NUMERICAL SOLUTION OF WAVE EQUATION USING HAAR WAVELET

Wavelet, being a powerful mathematical tool, has been widely used in image digital processing, quantum field theory and numerical analysis. Wave equation is a special partial differential equation arising in numerous engineering problems. Wave equation is a second order hyperbolic partial differential equation and it describes the phenomenon of wave vibration and wave propagation. Consider the partial differential equation of the form


Introduction
Wavelet, being a powerful mathematical tool, has been widely used in image digital processing, quantum field theory and numerical analysis. Wave equation is a special partial differential equation arising in numerous engineering problems. Wave equation is a second order hyperbolic partial differential equation and it describes the phenomenon of wave vibration and wave propagation. Consider the partial differential equation of the form url: www.acadpubl.eu § Correspondence author ∂ 2 u ∂t 2 = c 2 ∂ 2 u ∂x 2 + f (x, t), 0 < x < 1, t > 0, (1) with initial conditions and boundary conditions where u(x, t) represents the displacement of every point at time t, c 2 = F/σ, F represents the tension of every point in the media, σ represents medium density, F and σ are constant, so c 2 is also constant, f (x, t) represents the external force on per unit mass.
In Section 2, we briefly describe Haar wavelet method. In Section 3, we have described function approximation. Methods for solving wave equation have been presented in Section 4 and in Section 5, numerical examples have been solved using the Haar wavelet method to illustrate the efficiency and accuracy of present method over the method described in Shi [5].

Haar Wavelet Method
In recent decades the field of Haar wavelets for solving partial differential equations have attracted interest of researchers in several areas of science and engineering. A survey on differential equation is presented in Hariharan [6]. Wavelet analysis ia a new branch of mathematics and widely applied in differential and integral equations. Several methods have been proposed to find the numerical solution of different linear and nonlinear partial differential equations. Wavelets have been applied extensively in mathematical problems related to scientific and engineering fields. There are many wavelet families such as Daubechies wavelet [8], Hermite-type trigonometric wavelet and many more. In 1910, Haar [7] introduced a function which presents a rectangular pulse pair. After that various generalizations were proposed. Among all these wavelet families, it is the simplest orthonormal wavelet with compact support. Haar wavelet is Daubechies wavelet of order one. Hariharan et al. [3] presented the numerical solution of Fisher's equation using Haar wavelet. Further using Haar wavelet, Hariharan and Kannan [4] solved numerically Fitzhugh-Nagumo equation. Berwal et al. [2] presented the numerical solution of Telegraph equation using Haar wavelet.
Lepik [9] and [10] presented methods based on Haar wavelet for numerical solution of differential and integral equations. The Haar functions are an orthogonal family of switched rectangular waveforms where amplitudes can differ from one function to another. They are defined in the interval [0, 1].
where α = k m , β = k+0.  , we introduce the following notation: Here H 4 (x) is called Haar coefficient matrix. It is a square matrix of order 4, and is defined as H(i, l) = (h i (x l )), which has dimension 2M ×2M . Let us integrate equation (4), we get In the collocation points, equation (7) gets the form Q(i, l) = q i (x l ), where Q is a 2M ×2M matrix. Chen and Hsiao [1] presented this matrix in the form Q n = P n H n , where P n H n is interpreted as the product of the matrices P n and H n , called Haar integration and coefficient matrix, respectively. The operational matrix of integration P , which is a 2M square matrix, is defined by the relations: where n = 1, 2, 3, 4.... These integrals can be evaluated using equation (4) and first four of them are given below:- elsewhere. (11) elsewhere. (12)

Function Approximation
We know that all the Haar wavelets are orthogonal to each other: Therefore, they construct a very good transform basis. Any square integrable function y(x) in the interval [0, 1] can be expanded by a Haar series of infinite terms as: where the Haar coefficients c i are determined as: where i = 2 j + k, j≥0 and 0≤k < 2 j , xǫ[0, 1] such that the following integral square error ε is minimized: where m = 2 j and j = 0, 1, 2, 3, .... Usually the series expansion of (15) contains infinite terms. If y(x) is piecewise constant by itself or may be approximated as piecewise constant during each subinterval, then y(x) will be terminated at finite m terms. This means where the coefficients c m

Method for Solving Wave Equation
Consider the Wave equation (1) with initial conditions u(x, 0) = f 0 (x), ∂u ∂t (x, 0) = f 1 (x) and boundary conditions u(0, t) = g 0 (t), u(1, t) = g 1 (t). Assume thaẗ u ′′ (x, t) can be expanded in terms of Haar wavelets as follows: Integrating the above equation twice with respect to t from t s to t, and twice with respect to x, from 0 to x, we get, Differentiating the above equation twice with respect to t, we geṫ From the initial and boundary conditions, we have the following equations as: At x = 1 in the formulae (24), (25) and (26) and by using conditions, we have If the equations (27), (28) and (29) are substituted into equations (22) − (26), and the results are discriticized by assuming x → x l and t → t s+1 , we obtain +ġ 0 (t s+1 ), (33) (34) But, we know that hence, in the given schemë which leads us, that the time layer t s to t s+1 is used. From here, wavelet coefficients are calculated and solution of wave equation is obtained.

Numerical Examples
Example 1. Consider the wave equation of the form with initial conditions and boundary conditions u| x=0 = 0, u| x=1 = 0, t > 0.

Conclusion
It is concluded that Haar wavelet method is more accurate, simple, fast and computationally attractive than other known methods to solve wave equation.  For getting the necessary accuracy the number of calculation points may be increased. Further, it is also clear from the above examples that numerical results obtained by the present method are better than that obtained in Shi [5].