Wavelet Methods for Solving Fractional Order Differential Equations

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, andmanymore. In this paper, we reviewdifferentwaveletmethods for solving both linear andnonlinear 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.


Introduction to Wavelets
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 timefrequency 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) [1] first introduced the idea of wavelets as a family of functions constructed from translation and dilation of a single function called the "mother wavelet." Wavelet (): an oscillatory function () ∈  2 (R) with zero mean is a wavelet if it has the following desirable attributes: (1) smoothness: () is  times differentiable and its derivatives are continuous; (2) localization: () is well localized both in time and frequency domains; that is, () and its derivatives must decay rapidly.For frequency localization Ψ() must decay sufficiently fast as  → ∞ and that Ψ() becomes flat in the neighborhood of  = 0.
The flatness is associated with number of vanishing moments of (); that is, Ψ ()  = 0 or equivalently     Ψ () = 0 for  = 0, 1, . . .,  (1) in the sense that the larger is the number of vanishing moments the more is the flatness when  is small; Although most of the numerical methods have been successfully applied for many linear and nonlinear differential equations, they have also some drawbacks in regions where singularities or sharp transitions occur.In those cases the solutions may be oscillating and for accurate representation of the results adaptive numerical schemes must be used which complicates the solution.To overcome the above difficulty wavelet transform methods are quite useful.
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).[2].A set of subspaces {  } ∈ is said to be MRA of  2 (R) if it possess the following properties:

Multiresolution Analysis (MRA)
where  denotes the set of integers.Properties (3)-( 5) state that {  } ∈ is a nested sequence of subspaces that effectively covers  2 (R).That is, every square integrable function can be approximated as closely as desired by a function that belongs to at least one of the subspaces   .A function  ∈  2 () is called a scaling function if it generates the nested sequence of subspaces   and satisfies the dilation equation; namely, with   ∈  2 and  being any rational number.
For each scale , since   ⊂  +1 , there exists a unique orthogonal complementary subspace   of   in  +1 .This subspace   is called wavelet subspace and is generated by  , = (2   − ), where  ∈  2 is called the wavelet.From the above discussion, these results follow easily:

Fractional Calculus
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 [3] provide an excellent instrument for the description of memory and hereditary properties of various materials and processes.This is the main advantage of fractional derivatives in comparison with classical integer order models in which such effects are neglected.
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.

2.1.
Fractional Derivative and Integration.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 [3].Riemann-Liouville fractional derivative is not suitable for real world physical problems since it requires the definition of fractional order initial conditions, which have no physically meaningful explanation yet.Caputo introduced an alternative definition, which has the advantage of defining integer order initial conditions for fractional order differential equations.
2.1.1.Definition of Riemann-Liouville Integral.The most frequently encountered definition of an integral of fractional order is the Riemann-Liouville integral [3], in which the fractional integral operator   of a function () is defined as [3]: where Γ(⋅) is the well-known gamma function, and some properties of the operator   are as follows: () =  +  () , ( > 0,  > 0) , The following are two basic properties of the Caputo fractional derivative: For Caputo's derivative we have Similar to integer order differentiation, Caputo's derivative is linear: where  and  are constants, and it satisfies the so-called Leibnitz's rule   (()()) = ∑ ∞ =0 (   )  () () − (), where () is continuous in [0, ] and () has continuous derivatives sufficient number of times in [0, ].

Wavelet Methods for Fractional Differential Equations
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 [14].The wavelet based approximations of ordinary and partial differential equations have been attracting the attention, since the contribution of orthonormal bases of compactly supported wavelet by Daubechies and multiresolution analysis based Fast Wavelet transform algorithm by Beylkin et al. [15] gained momentum to make wavelet approximations attractive.

Haar Wavelets. 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  ∈ [0, 1) but in general case  ∈ [, ], we divide the interval [, ] into  equal subintervals, each of width Δ = ( − )/.In this case, the orthogonal set of Haar functions are defined in the interval [, ] by [6] where for  = 1, 2, . . ., ,  = 2  and  is a positive integer which is called the maximum level of resolution.Here  and  represent the integer decomposition of the index ; that is,  =  + 2  − 1, 0 ≤  <  and 1 ≤  < 2  + 1. [6].

Operational Matrix for General Order Integration
The integration of the   () = [ℎ 0 (), ℎ 1 (), . . ., ℎ −1 ()]  can be approximated by [16] where  is called the Haar wavelet operational matrix of integration which is a square matrix of -dimension.To derive the Haar wavelet operational matrix of the general order of integration, we recall the fractional integral of order (> 0) which is defined by Podlubny [3]: where R + is the set of positive real numbers.The Haar wavelet operational matrix   for integration of the general order  is given by where 0, elsewhere, where for  = 1, 2, . . ., ,  = 2  and  is a positive integer, called the maximum level of resolution.Here  and  represent the integer decomposition of the index .That is,  =  + 2  − 1, 0 ≤  <  and 1 ≤  < 2  + 1.

Function Approximation.
Any function () ∈  2 ([0, 1)) can be expanded into Haar wavelets by [6,10,16] where If () is approximated as piecewise constant in each subinterval, the sum in (24) may be terminated after  terms and consequently we can write discrete version in the matrix form as where Y and C   are the -dimensional row vectors.Here  is the Haar wavelet matrix of order  defined by where h 0 , h 1 , . . ., h −1 are the discrete form of the Haar wavelet bases.
The collocation points are given by Similarly, a function of two variables (, ) ∈  2 ([0, 1] × [0, 1]) can be approximated by discrete version in the matrix form of Haar wavelets as where (, ) is the discrete form of (, ) and  is the coefficient matrix of , and it can be obtained by the following formula: Since  is orthogonal, (29) becomes (30) [17].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:

Cubic B-Spline Basis Functions on 𝐻 2 (𝐼).
Let  = [0, ] be an interval with  > 4, and let  2 () be a Sobolev space which contains functions with square integrable second derivatives, and the homogenous Sobolev space  2 0 () can be defined by which is a Hilbert space equipped with inner product: Cai and Wang [18] present a multiresolution analysis (MRA) and a wavelet decomposition for  2 0 () by constructing scaling spline functions which have compact supports, where Then the scaling spaces Let is an unconditional basis of  2 0 (), which turns out to be a basis of continuous space  0 ().For nonhomogeneous Sobolev space  2 (), Cai and Wang [18] introduced boundary spline functions to deal with the values of functions at boundary points.

Chebyshev
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 () called the mother wavelet.When the dilation parameter  and the translation parameter  vary continuously, we have the following family of continuous wavelets as [13]  , ( If we restrict the parameters  and  to discrete values as  =  − 0 ,  =  0  − 0 ;  0 > 1,  0 > 0, where  and  are positive integer, the following family of discrete wavelets are defined as where  , () forms a wavelet basis for  2 (R).In particular, when  0 = 2 and  0 = 1, then  , () forms an orthonormal basis; that is, ⟨ , () ,  , ()⟩ =     . (66)
If the infinite series in (70) is truncated, then (70) can be written as where  and Ψ() are 2 −1  × 1 matrices given by Taking the collocation points as follows: we define the Chebyshev wavelet matrix Φ × as

Operational Matrix of the Fractional
Integration.The integration of the vector Ψ() defined in (73) can be obtained as where  is the 2 −1  × 2 −1  operational matrix for integration [13].
The functions   () are disjoint and orthogonal; that is, Similarly, Chebyshev wavelets may be expanded into an term block pulse functions (BPF) as where In [21], Kilicman and Al Zhour have proposed the block pulse operational matrix of the fractional integration   as follows: where with   = ( + 1) +1 − 2 +1 + ( − 1) +1 .Next we derive the Chebyshev wavelet operational matrix of the fractional integration.Let where matrix     ×  is called the Chebyshev wavelet operational matrix of the fractional integration.
Using (81) and (82), we have From ( 84) and (85) we get Then, the Chebyshev wavelet operational matrix of the fractional integration     ×  is given by Thus, we derive the operational matrix of fractional integration for the second kind Chebyshev wavelet.[22].In this section, we use Chebyshev wavelet operational matrices of the fractional integration to solve nonlinear fractional differential equation.We consider fractional Riccati equation

Application of Chebyshev Wavelet on FDE
subject to initial state (0) = 0. Let and then Solving the above nonlinear system of algebraic equations, we can find the vector  and consequently the solution for ().
If the infinite series in (97) is truncated, then (97) can be written as where  and Ψ() are 2 −1  × 1 matrices given by  ≡ [ 1,0 ,  In order to show the effectiveness of Legendre wavelet method for solving fractional differential equations, we consider the following numerical example [11]: such that The corresponding integral representation for (102) and ( 103) is Let  be approximated using Legendre wavelet as and then where m = 2 −1 .
By solving the linear system (107), we can find the vector  and hence consequently the solution ().

Application of Legendre Multiwavelet Galerkin Method on Nonlinear
Fractional Partial Differential Equations [12].

Conclusion and Discussion
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 [22].In this present analysis, a Legendre wavelet operational matrix of fractional order integration is obtained and is used to solve fractional differential equations numerically.It is worth mentioning that results agree well with exact solutions even for small values of  and .The method is very convenient for solving initial value problems as well as boundary value problems.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.