From Wavelet Analysis to Fractional Calculus: A Review

Abstract

In this note, we review some important results on wavelets, together with their main applications. Similarly, we present the main results on fractional calculus and their current applications in pure and applied science. We conclude the paper showing the close interconnection between wavelet analysis and fractional calculus.

1. Introduction

After the dawn of Fourier analysis, many scholars have carried out in-depth research on finding mathematical tools to conveniently deal with time and frequency properties of continuous waves in a simple and quick manner [1]. Starting from the Haar system analysis in the early 20th century to the fundamental results of Daubechies by the concept of multi-resolution analysis, wavelet analysis is nowadays a theory with applications in other mathematical fields, physics, engineering, etc. [2,3,4,5,6]. Taking into account these many applications, it is reasonable to wonder if wavelet analysis will continue to increase in the near future. In other words, we wonder if this theory has reached its maximum in terms of applications.

Likewise, fractional calculus is a powerful tool to describe processes both in pure and applied sciences. Historically, fractional calculus makes its appearance (approximately) four centuries ago with Leibniz. Fractional calculus is applied in different research areas such as pure and applied mathematics, science, engineering, industry, and technology. More precisely, and only with regard to mathematics, current research in fractional calculus concerns, among others, partial differential equations, integer and fractional order equations, numerical analysis, discrete mathematics, control theory, and probability. Quite recently, considerable attention has been paid to the fractional calculus of zeta functions. In general, fractional calculus of complex functions entails several problems [7]; therefore, it has not grown as fast as real fractional calculus. However, the fractional derivative of zeta functions is fairly easy to compute [8,9,10,11,12]. Furthermore, the fractional calculus of zeta functions has shown new relevant applications both in signal processing and dynamical systems [13,14].

In recent years, there has been a growing interest in the joint analysis of fractional calculus and wavelet analysis for real-world applications. More precisely, fractional calculus found several applications, especially in theoretical physics and image processing (e.g., Maxwell equations in fractal media [15]). In these applications, it is often possible to obtain a wavelet expansion of the basis functions. This is a first potential example of joint analysis. On the other hand, the introduction of local fractional calculus in some wavelet bases enables us to extend the local fractional derivative to non-smooth continuous functions (see [16]). Accordingly, the joint analysis of fractional calculus and wavelet analysis may shed some new light on several unsolved problems in pure and applied science.

The remainder of the paper is organized as follows. The next section provides a historical overview on wavelet analysis. Section 3 is devoted to the main applications of wavelet analysis. In Section 4, we give a brief historical introduction to fractional calculus. Section 5 introduces the reader to the current main applications of fractional calculus. Finally, Section 6 summarizes the results of this work and draws conclusions.

2. Historical Note on Wavelet Analysis

In this section, we give an overview of the main achievements in the history of wavelets based on insights from the researchers responsible for those contributions.

2.1. From Fourier Analysis to Haar Function Analysis

The theory stated in 1807 by the French mathematician J.-B. Joseph Fourier allowed researchers to define wavelets a century later. Fourier concluded that a complex function, usually coming from any periodic signal, can be expressed as a weighted sum of sine and cosine functions. Initially opposed by other important mathematicians and researchers of his time, such as his mentor Joseph-Louis Lagrange [17], Fourier ended up writing his work 15 years later in his book The Analytical Theory of Heat. In this book, he introduced an integral operator now called Fourier transform.

Nevertheless, the representation used by Fourier had a major disadvantage generated by the use of sinusoids as building blocks: this type of function has perfect support for the frequency domain and, therefore, for a spectral content analysis but not for the time domain. Furthermore, this domain is extended to the infinity when applied into Fourier transform, so it cannot be used for approximations of non-stationary signals whose spectral content changes with time. A solution to overcome this limitation is the signal segmentation by using time windows so that performing the analysis of each generated segment becomes feasible. This insight led to the short-time Fourier transform (STFT), discussed hereafter.

Introduced by Dennis Gabor in his article Theory of communication, published in 1946, STFT was proposed in such a way that a time-localized Fourier transformation is performed on a given interval represented by a fixed window, with this window being moved over time or space. Based on this movement, a Fourier transform can be performed on the entire signal, where the choice of the window function is critical to the time and frequency resolutions obtained as results. Despite this approach, STFT has one major disadvantage: the frequency content of the signal under analysis is generally and obviously unknown a priori, making the choice of a suitable window length difficult. In this scenario, the researchers were motivated to test other techniques for processing non-stationary signals, such as the constant Q transform (CQT).

Proposed independently by the American physicist Judith C. Brown in her 1991 paper Calculation of a constant Q spectral transform, CQT is a type of transform that uses geometrically-spaced center frequencies, making a correlation with exact musical notes. This technique transforms a time-domain signal into time-frequency domain in such a way that the frequency resolution is better for low frequencies and time resolution is better for high frequencies [18,19]. With these characteristics, CQT is highly suitable for applications involving musical signal analysis and for scenarios with a considerable merit (Q) factor that can force a filtering process to be performed on the input signals hundreds of times.

Notably, CQT has some particular characteristics and cannot be inverted due to the fact that the number of samples between calculations can be greater than the length of the analysis window, which is the same analysis window shown in the Fourier transform, in one high frequency situation. Compared, for instance, with the Fourier transform discretization, which is a useful tool for audio signal processing, this technique has a worse computational performance. Furthermore, the discovered data is more complex to handle compared to, for instance, the spectrogram produced by STFT at successive time intervals [20]. Some applications of this technique will be shown in the next section.

2.2. New Perspectives: The Beginning of the Wavelet Dawn

The first references to the term “wavelet” were made in the doctoral thesis of Alfred Haar, entitled On the theory of systems of orthogonal functions, in 1909. He discovered a set of rectangular orthonormal basis functions that led to the formulation of an entire family of small waves called Haar wavelets. This family is based on the Haar set and is considered one of the simplest wavelet families in the literature.

From 1930 to 1970, research on wavelet theory and Fourier transform helped to make time-frequency analysis interesting for scientists in many fields, not only signal processing. Starting from the introduction of the scaling function by Lévy, through the results of Littlewood and Paley on the energy location of a function, ending with the STFT of Gabor, these decades popularized and consolidated wavelets theory.

French physicist Paul Lévy discovered, during some research into random signals called Brownian motion, that the base function proposed by Haar was better than the Fourier base function for some applications related to Brownian motion. Furthermore, because different intervals can be scaled in the Haar set, a significant increase in precision occurs in contrast to the precision provided by the Fourier base function, which can only have an interval [$-\infty ,\infty$]. Therefore, if one needs to calculate, for instance, the multi fractal structure inherently present in the Brownian motion [21], an analysis of more than one interval is necessary, making Fourier inadequate. This is the first major milestone in the field of wavelets since the results of Haar.

After Lévy, another relevant result is that of Littlewood and Paley, which tried to calculate the energy location of a function. Some difficulties took place. In fact, the information needed to enter the Fourier coefficients to allow for this calculation is hidden in the Fourier series, being necessary to subject the series to specific manipulations. During the research, “dyadic blocks” were defined to lead with the Fourier function in an elegant way, decomposing the function under analysis, f, into a sum of functions $f_p$ with localized frequencies. An extension of this work, performed by the Polish mathematician Antoni Zygmund, allowed us to extend the one-dimensional periodic case of Littlewood and Paley to a Euclidean n-dimensional space. In this scenario, the “mother wavelet” $\psi \left(x\right)$ appeared, an achievement that has been used numerous times in the subsequent stages of the evolution of wavelet theory. Other individuals have also proposed and extended some important concepts for several wavelet families, such as Philip Franklin, Elias Stein, and Norman Ricker.

2.3. From Grossmann and Morlet to Modern Approaches

In the late 1970s, Jean Morlet, a geophysical engineer who worked in a French oil company, developed a technique for scaling and shifting the analysis window functions during the investigation of acoustic echoes, a type of signal that had high-frequency components. In his first attempts, an STFT was used to understand these echoes and try to find oil under the crust of the earth. However, he noted that keeping the window length fixed was inconvenient. One solution found was to keep the window function frequency constant while changing the window by stretching/compressing the function, allowing for the window functions to have compact support in both time and frequency. Due to the “small and oscillatory” nature of these windows, Morlet called their basis functions waves of constant form, implying that the wavelet transform had just been born.

Essentially, the work of Morlet was similar to what Haar did before, a common situation that has surfaced many times in the history of wavelets when one author rediscovered an entire mathematical approach proposed by other research a few decades earlier. Additionally, like Fourier, Morlet also faced much criticism from his fellow geophysicists. In 1980, looking for help in finding a rigorous basis for his work, Morlet met Grossmann, a theoretical physicist, who helped him formalize the transform and envisioned the inverse transformation without any loss of information. Yves Meyer noted that the work of Morlet was just a rediscovery. In fact, it was simply the paper of Calderón (1964) on harmonic analysis with a slightly different interpretation.

After this innovative work, a significant number of studies were introduced in the wavelet theory. The most important achievement was the proposition of multi-resolution analysis by Meyer and Stephane Mallat, a French electrical engineer and mathematician. Meyer first presented the article titled Orthonormal waves in 1989. The multi-resolution handles the design of the scaling function, allowing for other researchers and scientists to build their own base wavelets with the required rigor. As an example, Ingrid Daubechies, a Belgian physicist and mathematician, created, based on the concept of multi-resolution, her own family of wavelets in 1988. Her compactly supported orthonormal wavelet bases laid the foundations of modern wavelet theory.

3. Wavelet Analysis—An Application Overview

This section provides a summary of the main application areas of wavelet analysis.

3.1. Signal Processing and Image Compression

Alex Grossman, Yves Meyer, Ronald R. Coifman, and Stéphane Mallat are just a few examples of names which made wavelet analysis remarkably popular and important due to applications in several fields of the science. In particular, the first two scholars developed with Victor Wickerhauser very rapidly computerized searches by an enormous range of signal representations called wavelet packet [22,23]. This algorithm allowed, for instance, the FBI to compress a finger print data base of 200 terabytes into less than 20 terabytes. Hence, the wavelet packet saved millions of dollars in transmission and storage costs. In addition, Mallat produced a fast wavelet decomposition and reconstruction algorithm [24]. Over the course of time, it became the base for many wavelet applications in pure and applied science (e.g., see [25]).

Wavelet analysis also allowed to the development of several techniques in image fusion. In particular, each algorithm has its particular mathematical properties and leads to different image decompositions. Fusing two types of information (temporal and spectral), the discrete wavelet transform (DWT) plays a fundamental role in imagine fusion. In recent years, scientific community has continuously developed DWT-based techniques [26].

An interesting application of the wavelet analysis in image fusion is given by the discrete shapelet transform, recently created and presented to the scientific community [27]. This discrete transform estimates the degree of similarity between the signal under analysis and a pre-specified shape. Its work principle consists of a fractal-based criterion to redefine the original Daubechies’ DWTs. This technique allows us to obtain a time–frequency–shape joint analysis.

3.2. Electromagnetism

In recent years, there has been a growing interest in the application of wavelet analysis in electromagnetism [28].

First, Rao–Wilton–Glisson basic functions can be expanded in wavelet basis with an application in fractal-like antennas. The advantages from this approach are based on the close link between wavelet analysis and fractal geometry (e.g., flexibility, multi-band behavior). It is worth pointing out that the main numerical techniques for electromagnetic modeling can be rewritten obtaining the multi-resolution time domain method [29,30].

Secondly and more importantly, wavelet analysis has become popular and important due mainly to several applications in numerous and widespread fields of electromagnetism (remote sensing, scattering, etc.). In addition, wavelet analysis shows rather strikingly that contemporary mathematics is capable of providing ever more refined models for electromagnetic applications. In general, the combined choice of pre-fractal sets and wavelet bases relies on technical requirements and complexity of the electromagnetic problem.

3.3. PDEs and Integral Equations

Wavelet analysis is often used for numerical methods in integral equations and PDEs. In particular, Chebyshev wavelets are ad hoc wavelets to find the numerical solutions of integral equations and PDEs. For instance, Hyedari et al. introduced a numerical method based on Chebyshev wavelets for solution of PDEs with boundary conditions of the telegraph type [31]. Likewise, Chebyshev wavelets allow us to solve nonlinear systems of Volterra integral equations [32,33]. Moreover, wavelet analysis provides approximate solutions of stochastic Itô-Volterra integral equations by Chebyshev wavelets of second kind [34]. It is worth pointing out that wavelet analysis can be used for the numerical solutions of fractional differential equations (see, e.g., [35,36]). More generally, wavelet methods for solving integral equations and PDEs usually depend on the different operational matrices [37]. Finally, we note that Chebyshev wavelets gives sharp estimates of functions in Hölder spaces of order $\alpha$ [38].

3.4. Astronomy

In the late 21st century, hundreds of astronomical surveys were released using the term “wavelet” for filtering, object detection, image reconstruction, and so on [39]. Furthermore, in recent work, we can note that the wavelet transform is one of the favorite tools for automated analysis based on specific patterns within statistical distributions [40], aiding in detailed studies of flux variations in extra-galactic radio sources, detection of stellar events in deep space, and efficient ways to measure the same radio signals captured by different locations [41,42,43].

3.5. Acoustics

Wavelet analysis finds an application in acoustics due to the non-stationary nature of the acoustic emission. First, Newland dealt with harmonic wavelets showing their applications in acoustic [44,45]. Second, we note that the wavelet transform is able to extract characteristics sensitive to gradual flank wear [46]. Moreover, different studies on condition monitoring of industrial machines have been carried out with the specific aim to meet the demand. In this context, the wavelet transform has been applied to assist in the diagnosis of multi-fault bearings, especially in reducing noise of vibration data [47]. Furthermore, the wavelet transform has been used to monitor the general condition of the tool, improving other signal processing techniques such as STFT [48].

We note that CQT has also been applied in some interesting ways in acoustic signal analysis. In particular, this integral transform was used as input to an artificial neural network classifier in the classification of underwater targets due to the possibility of inferring information in sub-bands [49]. Additionally, CQT was inserted in the pre-processing phase in the classification of the audio scene, helping to obtain better results at the end of the extraction resource pipeline [50]. It is relevant to note that this context was extended to the classification of the acoustic scene [51].

We emphasize that STFT still represents a very active research area. For instance, this technique was used as a classification method in the analysis of vibration signals [52]. Furthermore, STFT was used as a pre-processing method for information extraction for reasons of simplicity [53,54].

3.6. Biomedicine

From the final decades of the 21st century, the importance of the wavelet transform in biomedical applications has increased. In this context, a relevant article showed that many concepts of wavelet theory can be shifted to the medical context, aiding in the analysis of, for instance, electrocardiography and magnetic resonance with equal or better performance compared to the current state of the art [55]. Moreover, a wavelet-based neuron activation function, neuron W, was applied to some medical datasets, proving that this method has significant advantages in dynamic data mining tasks due to simple computation and better accuracy and speed of processing information [56].

It is worth pointing out that CQT also has an important impact on medical applications. In fact, a constant Q non-stationary Gabor transform (CQ-NSGT), i.e., an invertible CQT, was employed to transform the 1D electrocardiogram signal on a 2D RGB image introduced in a convolutional neural network during the analysis of cardiovascular diseases [57]. More specifically, this technique was applied to congestive heart failure and arrhythmia. In this work, the authors also dealt with STFT and continuous wavelet transform (CWT). In this case, the non-stationary nature of the analyzed signals excluded the STFT as a possible choice. We note that CWT can be used, but the need for high computational time makes this technique unsuitable for low-to-medium hardware requirements. The authors showed that the CQ-NSGT met this restriction with satisfactory results. Furthermore, the CQT is proposed to process electroencephalogram (EEG) signals, improving previous methods based on the Fourier transform [58].

Finally, we underline that a rational discrete STFT was proposed for feature extraction for detection of epileptic seizures in EEG signals, showing better results compared to them of the classical STFT [59]. Likewise, STFT can be used as the pre-processing step in the EEG spectrogram to extract time frequency information, denoising and converting the EEG fragments into EEG spectrogram representations [60].

4. Historical Note on Fractional Calculus

This section gives a historical note on fractional calculus. Nowadays, fractional calculus is applied in various research fields, e.g., mathematical physics, engineering, theoretical and applied physics, and so on. Moreover, fractional calculus is applied where phenomena can be described by nonlinear models. Dynamical systems and linear time-invariant systems of fractional orders attract researchers from many areas of science and technology, going from mathematics to physics and engineering.

Fractional calculus is a new mathematical theory which dates back to the foundation of differential calculus. In fact, Leibniz (1695) was probably the first scholar to write some notes on derivatives of fractional order. Nevertheless, Liouville, Riemann, and Fourier collected, during the nineteenth century, all the basic knowledge about fractional calculus (see [61] and the references given there). Since the end of the nineteenth century and particularly in the last century, there has been a relevant interest in fractional calculus and its applications. More precisely, fractional calculus has been used in mathematics, physics, engineering, computer science and so on. Recently, it was applied to image processing and finance.

Clearly, it is relevant to understand the reasons why fractional calculus inspires many scholars nowadays with different scientific backgrounds. This depends on three main reasons. First and foremost, the operator of fractional derivatives does not satisfy all properties of the integer-order derivative. This implies the choice of additional axioms, and so there are different definitions for the same operator. In particular, this entails non-uniqueness in defining the fractional derivative. Secondly, since the origins of fractional calculus there has been a heated discussion on local and non-local derivatives. It is relevant to note that non-local derivatives need some boundary conditions, i.e., the problem deserves a global approach with different integro-differential operators. Thirdly and lastly, it is worth pointing out that the main operators of fractional calculus are based on special functions. These functions strongly depend on numerical methods for their computing; thus, modern computational improvements gave new life and interest to the research in fractional calculus.

Nevertheless, we are nowadays in a situation where fractional calculus has many applications in physics, engineering, etc., but there are still unsolved problems in the theory of fractional calculus. To draw a parallel by probability theory, fractional calculus can be viewed nowadays as the probability theory before 1933, the year in which Kolmogorov laid the foundations of probability theory by his thee axioms.

In the current scientific research, any problem which has a model involving an integer-order derivative can be generalized by the use of fractional calculus. Thus, we have a fractional model of this problem, which can depend on either a local or non-local approach, along with the choice of the fractional derivative among many possible alternatives. Moreover, recent developments in numerical methods have brought confusion in fractional modeling. Indeed, different bases of orthogonal functions imply different fractional models [62]. Quite recently, some scholars tried to extend local fractional calculus to irregular sets (e.g., fractal sets). This approach mixes fractional calculus, wavelet analysis and fractal geometry. Many scholars think that this approach may open new frontiers in research (e.g., signal processing, data science), allowing us to deal with self-similar unsmooth shapes. In fact, non-linear analysis of big data offers nowadays problems related to non-linear science. From a more general point of view, some of these problems can be solved using this technique. For more details, we refer the reader to [16,63,64].

5. Fractional Calculus—An Application Overview

This section is devoted to presenting an overview on the current applications of fractional calculus.

5.1. Control Theory

The classic type of control, i.e., the proportional–integral–derivative (PID) control, is mandatory in industry. Nevertheless, all new results in fractional calculus, fractional methods in control theory, may gain ground in applied environments in the coming years. In particular, fractional-calculus-based control can outperform the adoption of PID technologies in many areas, especially in industry [65]. In fact, this approach enables a reduction in control effort and resource usage. Moreover, fractional calculus can be applied in other fields of control theory, e.g., robotics [66], electronic throttle control in combustion engines [67], and control of a pumped storage unit [68].

5.2. Electromagnetism

Fractional electromagnetism is a generalization of the electromagnetism replacing the differential operations in Maxwell equations by their fractional counterparts. This approach has different applications. In particular, fractional electromagnetism is used whenever the media can be modeled as prefractal sets [15]. In particular, the laws of electromagnetism can be rewritten by fractional calculus giving solution to different electromagnetic problems [69]. Indeed, fractional calculus has been applied in the skin effect, which further motivated a new perspective towards the replacement of classical models by fractional-order mathematical descriptions [70]. It is worth mentioning that the application of a fractional-calculus-based approach allows a rich theoretical extension of integer order differentation, allowing, for instance, a new range of innovative potential in materials science and engineering [71].

5.3. Quantum Mechanics

Fractional quantum mechanics was born and quickly developed in the last 20 years as a field of quantum physics. In 2000, Nikolai Laskin introduced a new extension of a fractality concept in quantum physics, expanding the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths [72]. We note that the two main concepts of Fractional quantum mechanics are given by the fractional Schrödinger equation and the fractional path integral. In particular, the fractional Schrödinger equation indicates the existence of fractional quantum mechanics. During the years, the researchers developed different topics in fractional quantum mechanics (e.g., fractional statistical mechanics, $\alpha$-stable Lévy random process). For more details, we refer the reader to [73].

It is worth pointing out that at $\alpha =2$, Lévy motion becomes Brownian motion. Accordingly, fractional quantum mechanics generalizes the classical quantum mechanics as a particular case at $\alpha =2$. In fact, for $\alpha =2$, the quantum-mechanical path integral over the Lévy paths and the fractional Schrödinger equation simply become the Feynman path integral and classical Schrödinger equation, respectively [74,75].

5.4. Signal Theory

The non-integer order systems can describe dynamical behavior of materials and processes on a vast time and frequency. In the last year, fractional calculus has been applied to non-integer order systems. In particular, it is relevant to note that fractional systems exhibit both short and long term memory. These characteristics have great influence on the development and application of fractional systems in real-world applications.

It is worth noticing that, contrary to the classical case, the design of fractional systems has at least one extra degree of freedom, i.e., the fractional order. Nevertheless, this further difficulty provides to the designer the possibilities to obtain a more robust system. Recent trends in fractional systems concern circuits with fractional elements, fractional transmission lines, band-limited approximations, weighted summation of exponentials, etc. [76,77,78].

5.5. Cryptography

Up to about 40 years ago, a lot of engineers and scholars were doubtful with regards to the practical applications of chaotic behavior. Accordingly, the majority of the scientific community tried to not deal with chaos. Nevertheless, nowadays scholars look at the strange characteristics of chaotic behavior in a different way [79]. Moreover, chaos synchronization and chaos control have a relevant role in the security of modern communication systems, where cryptography plays the main role.

The application of chaotic behavior in cryptography determines a wide use of fractional models in information security. In particular, the recent trends of fractional calculus in cryptography concerns fractional-order hyperchaotic systems, evolution of chaotic systems, chaos-based cryptography, and fractional-chaos-based-cryptosystems [80,81,82,83].

5.6. Image Processing

Over the last decade, fractional calculus found a widespread application in image processing. In fact, fractional models own an additional degree of freedom, i.e., the fractional order of differentiation, which improves the system in robustness. Indeed, recent applications involve the use of fractional model image denoising, image edge detection, image segmentation, image compression, etc. [84,85].

It is worthy pointing out that fractional-order derivative-based techniques are used especially in computer vision. In fact, fractional calculus assures noise resilience. In addition to this, models based on fractional derivatives can preserve both high and low-frequency components of an image. In particular, the similarity of adjacent pixels can be influenced by different parameters (e.g., error, noise). Thus, the fractional model can help to solve this problem properly [86]. In the last years, fractional calculus of fuzzy valued functions provided relevant results in the enhancement of gray and color images [87].

5.7. Biology

In recent years, a trend has been observed in the research community in taking tools from fractional calculus for use in biological phenomena [88].

We recall that biological tissues are complex systems characterized by dynamic processes that occur at different length and time scales. Simple Debye models cannot adequately describe the behavior of these materials. Accordingly, the fractional version of simple Debye models are necessary to model biological phenomena at different scales. Moreover, anomalous diffusion is a characteristic of non-homogeneous systems that occurs in porous materials, nuclear magnetic resonance [89,90], etc. Linear and non-linear models of anomalous diffusion fail to recapitulate the phenomenon accurately. Some fractional order models have been proposed to solve this problem. In particular, in [91], a finite-difference-approximation to the fractional diffusion equation was proposed by the Riemann-Liouville derivative. Additionally, diffusion in biological tissue has also been characterized as anomalous, i.e., by Riemann-Liouville derivative.

Bio-systems can also be modeled by fractional calculus. This depends on the fractal nature of several bio-systems. More precisely, fractional calculus finds applications in modeling both the respiratory system and vegetables. Finally, fractional calculus also finds an application in drug distribution [92].

5.8. Epidemiology and COVID-19

Reliable data cannot always be guaranteed during epidemics and for natural endemic diseases. Moreover, fractional order systems can describe long-range temporal/spatial dependency analysis. Thus, fractional calculus can be used for a model of risk propagation in epidemiology. Nowadays, fractional models help epidemiologists in formulating the right choices for public health [93].

Some examples of applications of fractional calculus in epidemiology are those which adopt such information for disease control, including the control of the hepatitis B virus in rural Pakistan [94], modeling the dynamics of the dengue epidemic in the Cape Verde Islands [95], and analyzing the spread of COVID-19 considering the memory effect on the data [96].

6. Conclusions

This paper deals with fractional calculus, wavelet analysis, and their applications. In particular, we gave an extensive collection of the recent applications that these two mathematical theories found in other mathematical fields and applied sciences. In both cases, the importance and the numerousness of the applications indicate their natural ability to mimic nature. In addition, the evidence suggests that both these theories work out for fractal structure (or chaotic behavior). This entails that fractal geometry, wavelet analysis, and fractional calculus provide three different but equivalent methods to deal with multi-scale properties of general dynamical systems. More specifically, the combined choice of wavelet basis and definition of fractional derivative nowadays gives high-level performance models whenever the geometry of the problem is fractal-like or the behavior is chaotic.