [18] Hariharan, G., 2013. "An Overview of Haar Wavelet
[1] Stromberg, J.O., 1981. In Proceedings of Harmonic Method for Solving Differential and Integral Equations,"
Analysis, University of Chicago, 475-494. World Applied Sciences Journal, 23, 01-14
[2] Grossmann, A., and Morlet, J., 1984. SIAM J.Math.Anal. [19] Haar, A., 1910. "Zur Theorie Der Orthogonalen
15, 723. Funktionsysteme," Math. Ann. 69, 331-371.
[3] Meyer, Y., 1989. Analysis at Urbana 1: Analysis in [20] Hsiao, C.H. and S.P. Wu, 2007 "Numerical Solution of
Function Spaces Cambridge University Press, Cambridge. Time-Varying Functional Differential Equations" via Haar
[4] Mallat, S.G., 1989. Trans. Amer.Math.Soc.315,69. Wavelets. Appl. Math. Comput., 188 (1): 1049-1058.
IJCATM : www.ijcaonline.org
36
