The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations

Abstract

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

1. Introduction

Recently,there have been several schemes devoted to the solution of fractional differential equations. These schemes can be broadly classified into two classes, numerical and analytical ([1]). As we know, with the help of some special functions, such as Mittag-Leffler function and Green function, Miller and Ross have obtained the explicit representations of solutions of some classes of homogeneous linear fractional differential equations (FDEs) in [2]; through the use of the technique of Laplace and Fourier Transform, the analytical solutions have been given by Podlubny in [3]. The numerical scheme we have encountered can be divided into two groups. In the first group, the solution is approximated over the entire domain using approximating functions such as polynomials and orthogonal functions. In the second group, the entire domain is divided into several small domains like in a finite element technique, and the solution is obtained for variables at the node points ([1]).As Edwards et al. declaimed in [4], several forms of fractional differential equations have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solution. Thus, several papers have been presented in dealing with approximate numerical techniques for FDEs. Among these, the papers of Diethelm, Edwards, Ford, Freed, and Simpson are noteworthy (see, e.g., [1, 4–6]). One of these schemes which should be mentioned is the use of a Predictor-Corrector (more precisely, PECE) method in [6]. In order to obtain higher precision, they have replaced the PECE method by a P(EC)^𝑚𝐸 method with 𝑚≥2 in [7]. In particular, the PECE method is an important numerical scheme which has been applied in many fields; for example, Yang and Liu have applied the PECE method for simulating fraction order dynamical control system in [8]. In addition, several numerical schemes have also been proposed by other authors (see [9, 10]). In this paper, we present a new scheme which contains the features of both two groups by using the fractional B-splines wavelet. In paper [11], Unser and Blu have proved that the fractional B-splines generates a valid multiresolution analyses of 𝐿² for 𝛼>−1/2, which means that the orthogonal fractional B-splines could be obtained by applying the standard technique described in [12]. And they also obtained that the fractional B-splines deriving the asymptotic development of the 𝐿² have a fractional order of approximation.

In fact, as the theories of wavelets analyses improve day by day, the wavelet has become a powerful mathematical tool which widely used in signal processing, image compression and enhancement, pattern recognition, control systems, and other fields in the past two decades. But almost no papers or books have applied the theories of wavelets to solve the fractional differential equations. And our fundamental purpose of this paper is applying the fractional B-Splines wavelets to prove the existence and uniqueness of the solution of the nonhomogeneous linear fractional differential equations (also so-called linear multiterm fractional differential equations) with its initial conditions. Let us begin to discuss the solution of multiterms fractional ordinary differential equations with the following form: (𝑏_𝑛𝐷^𝛼𝑛+𝑏_𝑛−1𝐷^𝛼𝑛−1+⋯+𝑏₁𝐷^𝛼1)𝑦(𝑡)=𝑓(𝑡). (1.1) For convenience, we consider the initial values: [𝐷^𝛼𝑘−1𝑦(𝑡)]_𝑡=0=0,[𝐷^𝛼𝑘−2𝑦(𝑡)]_𝑡=0=0,…,[𝐷^{𝛼𝑘−𝑟𝑘}𝑦(𝑡)]_𝑡=0=0, (1.2) where 𝐷^𝛼 are taken in the Riemann-Liouville sense, 𝛼_𝑛>𝛼_𝑛−1>⋯>𝛼₁,𝑏_𝑛≠0,𝛼_𝑛≥1, and 0≤𝑟_𝑘−1≤𝛼_𝑘<𝑟_𝑘,𝑟_𝑘∈𝑁,𝛼_𝑘∈𝑅⁺,𝑏_𝑘∈𝑅,𝑘=1,…,𝑛. The function 𝑓(𝑡) belongs to the space 𝐿²(Ω); without loss of generality, in this paper, we consider the interval as Ω=[0,𝑇],𝑇∈𝑅⁺.

The plan of this paper is as follows. In Section 2, we recall the definitions of fractional derivative and integral and related properties which will be used in this paper, give the representation of Mittag-Leffler function and generalized Mittag-Leffler function, and then introduce the fractional B-splines and some related properties of wavelets. In Section 3, by applying the technique of the Laplace Transforms, and considering the proprieties of the generalized Mittag-Leffler function, we prove the lemma of the differential equations of arbitrary positive real order, which make sure the solution belongs to the space 𝐿²; by virtue of the solution that can be expressed as the form of wavelet series and the basis function that is the orthogonal fractional B-splines wavelet which yields the Riesz basis for the space 𝐿²,we can prove the uniqueness of the coefficients of the representation of the solution, which gain the uniqueness of the solution of the fractional differential equations and validate the representation of solution. Thus, we have finished the proof of the theorem. In Section 4, the asymptotic solution of the differential equations of fractional order 𝛼∈𝑄⁺ and corresponding truncated error will be discussed.

The present paper is essentially based on the works of the Unser and Blu in [11] and Podlubny in [3], to which more general classes of the fractional differential equations we shall refer in the following research. For more related review of fractional B-spline wavelet, see the papers of the Unser and Blu ([11, 13, 14]), which have discussed some more important wavelet properties, such as Riesz bounds and two-scale relation.

2. Preliminary and Definitions

2.1. Definitions

We may recall the definition of the Left Riemann-Liouville differential operators of arbitrary order 𝛼>0, which take the form ₀𝐷^𝛼_𝑡1𝑦(𝑡)∶=𝑑Γ(𝑚−𝛼)𝑑𝑡^𝑚𝑚𝑡₀𝑦(𝜏)(𝑡−𝜏)^{𝛼−𝑚+1}𝑑𝜏, (2.1) where 𝑚 is the integer defined by 𝑚−1≤𝛼<𝑚 (see [3]), and Γ(⋅) is gamma function. Similarly, the left Riemann-Liouville integral operators of order 𝛼>0 is defined as ₀𝐷_𝑡^−𝛼1𝑦(𝑡)∶=Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑦(𝜏)𝑑𝜏. (2.2) And then we should give the following expression for the Laplace transform of the Riemann-Liouville differential operators of the order 𝛼>0, which is 𝐿{₀𝐷^𝛼_𝑡𝑦(𝑡);𝑠}=𝑠^𝛼𝑌(𝑠)−^𝑚−1_𝑘=0𝑠^𝑘[₀𝐷_𝑡^{𝛼−𝑘−1}𝑦(𝑡)]_𝑡=0,(𝑚−1≤𝛼<𝑚). (2.3)

Let us now introduce the case of the Caputo differential operators of arbitrary order 𝛼>0, which is defined as ^𝐶₀𝐷^𝛼_𝑡1𝑓(𝑡)=Γ(𝑛−𝛼)^𝑡₀𝑓^(𝑛)(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼+1−𝑛},(𝑛−1<𝛼≤𝑛). (2.4) And the formula of its Laplace transform can be expressed as 𝐿{^𝐶₀𝐷^𝛼_𝑡𝑓(𝑡);𝑠}=𝑠^𝛼𝐹(𝑠)−^𝑛−1_𝑘=0𝑠^{𝛼−𝑘−1}𝑓^(𝑘)(0),(𝑛−1<𝛼≤𝑛). (2.5)

2.2. Mittag-Leffler Functions and Generalized

The Mittag-Leffler functions and its generalized forms have played a special role in solving the fractional differential equations. In this section, we just give the definition of the following series of representation of the Mittag-Leffler function 𝐸_𝛼(𝑧) with 𝛼>0, which validates in the whole complex plane: 𝐸_𝛼(𝑧)∶=^+∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+1),𝛼>0,𝑧∈𝐶. (2.6) And for the generalized Mittag-Leffler function, we use the following definition: 𝐸_𝛼,𝛽(𝑧)∶=^+∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+𝛽),𝛼>0,𝛽>0,𝑧∈𝐶. (2.7) Let us now consider the Laplace transforms of the function 𝑡^{𝛼𝑘+𝛽−1}𝐸_𝛼,𝛽(±𝑎𝑡^𝛼) (see [3]), which is defined by ^∞₀𝑒^−𝑠𝑡𝑡^{𝛼𝑘+𝛽−1}𝐸^(𝑘)_𝛼,𝛽(𝑎𝑡^𝛼)𝑑𝑡=𝑘!𝑠^𝛼−𝛽(𝑠^𝛼−𝑎)^𝑘+1,(Re(𝑠)>|𝑎|^1/𝛼). (2.8)

2.3. Fractional B-Splines Wavelet

Splines have had a significant impact on the early development of the theory of the wavelet transform (see [13]). And Unser and Blu have first mentioned the fractional B-Splines in [13], who extended Schoenberg's family of polynomial splines to all fractional degrees 𝛼>−1 and defined the fractional causal B-splines by taking the (𝛼+1)th fractional difference of the one-sided power function: 𝛽^𝛼₊1(𝑥)∶=▵Γ(𝛼+1)^+𝛼+1𝑥^𝛼₊=1Γ(𝛼+1)_𝑘≥0(−1)^𝑘𝑘𝛼+1(𝑥−𝑘)^𝛼₊, (2.9) where the one-side power function 𝑥^𝛼₊ is defined as follows: 𝑥^𝛼₊=𝑥^𝛼,𝑥≥0,0,otherwise. (2.10) Then we introduce the fractional B-splines autocorrelation sequence as follows: 𝛽_∗^2𝛼+1(𝑘)∶=⟨𝛽^𝛼(𝑥),𝛽^𝛼(𝑥−𝑘)⟩. (2.11) From [13], we know the explicit form of the fractional B-splines wavelet: 𝜓^𝛼₊(𝑥2)=_𝑘∈𝑍(−1)^𝑘2^𝛼_𝑙∈𝑍𝑙𝛽𝛼+1_∗2𝛼+1(𝑙+𝑘−1)𝛽^𝛼₊(𝑥−𝑘), (2.12) which yields a Riesz basis for 𝐿²(Ω). Thus, using the standard orthogonalize technique described in [12], we have 𝜑^𝛼₊√(𝜔)=2𝜓^𝛼₊(𝑤)(∑^∞_𝑘=0|𝜓^𝛼₊(𝜔+2𝑘𝜋)|²)^1/2, (2.13) where 𝜓^𝛼₊(𝑤) is the Fourier Transform of the function of 𝜓^𝛼₊(𝑥). Thus we obtain the orthogonal fractional B-splines 𝜑^𝛼₊(𝑥) which also yields a Riesz basis for 𝐿²(Ω). According to the theories of wavelet analyses, the function 𝑦(𝑡)∈𝐿²(Ω) can be expressed as 𝑦(𝑡)=^+∞_𝑘=−∞𝑐(𝑘)𝜑^𝛼₊(𝑡−𝑘), (2.14) where the coefficients 𝑐(𝑘) are constants.

3. The Existence and Uniqueness of the Solution ofMultiterms Fractional Ordinary Differential Equations

In this section, we will prove the existence and uniqueness properties of the solutions of the nonhomogeneous linear differential equations of arbitrary real order 𝛼>0.

3.1. An Important Lemma

Under the hypothesis of the existence for the solutions of (1.1), we have the following lemma.

Lemma 3.1. Let 𝛼_𝑛>𝛼_𝑛−1>⋯>𝛼₁, and 𝛼_𝑛≥1, and 𝑓(𝑡)∈𝐿²(Ω), then the solution of the initial value problem (1.1) and (1.2) is also in 𝐿²(Ω).

Proof. In order to prove this lemma, we divide the proof into two steps.
Firstly, we consider the case of 𝑛=1, then (1.1) can be rewritten as ₀𝐷^𝛼1_𝑡𝑦(𝑡)=𝑓(𝑡). (3.1) Taking the transform of (3.1) to both sides, we obtain 𝑦(𝑡)=₀𝐷^−𝛼1_𝑡𝑓(𝑡). (3.2) By virtue of the functions 𝑓(𝑡)∈𝐿²(Ω) and 𝐷^−𝛼∶𝐿²(Ω)→𝐿²(Ω) that is a bounded linear operator (see [15]), we can easily derive the function 𝑦(𝑡)∈𝐿²(Ω).
And then, let us consider the case of 𝑛>1.
To (1.1), using the formula (2.2) and taking the Laplace transform to both sides, we conclude that (𝑏₁𝑠^𝛼1+𝑏₂𝑠^𝛼2+⋯+𝑏_𝑛𝑠^𝛼𝑛)𝑌(𝑠)=𝐹(𝑠), (3.3) and then 𝑌(𝑠)=𝐹(𝑠)𝑏₁𝑠^𝛼1+𝑏₂𝑠^𝛼2+⋯+𝑏_𝑛𝑠^𝛼𝑛=𝐹(𝑠)𝐺(𝑠). (3.4) To prove the functions 𝑦(𝑡)∈𝐿²(Ω), we change the functions 𝐺(𝑠) into the following form: 1𝐺(𝑠)=𝑏_𝑛𝑠^𝛼𝑛+𝑏_𝑛−1𝑠^𝛼𝑛−111+(∑^𝑛−2_𝑘=0𝑏_𝑘𝑠^𝛼𝑘/(𝑏_𝑛𝑠^𝛼𝑛+𝑏_𝑛−1𝑠^𝛼𝑛−1)). (3.5) Then replacing the factor 1/(𝑏_𝑛𝑠^𝛼𝑛+𝑏_𝑛−1𝑠^𝛼𝑛−1) with (𝑏_𝑛⁻¹𝑠^{−𝛼𝑛−1})/(𝑠^{𝛼𝑛−𝛼𝑛−1}+𝑏_𝑛−1/𝑏_𝑛) in (3.5), and expanding the second factor into a formal of series, we have 𝐺(𝑠)=^∞_𝑚=0(−1)^𝑚𝑏_𝑛⁻¹𝑠^{−𝛼𝑛−1}(𝑠^{𝛼𝑛−𝛼𝑛−1}+𝑏_𝑛−1/𝑏_𝑛)^𝑚+1(^𝑛−2_𝑘=0(𝑏_𝑘𝑏_𝑛)𝑠^{𝛼𝑘−𝛼𝑛−1})^𝑚. (3.6) Using the method described in [3], we obtain the expression of 𝐺(𝑠), which is 1𝐺(𝑠)=𝑏_𝑛^∞_𝑚=0(−1)^𝑚_{𝑘0+𝑘1+⋯+𝑘𝑛−2𝑘=𝑚0,𝑘1,…,𝑘𝑛−2≥0}(𝑚;𝑘₀,𝑘₁,…,𝑘_𝑛−2)^𝑛−2_𝑖=0(𝑏_𝑖𝑏_𝑛)^𝑘𝑖𝑠^{−𝛼𝑛−1+∑𝑛−2𝑖=0(𝛼𝑖−𝛼𝑛−1)𝑘𝑖}(𝑠^{𝛼𝑛−𝛼𝑛−1}+𝑏_𝑛−1/𝑏_𝑛)^𝑚+1, (3.7) Substituting 𝐺(𝑠) in (3.4), considering the formula (2.8), and taking the inverse Laplace transform term-by-term, we obtain that the analytical solution of initial values problem (1.1) and (1.2) in the following form: 𝑦(𝑡)=𝑓(𝑡)∗𝑔(𝑡) (3.8) with 1𝑔(𝑡)=𝑏_𝑛^∞_𝑚=0(−1)^𝑚𝑚!_{𝑘0+𝑘1+⋯+𝑘𝑛−2𝑘=𝑚0,𝑘1,…,𝑘𝑛−2≥0}(𝑚;𝑘₀,𝑘₁,…,𝑘_𝑛−2)^𝑛−2_𝑖=0(𝑏_𝑖𝑏_𝑛)^𝑘𝑖𝑡^{(𝛼𝑛−𝛼𝑛−1)𝑚+𝛼𝑛+∑𝑛−2𝑗=0(𝛼𝑛−1−𝛼𝑗)𝑘𝑗−1}×𝐸_𝛼^(𝑚)_{𝑛−𝛼𝑛−1,𝛼𝑛+∑𝑛−2𝑗=0(𝛼𝑛−1−𝛼𝑗)𝑘𝑗}𝑏(−^𝑛−1𝑏_𝑛𝑡^{𝛼𝑛−𝛼𝑛−1}), (3.9) where (𝑚;𝑘₀,𝑘₁,…,𝑘_𝑛−2)=𝑚!/∏^𝑛−2_𝑖=0(𝑘_𝑖!) is the multinomial coefficient, the representation of 𝑓(𝑡)∗𝑔(𝑡) is the convolution of functions 𝑓(𝑡) and 𝑔(𝑡), and 𝐸^(𝑘)_𝜆,𝜇 is the 𝑘th derivative of the Mittag-Leffler function with parameters 𝜆 and 𝜇 (see [16]).
From the representation (3.9), the function 𝐸^(𝑚)_𝛼,𝛽(𝑧) is bounded in 𝑧∈𝐶, and the index satisfies (𝛼_𝑛−𝛼_𝑛−1)𝑚+𝛼_𝑛+∑^𝑛−2_𝑗=0(𝛼_𝑛−1−𝛼_𝑗)𝑘_𝑗−1≥0; so we can easily gain that the function 𝑔(𝑡) is bounded in Ω. And then ‖𝑦(𝑡)‖_𝐿2(Ω)=‖𝑓(𝑡)∗𝑔(𝑡)‖_𝐿2(Ω)≤‖𝑓(𝑡)‖_𝐿2(Ω)‖𝑔(𝑡)‖_𝐿2(Ω)≤𝐶‖𝑓(𝑡)‖_𝐿2(Ω), (3.10) it yields that the functions 𝑦(𝑡)∈𝐿²(Ω), and so we have finished the proof of the lemma.

3.2. The Proof of Existence and Uniqueness

Considering the solution of the fractional differential equation 𝑦(𝑡)∈𝐿²(Ω) and the fractional B-spline wavelet 𝜑^𝛼₊(𝑥) which generates a Riesz basis for 𝐿²(Ω), we can prove the following theorem.

Theorem 3.2. Let 𝛼_𝑛>𝛼_𝑛−1>⋯>𝛼₁, and 𝛼_𝑛≥1, and 𝑓(𝑡)∈𝐿²(Ω), then the initial value problems (1.1) and (1.2) have a unique solution. Further more, the solution has explicit representation of fractional B-splines wavelets series.

Proof. From Lemma 3.1, we have obtained the functions 𝑦(𝑡)∈𝐿²(Ω), which can be expressed as 𝑦(𝑡)=^+∞_𝑘=−∞𝑐(𝑘)𝜑^𝛼₊(𝑡−𝑘), (3.11) where the index satisfies 𝛼>0.
Then substituting (2.14) into (1.1), and taking the Fourier transform to both sides, we obtain 𝑐(̂𝜔)[𝑏₁(𝑖𝜔)^𝛼1+𝑏₂(𝑖𝜔)^𝛼2+⋯+𝑏_𝑛(𝑖𝜔)^𝛼𝑛]𝜓^𝛼₊(𝑤)=𝑓(𝜔), (3.12) where 𝑐(̂𝜔)=∑^+∞_𝑘=−∞𝑐(𝑘)𝑒^{−𝑖𝑘𝑤}, and (3.12) is equivalent to the following form: 𝑐(̂𝜔)𝜓^𝛼₊(𝑤)=𝑓(𝜔)[𝑏₁(𝑖𝜔)^𝛼1+𝑏₂(𝑖𝜔)^𝛼2+⋯+𝑏_𝑛(𝑖𝜔)^𝛼𝑛]. (3.13) And then, by taking the inverse Fourier transform to (3.13), we have ^+∞_𝑘=−∞𝑐(𝑘)𝜑^𝛼₊(𝑡−𝑘)=𝑓(𝑡)∗𝑔(𝑡). (3.14) Because the function 𝜑^𝛼₊(𝑡) is the orthogonal fractional B-splines, hence, the representation of the coefficient 𝑐(𝑘) is defined by 𝑐(𝑘)=⟨𝑓(𝑡)∗𝑔(𝑡),𝜑^𝛼₊(𝑡−𝑘)⟩, (3.15) where ⟨•,•⟩ is the inner product. Then substituting (3.15) in (2.14), we obtain the solution of the initial value problems (1.1) and (1.2).
Let us suppose that the initial value problems (1.1) and (1.2) have another solution, which can be expressed as 𝑦₁(𝑡)=^+∞_𝑛=−∞𝑑(𝑛)𝜑^𝛼₊(𝑡−𝑛). (3.16) Utilizing the similar scheme to 𝑐(𝑘), we acquire the representation of 𝑑(𝑛), which is 𝑑(𝑛)=⟨𝑓(𝑡)∗𝑔(𝑡),𝜑^𝛼₊(𝑡−𝑛)⟩. (3.17) Taking (3.17) into (3.16), we obtain the representation of 𝑦₁(𝑡). Obviously, we have 𝑦(𝑡)−𝑦₁(𝑡)=^+∞_𝑘=−∞𝑐(𝑘)𝜑^𝛼₊(𝑡−𝑘)−^+∞_𝑛=−∞𝑑(𝑛)𝜑^𝛼₊(𝑡−𝑛)=0; (3.18) it derives that 𝑦(𝑡)=𝑦₁(𝑡), which means that the solution of the initial value problems (1.1) and (1.2) is unique. Then substituting 𝑦₁(𝑡) with its coefficient in (1.1), we can easily check that the equation is correct. It indicates that the function 𝑦₁(𝑡) is a solution of the differential equation (1.1), which yields the existence of solution. Finally, we have completed the proof of the solution of existence and uniqueness of the nonhomogeneous linear differential equations of arbitrary order.

4. The Asymptotic Solution of the Fractional Differential Equations and Error Estimation

4.1. The Asymptotic Solution of the Fractional Differential Equations

The purpose of this section is to discuss the case of 𝛼_𝑖∈𝑄⁺ in (1.1), which appeared in most practical applications we have encountered, and show the asymptotic solution of the fractional differential equations and error estimation. In fact, the case of 𝛼_𝑖∈𝑄,𝑖=1,2,…,𝑛 in Theorem 3.2 is equivalent to the following corollary.

Corollary 4.1. Let 𝑚𝑣≥1, and 𝑓(𝑡)∈𝐿²(Ω), then the nonhomogeneous linear fractional differential equations (𝑎_𝑚𝐷^𝑚𝑣+𝑎_𝑚−1𝐷^{(𝑚−1)𝑣}+⋯+𝑎₀𝐷⁰)𝑦(𝑡)=𝑓(𝑡) (4.1) on the initial values [𝐷^{(𝑚−𝑘)𝑣−1}𝑦(𝑡)]_𝑡=0=0,[𝐷^{(𝑚−𝑘)𝑣−2}𝑦(𝑡)]_𝑡=0=0,…,[𝐷^{(𝑚−𝑘)𝑣−𝑟𝑘−1}𝑦(𝑡)]_𝑡=0=0, (4.2) where0≤𝑟_𝑘−1≤(𝑚−𝑘)𝑣<𝑟_𝑘,  𝑣∈𝑄⁺,  𝑟_𝑘∈𝑁,  𝑘=0,1,…,𝑚, have a unique solution.

In Section 3, we have proved the existence and uniqueness of the nonhomogeneous linear fractional differential equations of arbitrary positive real order. Obviously, it also satisfies the case of positive rational order. To (1.1), let 𝑞∈𝑍 be the least common multiple of the denominators of 𝛼_𝑖∈𝑄,𝑖=1,2,…,𝑛, so 𝛼_𝑖=𝑞_𝑖/𝑞, where 𝑞_𝑖∈𝑍,𝑖=1,2,…,𝑛. And to (4.1), let 𝑣=1/𝑞 and 𝑚−𝑖₀=𝑞_𝑛,𝑚−𝑖₁=𝑞_𝑛−1,…,𝑚−𝑖_𝑛−1=𝑞₁, then considering the conditions of 𝑎_{𝑚−𝑖𝑗}=𝑏_𝑛−𝑗,𝑎_𝑘=0,𝑘≠𝑚−𝑖_𝑗, where 𝑖_𝑗<𝑚,𝑖_𝑗∈𝑍,𝑗=1,2,…,𝑛−1. Hence, the initial value problems (4.1) and (4.2) have been changed into of the problems (1.1) and (1.2) with the order 𝛼_𝑖∈𝑄⁺, respectively, which means that Theorem 3.2 in the case of 𝛼_𝑖∈𝑄⁺ is equivalent to Corollary 4.1.

Similarity to the scheme of Theorem 3.2, we can easily prove Corollary 4.1 and give the representation of explicit solution of the initial value problems (4.1) and (4.2), which can be defined by 𝑦(𝑡)=^+∞_𝑘=−∞𝑐(𝑘)𝜑^𝛼₊(𝑡−𝑘), (4.3) where 𝑐(𝑘)=⟨𝑓(𝑡)∗𝑔(𝑡),𝜑^𝛼₊(𝑡−𝑘)⟩, and 𝑔(𝑡) is the inverse of the Laplace Transform of the function 𝐺(𝑠), where 1𝐺(𝑠)=𝑎_𝑚𝑠^𝑚𝑣+𝑎_𝑚−1𝑠^{(𝑚−1)𝑣}+⋯+𝑎₀. (4.4) To obtain the asymptotic solution of (4.1), we will give the explicit formulation of the function 𝑔(𝑡). Let 𝑃(𝑠^𝑣)=𝑎_𝑚𝑠^𝑚𝑣+𝑎_𝑚−1𝑠^{(𝑚−1)𝑣}+⋯+𝑎₀ and 𝑥=𝑠^𝑣; thus the 𝑃(𝑥) is a polynomial of the degree 𝑚. Moreover, we suppose that 𝛾₁,𝛾_2,…,𝛾_𝑗 are distinct zeros of 𝑃(𝑥) with the order of 𝑙₁,𝑙_2,…,𝑙_𝑗, respectively; by applying the theories of polynomials, the 𝑃(𝑥) can be rewritten in the following form 𝑃(𝑥)=𝑎_𝑚(𝑥−𝛾₁)^𝑙1(𝑥−𝛾₂)^𝑙2⋯(𝑥−𝛾_𝑗)^𝑙𝑗, (4.5) where ∑^𝑗_𝑖=1𝑙_𝑖=𝑚,𝑗,𝑙_𝑖∈𝑁,𝑖=1,2,…,𝑗, Then, substituting the function 𝑃(𝑥) in (4.2) and expanding it to the sum of partial fractions, we have 𝐺(𝑠)=^𝑗𝑙_𝑖=1^𝑖−1_𝑛=0𝐴_{𝑖,𝑙𝑖−𝑛}(𝑠^𝑣−𝛾_𝑖)^{𝑙𝑖−𝑛}, (4.6) where the coefficients 𝐴_{𝑖,𝑙𝑖−𝑛},𝑖=1,2,…,𝑗;𝑛=0,1,…,𝑙_𝑖−1, are constants.

Thus, taking the inverse Laplace Transform to (4.6) and using the formulation (2.8), we obtain explicit formulation of the function 𝑔(𝑡), defined by 𝑔(𝑡)=^𝑗𝑙_𝑖=1^𝑖−1_𝑛=0𝑡^{(𝑖+1)𝑣−1}(𝑙_𝑖−𝑛)!𝐴_{𝑖,𝑙𝑖−𝑛}𝐸^{(𝑙𝑖−𝑛−1)}_𝑣,𝑣(𝛾_𝑖𝑡^𝑣). (4.7) According to Unser and Blu in [13], the wavelet base generated by fractional B-splines wavelet 𝛽^𝛼₊(𝑥) is denseness of the representation in 𝐿²(Ω). Let 𝛽̇^𝛼₊(𝑥) be the inverse of Fourier Transform of 𝛽̇^𝛼₊(𝜔), where 𝛽̇^𝛼₊√(𝜔)=2𝛽^𝛼₊(𝑤)(∑^∞_𝑘=0|𝛽^𝛼₊(𝜔+2𝑘𝜋)|²)^1/2. (4.8) With the help of the theories of wavelet analyses, we know that the functions 𝛽̇^𝛼₊(𝑥) are an orthogonal fractional B-splines wavelet and dense in 𝐿²(Ω) with 𝛼>0. Thus the solutions of initial value problems (4.1) and (4.2) have the following form: 𝑦(𝑡)=^+∞_𝑘=−∞𝑐(𝑘)𝛽̇^𝛼₊(𝑡−𝑘). (4.9) Then substituting 𝑦(𝑡) in (4.1), we obtain the representation of the coefficients 𝑐(𝑘), defined by 𝑐(𝑘)=⟨𝑓(𝑡)∗𝑔(𝑡),𝛽̇^𝛼₊(𝑡−𝑘)⟩. (4.10) Finally, by combining (4.9) and (4.10), the obtained function 𝑦(𝑡) is the asymptotic solution of initial value problems (4.1) and (4.2) that we are looking for.

4.2. Order of the Error Estimation

To estimate the error of asymptotic solution of initial value problems (4.1) and (4.2), we introduce the following properties of the fractional B-spines 𝛽^𝛼₊(𝑥), which have been proved by Unser and Blu (see [13, Theorem 3.1]).

Proposition 4.2. For all 𝛼>0, we have 𝛽^𝛼₊(𝑥)=Γ(𝛼+2)sin𝜋𝛼𝜋𝑥^𝛼+2_𝑛≥1𝑒^{2𝑛𝑖𝜋𝑥}(2𝑛𝑖𝜋)^𝛼+11+𝑜(𝑥^𝛼+2), (4.11) when 𝑥 tends to +∞.

To (4.8), the function 𝑎(𝜔)=∑^∞_𝑘=0|𝛽^𝛼₊(𝜔+2𝑘𝜋)|² is 2𝜋-periodic and symmetric, and so we can restrict its study to 𝜔∈[0,𝜋]. In particular, one has 𝜔𝑎(𝜔)≥|sin𝑐2|^2𝛼+22≥(𝜋)^2𝛼+2, (4.12) since sin𝑐𝜔/2 is strictly decreasing over [0,𝜋] (see [13]).

Combining (4.12) and (4.8) and taking the inverse of Fourier Transform, we obtain 𝛽̇^𝛼₊(𝑡−𝑘)≤(2𝜋)^𝛼+1𝛽^𝛼₊(𝑡−𝑘). (4.13) Then according to (4.10), (4.11), and (4.13), the asymptotic solution 𝑦(𝑡) can be defined by 𝑦(𝑡)=^+∞_𝑘=−∞⟨𝑓(𝑡)∗𝑔(𝑡),𝛽̇^𝛼₊(𝑡−𝑘)⟩𝛽̇^𝛼₊(𝑡−𝑘)≤‖𝑓(𝑡)∗𝑔(𝑡)‖_𝐿2^+∞_𝑘=−∞|𝛽̇^𝛼₊(𝑡−𝑘)|². (4.14) To calculate the truncated error of the asymptotic solution, let 𝑦_𝑁(𝑡) be the truncated sum (|𝑘|≤𝑁) corresponding to asymptotic solution (4.8), where 𝑦_𝑁(𝑡)=^𝑁_𝑘=−𝑁⟨𝑓(𝑡)∗𝑔(𝑡),𝛽̇^𝛼₊(𝑡−𝑘)⟩𝛽̇^𝛼₊(𝑡−𝑘). (4.15) Thus the truncated error 𝑦^∗(𝑡) will be obtained as follows: 𝑦^∗(𝑡)=𝑦(𝑡)−𝑦_𝑁=(𝑡)_|𝑘|>𝑁⟨𝑓(𝑡)∗𝑔(𝑡),𝛽̇^𝛼₊(𝑡−𝑘)⟩𝛽̇^𝛼₊(𝑡−𝑘)≤‖𝑓(𝑡)∗𝑔(𝑡)‖_|𝑘|>𝑁|𝛽̇^𝛼₊(𝑡−𝑘)|², (4.16) where the function 𝑔(𝑡) is bounded in Ω and 𝑓(𝑡)∈𝐿²(Ω); with the help of (4.12), the inequality of (4.16) can be amplified, which means that 𝑦^∗2(𝑡)≤(𝜋)^2𝛼+2‖𝑓(𝑡)∗𝑔(𝑡)‖_|𝑘|>𝑁|𝛽^𝛼₊(𝑡−𝑘)|². (4.17) Then substituting (4.10) in (4.16), we have 𝑦^∗2(𝑡)≤(𝜋)^2𝛼+2‖𝑓(𝑡)∗𝑔(𝑡)‖_|𝑘|>𝑁|Γ(𝛼+2)sin𝜋𝛼𝜋(𝑡−𝑘)^𝛼+2_𝑛≥1𝑒^{2𝜋𝑖𝑛𝑥}(2𝑛𝑖𝜋)^𝛼+11+𝑜((𝑡−𝑘)^𝛼+2)|². (4.18) To the right side of (4.17), we divided the representation into three parts for discussion. Firstly, because the function 𝑔(𝑡) is bounded in Ω and 𝑓(𝑡)∈𝐿²(Ω), there exists a constant 𝐶₁ which holds ‖𝑓(𝑡)∗𝑔(𝑡)‖_𝐿2≤𝑐‖𝑓(𝑡)‖_𝐿2≤𝐶₁. (4.19) Secondly, we consider the series function |_𝑛≥1𝑒^{2𝜋𝑖𝑛𝑥}(2𝑛𝑖𝜋)^𝛼+1|²≤_𝑛≥1|𝑒^{2𝜋𝑖𝑛𝑥}(2𝑛𝑖𝜋)^𝛼+1|²≤_𝑛≥11(2𝑛𝜋)^2𝛼+2, (4.20) which is convergent in Ω; thus there exists a constant 𝐶₂ which satisfies |_𝑛≥1𝑒^{2𝜋𝑖𝑛𝑥}(2𝑛𝑖𝜋)^𝛼+1|²≤𝐶₂. (4.21) At last, we denote a constant 𝐶₃ which defined by 𝐶₃=2^2𝛼+2Γ²(𝛼+2)sin²𝜋𝛼𝜋^2𝛼+4. (4.22) Then combining (4.19), (4.21), and (4.22), the inequality of (4.18) can be amplified to the following form: 𝑦^∗(𝑡)≤𝐶₁𝐶²₂𝐶_3|𝑘|>𝑁|1(𝑡−𝑘)^𝛼+21+𝑜((𝑡−𝑘)^𝛼+2)|²≤𝐶₁𝐶²₂𝐶_3|𝑘|>𝑁|1+𝐶₀(𝑡−𝑘)^𝛼+2|², (4.23) when |𝑡−𝑘| tends to +∞.

Thus, from the above discussion, it is evident to derive that the truncated error converges as ∑_|𝑘|>𝑁𝐶/(𝑘−𝑡)^2𝛼+4, where 𝐶=𝐶₁𝐶²₂𝐶₃(1+𝐶₀)². Note that we should choose a suitable 𝑁 which should be much more greater than 𝑇, where, 𝑡∈Ω=[0,𝑇]. Finally, we have obtained the error order of the asymptotic truncated sum in theory.

Remark 4.3. Noting that the process of the proof of the existence and uniqueness of the solution of the initial value problems (1.1) and (1.2) in the case of Riemann-Liouville fractional differential operator, and considering the formula of the Laplace transform of Caputo differential operator, we can be easily replaced the case of Riemann-Liouville fractional differential operator by Caputo sense with its initial values. It means the following corollary is correct.

Corollary 4.4. Let 𝛼_𝑛>𝛼_𝑛−1>⋯>𝛼₁>0, and 𝛼_𝑛≥1, and 𝑓(𝑡)∈𝐿²(Ω), then the multiterms fractional ordinary differential equations (𝑏_𝑛𝐷^𝛼𝑛+𝑏_𝑛−1𝐷^𝛼𝑛−1+⋯+𝑏₁𝐷^𝛼1)𝑦(𝑡)=𝑓(𝑡) (4.24) with its initial values 𝑦^(𝑘)(0)=0,(𝑘=0,1,2,…,⌈𝛼_𝑛⌉−1), (4.25) where ⌈𝛼_𝑛⌉=max{𝑚∣𝑚≤𝛼_𝑛,𝑚∈𝑍}, have a unique solution.

5. Conclusion

In this paper, we have proved the existence and uniqueness of the solution of the differential equations of arbitrary positive real order. And the representation of the solution of (1.1) has been given in the process of proof. We have obtained the asymptotic solution of the differential equations of fractional order 𝛼∈𝑄⁺ and corresponding error estimation. The most notable feature is the order of the asymptotic truncated error, namely, (2𝛼+4)th, which is effective to calculate the numerical solution of (4.1). In particular, the case of Riemann-Liouville differential operator is replaced by Caputo sense with its initial values in Corollary 4.4. Similarly to the scheme of the proof as showen in the paper, by considering the relationship between Riemann-Liouville and Caputo differential operator, we can easily complete the proof of the corollary in the case of the Caputo differential operator, which shows that the method we have discussed can be applied more widely.