16.1 Introduction

Wavelet method has been proven to be an efficient tool in analyzing dynamic systems and differential equations arising in other science and engineering problems. A wave‐like oscillation having amplitude beginning at zero that monotonically increases and then decreases back to zero is referred to as wavelet. Wavelets also serve as a tool in analysis of transient, nonstationary, and time‐variate phenomena. A wavelet series is often represented in terms of complete square‐integrable function or set of orthonormal basis functions. The broad classification of wavelet classes is considered as discrete, continuous, and multiresolution‐based wavelets. Continuous wavelets are projected over continuous function space whereas discrete wavelets are often considered over discrete subset of upper half plane. While analyzing discrete wavelets, only a finite number of wavelet coefficients are taken into consideration which sometimes makes them numerically complex. In such cases, multiresolution‐based wavelets are preferred.

In mid‐1980s, orthonormal wavelets were initially studied by Meyer [1] over real line ℜ. Gradually, compactly supported orthogonal wavelets viz. Daubechies wavelets [2] were constructed by Ingrid Daubechies. Daubechies wavelets are sufficiently smooth, possess orthogonality, and have compact support. A detailed study of wavelets and their applications may be found in Refs. [3–6]. Chen et al. [7] studied the computation of differential‐integral equation using the wavelet–Galerkin method involving Daubechies wavelet and its derivatives or integrals. But, Daubechies wavelets are rather difficult in handling due to lack of explicit expression whereas Haar wavelets are computationally easier. As such, the next section discusses preliminaries related to Haar wavelet only.

16.2 Haar Wavelet

In 1990, the concept of Haar wavelet [8] was introduced by Hungarian mathematician, Alfred Haar. Later, the pioneer work for system analysis was performed by Chen and Hsiao [9], who derived an operational matrix for integral of Haar function vector. Solutions of differential equations using Haar wavelets have been developed by various authors as given in Refs. [10–12] and the references mentioned therein. Further, a review of the Haar wavelet method for solving differential equations along with its advantage, applications, and comparative studies with other wavelets have been discussed in detail by Hariharan and Kannan [13].

Generally, the family of Haar wavelets [11, 13 ,14] h_i(x) for x ∈ [0, 1] is considered as

(16.1)

where the index i is calculated in terms of wavelet level m and translational parameter k using i = m + k + 1. Here, the resolution level of the wavelet is computed using integer m = 2^j for j = 0, 1, …, J such that J denotes the maximal resolution level. Further, the translation parameter is computed using k = 0, 1, …, m − 1. In this regard, the minimal and maximal values of index i are obtained as i = 2 and i = 2M = 2^J + 1, respectively. The value of h_i(x) at i = 1 may be assumed for x ∈ [0, 1] as the scaling function given by

(16.2)

Let us now calculate the Haar wavelet functions using Eq. (16.1) over 0 ≤ x < 1.

1. Initially, the values of M and j are obtained for maximal resolution level J = 0 as follows:
   

   and

   j = 0, 1, …, J ⇒ j = 0.

   Substitute j = 0 in m = 2^j to obtain the value of m = 1. Then, the value of k is obtained using k = 0, 1, …, m − 1 as k = 0. Index i = 2 is computed by substituting m = 1 and k = 0 in i = m + k + 1. Subsequently, the obtained values of i, k, and m are substituted in Eq. ( 16.1 ) to obtain Haar wavelet functions.

   Further, using Eqs. ( 16.1 ) and (16.2) over 0 ≤ x < 1, the first two Haar wavelets are computed as

   
2. Accordingly, the first four Haar wavelet functions may be generated for J = 1 as follows:

   The values of M, j, m, k_, and i for maximal resolution level J = 1 are obtained using,

   

   and

   

   respectively. By plugging i = 2, 3, 4, k = 0, 1, and m = 1, 2, the subsequent Haar wavelets are accordingly obtained. Then, using Eqs. ( 16.1 ) and ( 16.2 ) over 0 ≤ x < 1, the first four Haar wavelet functions are obtained as

   

   It is worth mentioning that the initial two Haar wavelets generated for J = 1 are equivalent to the Haar wavelets obtained for J = 0. Similarly, higher Haar wavelet functions may be obtained using Eqs. ( 16.1 ) and ( 16.2 ) for J > 1. Further, the integrals of wavelets given by Eq. ( 16.1 ) may be calculated using [ 11 , 14 ],

   (16.3)

   and

   (16.4)

   As such, the first four Haar wavelets and their integrals using Eqs. ( 16.1 )–(16.4) are obtained and incorporated into Table 16.1.

Haar functions, hi(x)	Integral, pi(x)	Integral, qi(x)

Accordingly, the respective plots of Haar wavelets and their integrals have been depicted in Table 16.2.

i	Haar functions, hi(x)	Integral, pi(x)	Integral, qi(x)
1
2
3
4

Wavelets serve as a very powerful tool for solving differential equations. In this context, only a basic idea of Haar wavelet is incorporated into this section. Further, just to have an overview of how to handle ordinary differential equation using Haar wavelets, a preliminary procedure is demonstrated. In this regard, Haar wavelet‐based wavelet–collocation method [14] for solving ordinary differential equation has been discussed in the next section.

16.3 Wavelet–Collocation Method

Let us consider a second‐order ordinary differential equation,

(16.5)

subject to initial conditions u(0) and u′(0) over the domain Ω. In order to solve the differential equation 16.5), highest‐order derivative is expressed in terms of linear combination of Haar wavelet functions [14] as

(16.6)

where c_i for i = 1, 2, … are the unknown wavelet coefficients yet to be determined. Accordingly, the lower derivative u′(x) and the solution u(x) are expressed in terms of integrals of Haar wavelet functions as

(16.7)

and

(16.8)

Now, substitute Eqs. (16.6)–(16.8) in the governing differential equation 16.5) to obtain

(16.9)

Then, the collocation points for solving Eq. ( 16.5 ) using the Haar wavelet–collocation method may be obtained using [ 10 , 11 , 14 ]

(16.10)

where r = 1, 2, …, 2M. Generally, the technique of Haar wavelet–collocation method transforms the differential equations to system of algebraic equations. In this regard, the system of equations in terms of unknown coefficients c_i is obtained for the collocation points x_r given by Eq. (16.10). Finally, the approximate solution is computed using Eq. ( 16.8 ) as

(16.11)

The effectiveness of the above‐mentioned procedure is illustrated using an example problem.

x	Haar wavelet–collocation solution,	Exact solution, u(x)
	0.9910	0.9919
	0.9162	0.9200
	0.7474	0.7568
	0.4572	0.4761

Exercise

1. 1 Solve the linear differential equation subject to initial condition u(0) = 0 and .
2. 2 Evaluate the Duffing equation with respect to initial conditions u(0) = 0 and for t ∈ [0, 1].

References

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2. 2 Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics 41 (7): 909–996.
3. 3 Daubechies, I. (1992). Ten Lectures on Wavelets, vol. 61. Philadelphia, PA: SIAM.
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8. 8 Haar, A. (1910). Zur theorie der orthogonalen funktionensysteme. Mathematische Annalen 69 (3): 331–371.
9. 9 Chen, C.F. and Hsiao, C.H. (1997). Haar wavelet method for solving lumped and distributed‐parameter systems. IEE Proceedings‐Control Theory and Applications 144 (1): 87–94.
10. 10 Lepik, Ü. (2005). Numerical solution of differential equations using Haar wavelets. Mathematics and Computers in Simulation 68 (2): 127–143.
11. 11 Lepik, Ü. (2007). Numerical solution of evolution equations by the Haar wavelet method. Applied Mathematics and Computation 185 (1): 695–704.
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13. 13 Hariharan, G. and Kannan, K. (2010). Haar wavelet method for solving some nonlinear parabolic equations. Journal of Mathematical Chemistry 48 (4): 1044–1061.
14. 14 Nayak, S. and Chakraverty, S. (2016). Interval wavelet method in solving diffusion equations, generalized and hybrid set structures and applications for soft computing. In: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing (ed. S.J. John), 457–472. Hershey, PA: IGI Publication.
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