Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. The fractional calculus has gained considerable importance during the past decades mainly due to its application in diverse fields of science and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry, and many more. In this paper, we review different wavelet methods for solving both linear and nonlinear fractional differential equations. Our goal is to analyze the selected wavelet methods and assess their accuracy and efficiency with regard to solving fractional differential equations. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on various wavelets in order to solve differential equations of arbitrary order.

Wavelet theory is a relatively new and emerging area in mathematical research. Wavelet methods have been used to develop accurate and fast algorithms for solving numerically integral and differential equations, especially those whose solutions are highly localized in position and scale. The concept of “wavelets” originated from the study of time-frequency signal analysis, wave propagation, and sampling theory. The main reason behind the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in joint time and frequency domain. In 1982, Jean Morlet, in collaboration with a group of French engineers, discovered the idea of wavelets transform for the analysis of nonstationary signals (signals containing transients and fractal structures).

Wavelet method is an exciting method for solving difficult problems in mathematics, physics, and engineering, with modern applications in diverse fields such as wave propagation, data compression, image processing, pattern recognition, computer graphics, the detection of aircraft, and submarines and improvement in CAT scans and other medical technology. Morlet (1982) [

Wavelet

smoothness:

localization:

the admissibility condition

The wavelet algorithms for solving differential equations usually are based on the collocation method. From the beginning of 1980s wavelets have been used invariably for the solution of differential equations. Most of the wavelet algorithms can handle easily periodic boundary conditions. In the present paper, the main idea is to apply wavelet methods, namely, Haar wavelet method, Legendre wavelet methods, Chebyshev wavelet method, B-spline wavelet, and so forth, for solving fractional differential equations (FDE).

A set of subspaces

For each scale

In the last few decades many authors pointed out that derivatives and integrals of noninteger order are very suitable for the description of properties of various real phenomena. Fractional derivatives [

The mathematical modelling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally lead to differential equations of fractional order and to the necessity of solving such equations. However, effective general methods for solving them cannot be found even in the most useful works on fractional derivatives and integrals.

Recently, orthogonal wavelets bases are becoming more popular for numerical solutions of differential equations due to their excellent properties such as ability to detect singularities, orthogonality, flexibility to represent a function at different levels of resolution, and compact support. In recent years, there has been a growing interest in developing wavelet based numerical algorithms for solution of fractional differential equations. Wavelets have been successfully applied for the solutions of ordinary and partial differential equations, integral equations, and integrodifferential equations of arbitrary order. Therefore, the main focus of the present paper is the application of different wavelet techniques for solving differential equations of arbitrary order.

A computational approach to solve fractional differential equation is an essential work in scientific research. Some methods [

The fractional calculus was first anticipated by Leibnitz, who was one of the founders of standard calculus, in a letter written in 1695. This fractional calculus involves different definitions of the fractional operators such as the Riemann-Liouville fractional derivative, Caputo derivative, Riesz derivative, and Grunwald-Letnikov fractional derivative [

The most frequently encountered definition of an integral of fractional order is the Riemann-Liouville integral [

The fractional derivative introduced by Caputo [

The Grunwald-Letnikov fractional derivative is defined as [

Wavelet analysis is a numerical concept which allows representing a function in terms of a set of basis functions, called wavelets, which are localized both in location and scale. Wavelets used in this method are mostly compact support introduced by Daubechies [

Haar wavelet functions have been used from 1910 and were introduced by the Hungarian mathematician Alfred Haar. Haar wavelets (which are Daubechies wavelets of order 1) consist of piecewise constant functions on the real line that can take only three values, that is, 0, 1, and −1, and are therefore the simplest orthonormal wavelets with a compact support. Haar wavelet method is to be used due to the following features: being simpler and fast, flexible, and convenient, having small computational costs, and being computationally attractive. The Haar functions are a family of switched rectangular wave forms where amplitudes can differ from one function to another. These properties of Haar wavelets are utilized to reduce the computation of integral equations to some algebraic equations.

Usually the Haar wavelets are defined for the interval

The integration of the

The Haar wavelet operational matrix

Any function

Here

The collocation points are given by

In order to show the effectiveness of Haar wavelet method for solving fractional differential equations, we consider the following numerical example of variable coefficient fractional convection diffusion equation:

Since

Let

Let

Any function

Consider the fractional differential equation of the form

Now (

The Legendre wavelet and Chebyshev wavelet are constructed from their corresponding polynomials. These wavelets are useful tools in the numerical computations. The second kind Chebyshev polynomials have many good properties and are widely applied in different disciplines.

Wavelets constitute a family of functions constructed from dilation and translation of a single function

Chebyshev wavelets

If the infinite series in (

The integration of the vector

We know the Riemann-Liouville fractional integral operator

Also we define a

The functions

In [

Next we derive the Chebyshev wavelet operational matrix of the fractional integration. Let

Using (

In this section, we use Chebyshev wavelet operational matrices of the fractional integration to solve nonlinear fractional differential equation. We consider fractional Riccati equation

Let

Legendre wavelet

The Legendre polynomials of order

A function

If the infinite series in (

In order to show the effectiveness of Legendre wavelet method for solving fractional differential equations, we consider the following numerical example [

In this section, the application of Legendre multiwavelet Galerkin method for providing approximate solutions for initial value problems of fractional differential equations has been discussed. Legendre multiwavelet

A function

Consider the nonlinear fractional partial differential equation

Let

The expansion coefficients

In this work, we derive the operational matrices of the fractional integration for Haar wavelet, Legendre wavelet, and Chebyshev wavelets. We have examined several wavelet methods to solve fractional differential equations. In this present paper, we have applied cubic B-spline wavelets for solving fractional differential equations. In order to increase the accuracy of the approximate solution, it is necessary to apply higher order spline wavelet method. The Legendre multiwavelet method can be applied for providing approximate solutions for initial value problems of fractional nonlinear partial differential equations. The main characteristic of this approach is using the properties of Legendre multiwavelet together with the Galerkin method to reduce the nonlinear fractional order partial differential equations (NFPDEs) to the nonlinear system of algebraic equations. Similarly, the Chebyshev wavelets, constructed from the corresponding Chebyshev polynomials, are quite efficient in dealing with the nonlinear FDE like Riccati equation [

Using these wavelet methods, the fractional differential equations have been reduced to a system of algebraic equations and this system can be easily solved by any usual methods. Haar wavelet method can also be applied to the fractional differential equations by reducing into a system of algebraic equations. These methods give more accuracy if we increase the order or the level of resolution. The approximate solutions by these aforesaid methods highly agree with exact solutions.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research work was financially supported by SERB, DST, Government of India, under Grant no. SR/S4/MS.:722/11.