Values at node points (a) linear spline, (b) quadratic B-spline, and (c) cubic B-spline.

## Abstract

In this Chapter, continuous Haar wavelet functions base and spline base have been discussed. Haar wavelet approximations are used for solving of differential equations (DEs). The numerical solutions of ordinary differential equations (ODEs) and fractional differential equations (FrDEs) using Haar wavelet base and spline base have been discussed. Also, Haar wavelet base and collocation techniques are used to approximate the solution of Lane-Emden equation of fractional-order showing that the applicability and efficacy of Haar wavelet method. The numerical results have clearly shown the advantage and the efficiency of the techniques in terms of accuracy and computational time. Wavelet transform studied as a mathematical approach and the applications of wavelet transform in signal processing field have been discussed. The frequency content extracted by wavelet transform (WT) has been effectively used in revealing important features of 1D and 2D signals. This property proved very useful in speech and image recognition. Wavelet transform has been used for signal and image compression.

### Keywords

- Haar wavelet
- continuous wavelet function
- wavelet transform
- B-cubic spline base

## 1. Introduction

Wavelets are special mathematical functions which have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. The fields of applied mathematics such as quantum physics, seismic geology and electrical engineering have used and developed independently wavelets during last twenty years ago which leads to new wavelet applications such as image compression, radar, and earthquake prediction. Haar wavelet was initiated and independently developed by some authors. Wavelets can be summarized as a family of functions constructed from transformation and dilation of a single function called mother wavelet. From various types of continuous and discrete wavelets, Haar wavelet is the discrete type of wavelet which was first proposed and the first orthonormal wavelet basis is the Haar basis. Differential equations (DEs) are most important tools in mathematical models for physical phenomena. Many basis used to approximate the solutions of DEs. Haar wavelet is simple basis used to approximate the solution of DEs. [1] established a simple numerical method based on Haar wavelet operational matrix of integration for solving two dimensional elliptic partial differential equations (PDEs) of the form

### 1.1 Objectives of chapter

This chapter aims at achieving the following objectives:

To introduce continuous wavelet functions.

To use Haar wavelet approximations in solving of differential equations (DEs).

To imply Haar wavelet functions to approximate the solutions of DEs of fractional-order.

To compare Haar wavelet base with cubic spline base.

To study wavelet transform as a mathematical approach.

To discuss the discrete wavelet transform (DWT).

To study the applications of wavelet transform in signal processing field.

### 1.2 Scope of study

This chapter entailed the studying of continuous wavelet functions and Haar wavelet approximations. Wavelet transform introduced as a mathematical approach with some of applications of wavelet transform which is widely used in signal processing field. The approximation of DEs using Haar wavelet base was implemented with comparing to B-cubic spline base.

## 2. Preliminary

In this section, we introduce the definitions of two types of continues Haar wavelet functions and linear, quadratic and cubic spline functions base.

### 2.1 Continues Haar wavelet functions

Haar functions have been introduced by Hungarian mathematician. The orthogonal set of Haar functions is defined as a square waves with magnitude of

for

To approximate

Then,

for

All Haar wavelets are orthogonal to each other:

### 2.2 Spline functions

The spline is used to refer to a wide class of smooth functions that are used in applications requiring data interpolation [9, 10]. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizers of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints. Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful definitions of the roughness measure, the spline functions are found to be finite dimensional in nature, which is the primary reason for their utility in computations and representation. For the rest of this section, the focus is entirely on one-dimensional, polynomial splines and the use of the term spline in this restricted sense. The base

#### 2.2.1 Linear spline

The simplest spline is a piecewise polynomial function, with each polynomial having a single variable. The spline S takes values from an interval

for each

0 | |||

1 | |||

0 | |||

(a) | |||

2 | |||

1 | 0 | ||

− | |||

(b) | |||

0 | 0 | 0 | |

- | |||

1 | 0 | - | |

- | - | ||

0 | 0 | 0 | |

(c) |

#### 2.2.2 Quadratic B-spline

Quadratic B-spline is quadratic B-Spline polynomial

See Figure 1(b) and Table 1(b).

#### 2.2.3 Cubic B-spline

Many researchers used B-cubic spline base which defined as follows:

Consequently,

To construct cubic spline base in which satisfy the boundary conditions

See Figure 1(c), (d) and Table 1(c).

## 3. Approximation of differential equations (DEs)

Mathematics has several tools to describe the problems in real life, engineering and science. ODEs and PDEs are significant tools in applied mathematics. They played significant rule in describing the mathematical models in applications of engineering, science and economics. High-order DE arises in some fields of engineering and science such as nonlinear optics and quantum mechanics. The approximated solutions of DEs should be studied when the ODEs and PDEs have no analytical solutions or it is very difficult to find the analytical solutions. The numerical or approximated solutions of DEs are very important in scientific computation, as they are widely used to model real life problems. In this section, we have studied the approximation solutions of DEs using spline and Haar wavelet bases (Table 2).

0 | 0.25 | 0.5 | 0.75 | 1 | |
---|---|---|---|---|---|

5 | 0 | 1.3345e-3 | 0.0015 | 5.0673e-3 | 3.6339e-3 |

0 | 1.311e-3 | 0.0005 | 5.0683e-3 | 3.6229e-3 | |

10 | 0 | 1.3232e-5 | 2.6342e-5 | 1.5634e-6 | 4.1443e-5 |

0 | 1.3211e-5 | 2.1212e-5 | 1.2341e-6 | 4.0101e-5 | |

50 | 0 | 2.3416e-7 | 1.6611e-7 | 5.1126e-7 | 2.1233e-7 |

0 | 2.1414e-7 | 1.2211e-7 | 5.2233e-7 | 2.1266e-7 | |

100 | 0 | 4.9383e-8 | 3.4453e-8 | 5.0347e-8 | 6.4332e-7 |

0 | 4.9121e-8 | 3.4564e-8 | 5.0111e-8 | 6.4222e-7 |

### 3.1 Approximation of ordinary differential equations (ODEs)

In this section, we have studied approximation solutions of ODEs using spline and Haar wavelet bases.

#### 3.1.1 Approximation of DEs using spline functions

In this section, we have introduced the linear, quadratic and cubic B-spline base and their applications in solving ODEs. The operational matrices of the fractional-order integration of the B-spline base has been studied.

#### 3.1.2 Rayleigh-Ritz Metod

Rayleigh-Ritz metod is variational technique for solving boundary value problems (BVPs) which is the first reformulated as a problem of choosing, from set of all sufficiently differentiable functions satisfying the boundary conditions, the function to minimize a certain integral. To describe the Rayleigh-Ritz method, we consider the approximation of the solution to a linear of two-boundary value problem from beam-stress analysis. This BVP is described by the following DE:

with boundary conditions

The DE describes the deflection

An approximation

to the solution

for i = 1,2,3,…,n.

Lastly, we have obtained the linear system of equations

and

for

** Example 3.1**[8]

Consider

subject to the initial condition is

with the exact solution

Let n = 5, then,

and

for

** Example 3.2**[8].

Consider

subject to the initial condition is

with the exact solution

Let n = 10, then,

and

for

#### 3.1.3 Analysis of collection method

Let the differential operator

Suppose

Put the approximation of

### 3.2 The quadratic B-Spline Base

Consider the quadratic B-spline Base

Suppose

subject to the initial condition is

** Example 3.3**[8].

subject to the initial condition is

The coefficients are

Consider the quadratic B-spline base, then, the matrix of coefficients has the following formula:

and

for

### 3.3 Approximation of DEs using Haar wavelet Functions Base

We introduce the Haar wavelet technique for solving general linear first-order ODEs [11].

#### 3.3.1 First-order linear ODEs

Consider the following general linear first-order ODE:

Substituting

We assume that

where

Substituting Eqs. (19) and (20) in Eq. (17), we get the following system of equation:

Put

and

for

### 3.4 Approximation of fractional differential equations (FrDEs)

In this section, we have studied approximation of DEs using spline and Haar wavelet bases.

#### 3.4.1 Operational matrix of the fractional order integration of the B-Spline Base

In this section, we have evaluated the operational matrices of the fractional-order integration of the linear, quadratic and cubic B-spline Base.

#### 3.4.2 Linear spline

This subsection examines the cubic linear spline operational matrix

where

#### 3.4.3 Quadratic B-spline

This subsection introduced the quadratic B-spline operational matrix

where

#### 3.4.4 Cubic B-spline

This subsection introduced the cubic B-spline operational matrix FS

where

where * x*=

_{i}

*;*x−i

*= 1,2,3.*i

### 3.5 Numerical Solutions of fractional differential equations

#### 3.5.1 Numerical solutions of fractional differential equations using Haar base

We will introduce the Haar wavelet technique for solving FrDEs.

** Example 3.4**[8].

Consider the general fractional-order linear DE

subject to initial conditions

Substituting

We assume that

If

Substituting Eqs. (28) and (29) in Eq. (26), we get

If

and

Substituting Eqs. (28) and (37) in Eq. (26), we get

Put

and

for

#### 3.5.2 Numerical solutions of fractional differential equations using B-spline base

We will introduce the B-spline technique for solving FrDE (26).

Consider the quadratic B-spline base

Suppose

We assume that

If

Substituting Eqs. (34) and (35) in Eq. (26), we get

If

Substituting Eqs. (28) and (29) in Eq. (26), we get

and

Put

and

for

#### 3.5.3 Numerical solution of fractional Lane differential equation

We generalize the definition of Lane-Emden equations up to fractional order as following:

with the initial condition

#### 3.5.3.1 Analysis of numerical method of fractional Lane differential equation

[15] studied the solution of DEs based on Haar operational matrix, [16] studied the solution of DEs using Haar wavelet collocation method, [17] studied the numerical solution of DEs by using Haar wavelets, [18] used Haar wavelet approach to ODEs, [19] solved the fractional Riccati DEs using Haar wavelet while [14] studied the fractional DEs of Lane-Emden type numerically by method of collocation. [20] introduced an operational Haar wavelet method for solving fractional Volterra integral equations, [21] solved fractional integral equations by the Haar wavelet method, [22] used Haar wavelet-quasi linearization technique for fractional nonlinear DEs, [21] solved the fractional integral equations by the Haar wavelet method, [4] used Haar wavelet method for solving fractional PDEs numerically. In Eq. (41), consider

However,

Hence,

If we consider

### 3.6 Comparison study using numerical collection method

Collocation method for solving DEs is one of the most powerful approximated methods. This method has its basis upon approximate the solution of FrDEs by a series of complete sequence of functions, a sequence of linearly independent functions which has no non-zero function perpendicular to this sequence of functions. In general,

where

Let

** Example 3.5**[8]

Consider

with the boundary conditions

The exact solution is

** Example 3.6**[8]

Consider

with the boundary conditions

The exact solution is

** Example 3.7**[8]

Consider

with the boundary conditions

The exact solution is

## 4. Wavelet transform (WT)

Fourier transform (FT) of a time signal

However, FT cannot reveal the time information associated with a specific frequency. This drawback enhanced research in the time-frequency domain [23]. One of the most important time-frequency distributions (TFD’s) is the wavelet transform (WT), which is a time-frequency representation of signals. While not all TFD’s are invertible, a big advantage of WT over many other TFD’s is invertibility. WT proved to be successful in revealing spectral features of signals. Instead of sustainable waves like sinusoidal waves as in the case of Fourier transform, WT is based on decomposing signals using decaying waves (small waves, or wavelets), all are shifted and dilated versions of a specific wavelet called mother wavelet. The continuous wavelet transform (CWT) of a signal

where

where

where

and

This fact gives another equivalent expression for

where

Generally speaking, Fourier transform

The WT is invertible, giving it a great advantage in applications:

where

### 4.1 Some applications of the wavelet transform

The frequency content extracted by wavelet transform (WT) has been effectively used in revealing important features of 1D and 2D signals. This property proved very useful in speech and image recognition [25]. Also, the orthogonality of WT paved the way for using WT in orthogonal frequency division multiplexing (OFDM), a pivotal technique for 4th and 5th generations of digital communication [26]. In addition to that, WT proved to put high focus on the low-frequency part of the signal, in which most of the information resides, hence, WT has been used for signal and image compression [27]. The compression process can be performed using hard-thresholding of the WT as follows:

where

### 4.2 Noise removal using WT

An important application of the Wavelet Transform is noise removal from signals and images. As most of the information content of real-life signals is in the low-frequency regions, removal of high frequency regions in the WT of signals can help in removing the majority of noise. This can be done via thresholding WT coefficients or by removing the details coefficients of WT and considering only the approximation coefficients of WT. This property of separating low-frequency content from high-frequency content in the WT is mainly due to the filtering involved in the structure of WT as explained above. Noise removal using WT is more efficient for 1D signals corrupted by 1D noise process, where the 2D structure of WT in joint time-frequency domain can spread the 1D noise effect into a 2D plane, hence the noise power is greatly reduced. For noise removal, a soft-threshold can be used to cut out high-frequency coefficients as follows:

where

Figure 3 shows the use of WT to denoise an image, while Figure 4 shows the denoising of 1D signal using WT, where WT is performed on MATLAB via the wavelet Daubechies 3,

## 5. Discussion and conclusion

The numerical solutions of differential equations using Haar wavelet technique have been studied. Haar wavelet technique is used to approximate the solutions of DEs. The results which obtained form numerical solutions of ordinary differential equations as well as fractional differential equations by Haar collection method are compared with spline base. The numerical results have clearly shown the advantage and the efficiency of the techniques in terms of accuracy and computational time. Special initial value problem of Lane-Emden equation has been solved to show the applicability and efficacy of the Haar wavelet method. Wavelet transform as a mathematical approach has been studied and the applications of wavelet transform in signal processing field have been introduced. The wavelet transform has been effectively used to reveal on the features signals and the compression of signal and image.