Applications of Bi-framelet Systems for Solving Fractional Order Differential Equations

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.


INTRODUCTION
In recent years, fractional differential equations (FDEs) have been widely used in the development of many modern problems in engineering practices and applied sciences. For example, it has applications in the modeling of earthquakes, 1 economics, 2 fluids, 3 dynamic of viscoelastic materials, 4 and many other disciplines, e.g. see Refs. 5-8. Fractional calculus is a mathematical area that studies and analyzes the properties of the derivative and integration of a non-integer orders. In particular, this area is getting more attention from many researchers to develop new methods for solving differential equations involved by fractional order. Atangana is unique among fractals-theorists in his ability to bring to bear new definitions, theory and ideas on some of the most intractable issues on FDEs. He is the founder of the fractional calculus with nonlocal and non-singular kernels popular in applied mathematics today and has achieved and contributed significantly to the numerical and pure analysis of FDEs. Atangana et al. defined the well-known Atangana-Baleanu fractional derivative definition that describes the complicated problems related to the power and exponential laws and free of singularities. [9][10][11][12][13][14][15][16][17] Note that, in general, the exact solution of most the FDEs does not exist. Therefore, examining and developing new numerical methods is very important. For example, Laplace and Fourier techniques were proposed, 18,19 Adomian decomposition method in Refs. 20-23, variational iteration method in Ref. 24, and other methods can be found in Refs. 25-28. Interested readers should consult other references therein to have an extra knowledge of the other used methods.
In the context of numerical and computational mathematics, framelet systems have proven as a powerful tool on tackling issues related to the numerical and computational framework. The main aim of this work is to shed some lights on the benefits of using bi-framelets in the area of the FDEs. Some of the FDEs we consider are as follows: for some n ∈ N where n − 1 < α ≤ n, c(t), f(t) are known square integrable functions, D α * u is Caputo FDE (see Definition 1.1) operator of u, and u(t) is the unknown function to be approximated. Definition 1.1. For a real function u(t) where t, α > 0, and n ∈ N, we have the following: • The Caputo's fractional derivative of order α is defined by • The Riemann-Liouville fractional derivative of order α is defined by • The Riemann-Liouville fractional integral operator (FIO) of order α is defined by Note that, , and for t > 0,

AN OVERVIEW OF QUASI-AFFINE BI-FRAMELET
Let us recall some definitions and facts to clarify the concept of bi-framelets. It is known that orthonormal bases are non-redundant systems, while their redundant setup are known as frames. The redundancy here is very useful in many applications such as the error recovery/correction in transmission of data. Frames are more general than orthogonal systems and provide better representations. They were introduced by Duffin and Schaeffer in Ref. 29. A big development later on has been achieved in Refs. 30-32.
If C 1 = C 2 = 1, it is called a Parseval frame. The importance of such tightness in the framelet system is that they provide a simple and better reconstructions for the elements of the space L 2 (R). In framelet analysis, we say that the function φ is a refinable function if where h 0 is a finitely supported sequence (called the low pass filter of φ) such that Note that, in frequency domain, Eq. (4) can be written asφ whereφ is the Fourier transform of φ.
is a pair of quasi-affine bi-framelet system of L 2 (R) if both X(Ψ) and X(Ψ) are framelet systems in L 2 (R) and Note that, Eq. (5) is also satisfied for all elements of Ψ andΨ with finitely supported sequences called high pass filters, h [k], andh [k], ∈ E, respectively. For simplicity, we define ψ 0 = φ andψ 0 =φ. Thus, for ξ ∈ R, we have Therefore, given a quasi-affine bi-framelet system X(Ψ), we can find a subset of L 2 (R), X(Ψ), such that (similar to Eq. (3)) In particular, if φ = φ and ψ =ψ , ∀ ∈ E, then it is called a tight framelet system for L 2 (R). So, we have the following equation: From Eq. (7), for a bi-framelet system, we have where Thus, we have the following quasi-affine bi-framelet representation: where a j,k = g,ψ j,k . Note that, the coefficients a j,k is not unique but it is one of the best choices for a better simulation. Hence, one can consider the following truncated representation from Eq. (11) for f : Sparse representation for a smooth function is of interest in many applications. Therefore, to have such sparsity, it is crucial for h to have high vanishing moments, where a function ψ has a vanishing moments of order s if t k , ψ = 0, for all k = 0, 1, . . . , s − 1.
In literature, there are many principles to construct bi-framelets, such as the the mixed unitary extension principles (MUEP) (see e.g. Refs. 33-36 and references therein). In this paper, we use the generalization of the MUEP, namely, the mixed oblique extension principle (MOEP) presented in Ref. 35. By doing this, we will present some examples of bi-framelet systems that have high vanishing moments. However, to have such property, it is required to have some required constraints on then, the system (X(Ψ), X(Ψ)) defined in Definition 2.2 forms a quasi-affine bi-framelet system of L 2 (R).
The MOEP provides a way to construct biframelets from refinable functions and it gives us a better approximation orders and reconstruction. Framelets have a great deal of use in many applications due to the features of redundancy (by increasing the number r), and many other properties (see e.g. Ref. 37). In this paper, we use analytic expressions of bi-framelets with high redundancy generated via the MOEB using some analytic refinable functions called B-splines. B-splines are of importance in harmonic theory and have wide range of use for many applications in approximation analysis. It is defined using the convolution product as follows.

Definition 2.3 (Ref. 37).
For m ∈ N, the Bspline of order m, B m , is defined as 1) , and

SYSTEMS OF QUASI-AFFINE BI-FRAMELETS
In this section, we use the MOEP to construct quasi-affine bi-framelet systems using B-splines and use it to solve some examples of FDEs.
System A (HAAR bi-framelet). Consider the B-spline of order 1, Hence, the system (X(Ψ), X(Ψ)) is a bi-framelet for L 2 (R). Figure 2 shows the graphs of the generators, ψ 1 andψ 1 in the time domain.

System B (Linear bi-framelet).
For m = 2, consider the linear B-spline for φ andφ with the following filters:ĥ , Then, this set of filters satisfies the conditions of MOEP and therefore the system (X(Ψ), X(Ψ)) forms a quasi-affine bi-framelet system for L 2 (R). The generators of this system are depicted in Fig. 3.

System C (Cubic-Linear bi-framelet).
Consider the (cubic) B-spline of order 4, φ = B 4 , with the following filter: and the (linear) B-spline of order 2,φ = B 2 , with the following filter: Depending on the MOEP, we define explicitly in the time domain the generators {ψ ,ψ , = 1, 2}, that generate a bi-framelet system, namely o t h e r w i s e , Fig. 3 The graphs of the generators in System B.

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Fractals 2020. 28 Fig. 4 The graphs of the generators in System C.
The graphs of these generators are shown in Fig. 4.
System D. Now lets consider the B-spline of order four for both φ andφ. Again, by applying the setup of the MOEP, we are able to find the generators explicitly as follows: o t h e r w i s e ,

o t h e r w i s e .
We depict the graphs of these generators in Fig. 5.

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Fractals 2020. 28 Fig. 5 The graphs of the generators in System D.

QUASI-AFFINE BI-FRAMELET OPERATIONAL MATRIX OF FDES
In this section, we introduce the quasi-affine biframelet operational matrix of the FDEs defined in Eq. (1) using the function approximation given by Eq. (12). For simplicity, we present the bi-framelet operational matrix using System A (Haar bi-framelet system X({ψ 1 , ψ 2 })). For other types of bi-framelet systems that generated by higher order of B-splines given in Sec. 3, the procedure will be the same.
Proof. Based on the upper bound of the biframelet system using Bessel property, we have But, Using the mean value theorem for integration, one can find ξ 1 , ξ 2 such that ξ 1 < ξ 2 and that Then, the result is concluded.

Applications of Bi-Framelet Systems for Solving Fractional Order Differential Equations
where the Haar bi-framelet vector A n (A T n is the transpose of the bi-framelet vector A n ) and Haar bi-framelet function vector G n (t) are given as Consider the collocation points where n p is the length of interval of ψ j,k that covers the compact support of the bi-framelet system at hand, for example, for the case of Haar bi-framelet system, n p = 2 n+1 (2n + 1). Define the operational bi-framelet matrix as follows: Now, considering the system defined by Eq. (1) and in order to solve it using a bi-framelet system via the discretized points, we substitute the truncated expansion given in Eq. (13). Thus where I α is the FIO defined by Eq. (2). For i = 1, . . . , n p , Eq. (14) can be written as where ..,np is a matrix function vector of order n p × 1. The coefficient bi-framelet vector A n in Eq. (15), and so the approximated solution, can be calculated using, for example, Mathematica software.

NUMERICAL APPLICATIONS
In this section, we use the bi-framelet systems presented in Sec. 3 to illustrate and show the efficiency of the proposed method by using some examples of FDEs. Example 1. Consider the following FDEs: f (t) = 1.10773 The exact solution for this FDE is u(t) = t.
By applying the procedure in Sec. 4 and solving the resulting system, we obtain the approximated solution of the FDE using System A, for example, when n = 12, 13, respectively, we have x, 0 < x ≤ 1 8 .

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Fractals 2020. 28 We also present the matrix L np×np (n = 1) in Eq. (15) using System A where  Fig. 6 Numerical and exact solution with the error graph using System A for n = 13.

Fig. 7
Numerical and exact solution with the error graph using System B for n = 3.

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Fractals 2020. 28     The error graph using System A for n = 1, 2, 3, respectively. The exact solution for the given FDE is u(t) = 1 2 (2t 4 − t 3 ). Again, based on on the numerical scheme presented in Sec. 4, we conclude the following numerical and graphical results based on System A through D in Tables 3 and 4, Figs. 8 and 9. Figure 10 shows the convergence behavior of the numerical results obtained in Examples 1 and 2 based on the Systems A through D.