30 Oct 2015: Bhrawy AH, Taha TM, Alzahrani EO, Baleanu D, Alzahrani AA (2015) Correction: New Operational Matrices for Solving Fractional Differential Equations on the Half-Line. PLOS ONE 10(10): e0142275. https://doi.org/10.1371/journal.pone.0142275 View correction
In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.
Citation: Bhrawy AH, Taha TM, Alzahrani EO, Baleanu D, Alzahrani AA (2015) New Operational Matrices for Solving Fractional Differential Equations on the Half-Line. PLoS ONE 10(5): e0126620. https://doi.org/10.1371/journal.pone.0126620
Academic Editor: Matthew Joseph Simpson, Queensland University of Technology, AUSTRALIA
Received: October 28, 2014; Accepted: April 5, 2015; Published: May 21, 2015
Copyright: © 2015 Bhrawy et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: This paper was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (14-135-35-RG).
Competing interests: The authors have declared that no competing interests exist.
FDEs describe accurately many models in science and engineering such as bioengineering applications, porous or fractured media, electrochemical processes, viscoelastic materials [1–7]. Indeed most of FDEs do not have exact solutions. Therefore, there have been great attempts to develop numerical methods to solve them. Several analytical and numerical techniques for solving FDEs are proposed in [8–19].
Spectral methods are efficient techniques for solving differential equations accurately see for instance [20–27]. Bhrawy and Abdelkawy  proposed the formulation of Jacobi pseudospectral scheme for solving multi-dimensional fractional Schrodinger equations subject to different boundary conditions. The operational matrices for fractional variable-order of the derivative and integral of Jacobi polynomials were derived and used based on Jacobi tau scheme to solve the variable-order FDEs . Recently, new accurate Petrov-Galerkin spectral solutions for FDEs are developed and analyzed in . Moreover, spectral pseudospectral technique was investigated in  to approximate the solution of fractional integro-differential equation.
In the context of numerical methods for solving differential equations in the half-line, the first attempts to use Laguerre polynomials in the implementation of spectral methods to solve differential equations was the work of Gottlieb and Orszag . After that, a series of published papers have appeared describing a range of various spectral methods based on Laguerre basis functions. Mikhailenko  developed an efficient algorithm based on the spectral Laguerre approximations of temporal derivatives for time-dependent problems. The authors of  proposed a new orthogonal family of generalized Laguerre functions to approximate the solution of differential equations of degenerate type. Xiao-Yong and Yan  investigated a pseudospectral scheme based on a class of modified generalized Laguerre to introduce a very efficient method for solving second-order differential equation in a long-time interval. Gulsu et al.  presented the Laguerre collocation method for solving a class of delay difference equations. Tatari and Haghighi  proposed an efficient mixed spectral collocation scheme to solve initial-boundary value problems in which Legendre and generalized Laguerre polynomials were used to discretize space and time variables.
On the other hand, results on numerical methods for FDEs seem to be lacking in the literature. In recent years, some authors have presented the generalized and modified generalized Laguerre spectral tau and collocation techniques for solving several types of linear and nonlinear FDEs on the half-line, (see [37, 38] and the references therein). However, it is also a very important task to develop the spectral techniques to obtain highly accurate solutions of FDEs on the half-line. Therefore, we present in this article a new family of orthogonal functions defined on the half-line namely, fractional-order generalized Laguerre functions.
In the present paper, we aim to construct the fractional-order generalized Laguerre operational matrices, of fractional derivative and integration, which are used to produce two efficient fractional-order generalized Laguerre tau schemes for solving numerically linear FDEs with initial conditions. We also aim to propose a new fractional-order generalized Laguerre collocation (FGLC) scheme for approximating the solution FDE of order ν (0 < ν < 1) with nonlinear terms. This approach is based on the operational matrix of fractional derivatives of these new functions, in which the nonlinear FDE is collocated at the N zeros of the fractional-order generalized Laguerre functions (FGLFs) defined on the interval (0, ∞). The resulting algebraic equations plus one algebraic equation (obtained from the initial condition), constitute (N+1) nonlinear algebraic equations. These equations may be solved by the Newton’s iterative technique to find the unknown fractional-order generalized Laguerre functions coefficients. We extend the application of FGLC method based on FGLFs to solve a system of linear FDEs with fractional orders less than 1. Several numerical examples are implemented to confirm the high accuracy and effectiveness of the new methods for solving FDES of fractional order ν (0 < ν < 1).
The remainder of this paper is organized as follows: we start by presenting some necessary definitions of the fractional calculus theory. In Section 3, we define the fractional-order generalized Laguerre functions. Section 4 is devoted to derive the main theorem of the paper which provides explicitly an operational matrix of fractional-order derivatives of the FGLFs. In Section 5, we derive an operational matrix of fractional-order integrals of the FGLFs. In Section 6, we apply the spectral methods based on the derived operational matrices FGLFs for solving FDEs and systems of FDEs including linear and nonlinear terms of fractional order less than 1. Several examples to illustrate the main ideas of this work are presented in Section 7. Finally Section 8 outlines the main conclusions.
Preliminaries and Notations
We start this section by reviewing some definitions of fractional derivatives and integrals which will be employed in the sequel.
Definition 2.1. The Riemann-Liouville integral Jν f(x) and the Riemann-Liouville fractional derivative Dν f(x) of order ν > 0 are defined by (1) and (2) respectively, where m−1 < ν ≤ m, m ∈ N+ and Γ(.) denotes the Gamma function.
Convert multi-order FDE into a system of FDE
Consider the multi-order FDE (7) where m < ν ≤ m+1, 0 < δ1 < δ2 < … ≤ δn < ν. This equation may be converted to a system of FDEs, as follows. Let u1 = u and assume (8) Case (i) If m−1 ≤ δ1 < δ2 ≤ m, then assume (9) Cases (ii) Consider m−1 ≤ δ1 < m ≤ δ2. If δ1 = m−1, then assume . If m−1 < δ1 < m ≤ δ2, then assume (10) similar steps can be converted the initial value problem Eq (7) to a system of FDE.
Fractional-Order Generalized Laguerre Functions
We recall below some relevant properties of the generalized Laguerre polynomials (Szegö  and Funaro ). Let Λ = (0, ∞) and w(α)(x) = xα e−x be a weight function on Λ. Consider the following inner product and norm
Next, let be the well-known generalized Laguerre polynomials. We know from  that for α > −1, (11) where and .
Definition of FGLFs
Now, we define a new fractional orthogonal functions based on generalized Laguerre polynomials to obtain the solution of FDEs more accurately. The FGLFs may be given by considering the change of variable t = xλ and λ > 0 on generalized Laguerre polynomials. Let the FGLFs be denoted by , thanks to Eq (11), then can be obtained from (15) where and .
According to Eq (13), the analytic form of of degree iλ is given by (16)
Proof. The proof of this lemma can be accomplished directly by using the definition of FGLFs and the orthogonality property of generalized Laguerre polynomials.
The approximation of functions
Let , then u(x) may be expressed in terms of FGLFs as (18) In practice, only the first (N+1)-terms fractional-order generalized Laguerre functions are considered. Then we have (19) where the fractional-order generalized Laguerre coefficient vector C and the fractional-order generalized Laguerre vector ϕ(x) are given respectively by (20)
Definition 3.1 (Generalized Taylor’s formula). Suppose that Dkλ u(x) ∈ C[0, L] for k = 0, 1, …, N, then we have where 0 < η ≤ x, ∀x ∈ [0, L], Also, one has where Eλ ≥ |D(N+1)λ u(η)|. In case of λ = 1, the generalized Taylor’s formula is the classical Taylors formula.
Now, the following Theorem presents an upper bound for estimating the error based on the expansion in terms of FGLFs.
Theorem 3.2 Suppose that Dkλ u(x) ∈ C[0, L] for k = 0, 1, …, N, (3+2N+α) > 0 and If uN(x) = CT ϕ(x) is the best approximation to u(x) from , then the error bound is presented as follows where Eλ ≥ |D(N+1)λ u(x)|, x ∈ [0, L].
Proof. Considering the generalized Taylors formula where 0 < η ≤ x, ∀x ∈ [0, L], making use of Definition 3.1, we obtain Since uN(x) = CT ϕ(x) is the best approximation to u(x) from , then by the definition of the best approximation, we have It turns out that the previous inequality is also true if Accordingly, we obtain (21) Now by taking the square roots, the theorem can be proved. Hence, an upper bound of the absolute errors is obtained for the approximate and exact solutions.
Fractional-Order Generalized Laguerre Operational Matrix of Fractional Derivatives
Let , then u(x) may be expressed in terms of fractional-order generalized Laguerre functions as (22) In practice, only the first (N+1)-terms fractional-order generalized Laguerre functions are considered. Then we have (23) where the fractional-order generalized Laguerre coefficient vector C and the fractional-order generalized Laguerre vector ϕ(x) are given respectively by (24) then the derivative of the vector ϕ(x) can be expressed by (25) where D(1) is the (N+1)×(N+1) operational matrix of first-order derivative. If we define the q times repeated differentiation of fractional-order generalized Laguerre vector ϕ(x) by Dq ϕ(x). (26) where q is an integer value and D(q) is the operational matrix of differentiation of ϕ(x).
Theorem 4.1 Let ϕ(x) be fractional-order generalized Laguerre vector defined in Eq (24) and also suppose 0 < ν < 1 then (27) where D(ν) is the (N+1) × (N+1) operational matrix of fractional derivative of order ν in the Caputo sense and is defined as follows: (28) where
Proof. The analytic form of the fractional-order generalized Laguerre functions of degree i is given by Eq (16), Using Eqs (4), (5) and (16) we have (29) Now, approximate xλk−ν by N+1 terms of fractional generalized Laguerre series yields (30) where bj is given from Eq (22) with u(x) = xλk−ν, and (31) Employing Eqs (29)–(31) we get (32) where
Accordingly, Eq (32) can be written in a vector form as follows: (33)
Eq (33) leads to the desired result.
Fractional-Order Generalized Laguerre Operational Matrix of Fractional Integration
We aim to construct an operational matrix of fractional integration for fractional-order generalized Laguerre vector.
If Jq ϕ(x) is the q (q is an integer value) times repeated integration of fractional-order generalized Laguerre vector ϕ(x), then (34) where P(q) is the operational matrix of classical integration of ϕ(x).
Theorem 5.1 Let ϕ(x) be the fractional-order generalized Laguerre vector and 0 < ν < 1 then (35) where P(ν) is the (N+1) × (N+1) operational matrix of fractional integration of order ν and 0 < ν < 1 in the Riemann-Liouville sense and is defined as follows: (36) and (37)
Proof. From Eqs (16) and (3), we have (38) The approximation of xkλ+ν using N+1 terms of fractional-order generalized Laguerre series, yields (39) where cj is given from Eq (22) with u(x) = xkλ+ν, that is (40) Thanks to Eqs (38) and (39), gives (41) where The vector form of Eq (41) is (42) Eq (42) leads to the desired result.
Application of Fractional-Order Generalized Laguerre Operational Matrices for FDEs
The main aim of this section is to propose two different ways to approximate linear FDEs using the fractional-order generalized Laguerre tau method based on fractional-order Laguerre operational matrices of differentiation and integration such that it can be implemented efficiently. Also, we propose a new collocation method for solve nonlinear FDEs and systems of FDEs based on the fractional-order generalized Laguerre ffunctions.
Operational matrix of fractional derivatives
A direct solution technique is proposed here, to solve linear FDEs using the fractional-order generalized Laguerre tau method in combination with FGLOM.
Now we will implement an efficient algorithm to solve the fractional initial value problem; Eqs (43)–(44). We approximate u(x) and g(x) by fractional-order generalized Laguerre polynomials as (45) (46) where vector G = [g0, …, gN]T is known and C = [c0, …, cN]T is an unknown vector.
By using Theorem 4.1 (relation Eqs (27) and (45)) we have (47) Employing Eqs (45)–(47), the residual RN(x) for Eq (43) can be written as (48) The application of spectral tau scheme, see , provides a system of (N) linear equations, (49) Substituting Eq (45) in Eq (44) yields (50)
Operational matrix of fractional integration
Here, the fractional-order generalized Laguerre tau scheme in conjunction of the derived operational matrix is proposed for solving the linear FDEs. The basic steps of such scheme are: (i) The aforementioned fractional differential equation is converted into a fractional integrated form equation by making use of fractional integration for this equation. (ii) Subsequently, this integrated form equation is approximated by expressing the numerical solution as a linear combination of fractional-order generalized Laguerre functions. (iii) Finally, the problem is transformed into a system of algebraic equations by using the operational matrix of fractional integration of fractional-order generalized Laguerre functions.
In order to show the importance of FGLOM of fractional integration, we apply it to solve the following FDE: (51) with initial condition (52) where γ is a real constant coefficient and also 0 < ν ≤ 1. Moreover, Dν u(x) denotes the Caputo fractional derivative of order ν for u(x) and the value u0 describes the initial condition of u(x). If we apply the Riemann-Liouville integral of order ν on Eq (51) and after making use of Eq (6), we get the integrated form of Eq (51), namely (53) this implies that (54) where Now, approximating u(x) and g(x) by employing the fractional-order generalized Laguerre functions as (55) (56)
In virtue of Theorem 5.1 (relation Eq (35)), the Riemann-Liouville integral of order ν of Eq (55), can be obtained from (57) Employing Eq (55) the residual RN(x) for Eq (54) can be written as (58) Finally, applying the spectral tau method to the residual gives (59) Also from Eq (55) into Eq (52) yields (60) Eqs (59) and (60) generate N of linear equations.
Nonlinear initial FDEs
Regarding the nonlinear fractional initial value problems on the semi-infinite domain, we investigate the spectral fractional-order generalized Laguerre collocation FGLC scheme in combination with FGLOM of fractional derivative to obtain an accurate approximate solution uN(x). The problem is collocated at N nodes of the fractional-order generalized Laguerre-Gauss interpolation defined on Λ. The resulting equations along with the algebraic equation resumed form the initial condition consist an algebraic system of (N+1) equations which may be solved numerically by Newton’s iterative method.
Consider the nonlinear FDE (61) with initial conditions Eq (44), where F can be nonlinear in general.
In order to use FGLOM for this problem, we first expand u(x) and Dν u(x) as Eqs (45) and (47) respectively. By substituting these approximations into Eq (61) we have (62) Substituting Eqs (45) and (26) into Eq (44), we obtain (63) Collocating Eq (62) at the zeros of the fractional-order Laguerre functions provides N equations together with one equation from Eq (63) consist a system of N+1 nonlinear equations. Consequently, the solution uN(x) may be archived by implementing Newton’s iterative scheme.
Corollary 6.1 In particular, the special case for generalized Laguerre polynomials may be obtained directly by taking λ = 0 in the fractional-order Laguerre functions, which are denoted by . However, the classical Laguerre polynomials may be achieved by replacing λ = 1 and α = 0, which are used most frequently in practice and will simply be denoted by Li(x).
FGLC method for solving systems of FDEs
The fractional derivatives can be expressed in terms of the expansion coefficients aij using Eq (27). The implementation of fractional generalized Laguerre collocation method to solve Eqs (64)–(65) is to find uiN(x) ∈ QN(Λ) such that (67) is satisfied exactly at the collocation points , i = 1, ⋯, n, which immediately yields (68) with Eq (65) written in the form (69) This means the system Eq (64) with its initial conditions have been reduced to a system of n(N+1) nonlinear algebraic Eqs (68)–(69), which may be solved by using any standard iteration technique.
We present in this section, several illustrative examples by implementing the proposed spectral algorithms in this article. These examples are chosen such that their exact solutions are known. The results for these examples demonstrate that the proposed methods are accurate, effective and convenient.
Tables 1 and 2 list the values of c0, c1, c2, c3, c4, c5 and c6 with different choices of α and two choices of ν = 1/3 and ν = 1/4. Indeed, we can achieve the exact solution of this problem with all choices of the fractional-order generalized Laguerre parameter α.
Thus we can write Table 3 presents the values c0, c1, c2, … and c6 for several choices of α. Indeed, we can achieve the exact solutions of this problem for all choices of the fractional-order generalized Laguerre parameters α.
The solution of this problem is obtained by applying the technique described in Section 6.2 based on the FGLOM of fractional integration. The maximum absolute error for and various choices of N and α are shown in Table 4. Moreover, the approximate solution obtained by the proposed method for and two choices of N is shown in Fig 1 to make it easier to compare with the analytic solution. From this figure, we see the coherence of the exact and approximate solutions.
Table 5 shows the absolute error function of using spectral fractional-order generalized Laguerre collocation FGLC scheme in combination with FGLOM of fractional derivative with ν, λ and two choices of α at N = 16 in the interval [0, 40]. Fig 2 displays the absolute error function for N = 6, α = 0, and γ = 0.1
We convert Eq (75) into a system of FDEs by changing variable u1(x) = u(x) obtaining: (76) with initial conditions (77)
The maximum absolute error for y(x) = y1(x) using FGLC method at N = 4 and various choices of α are shown in Table 6. It is clear that the approximate solutions are in complete agreement with the exact solutions.
We convert Eq (78) into a system of FDEs by changing variable u1(x) = u(x) obtaining: (79) with initial conditions (80)
In Table 7, we list the results obtained by the fractional-order generalized Laguerre generalized collocation (FGLC) method with various choices of α, N = 10, and ν = λ = 0.5. The present method is compared with the shifted Chebyshev spectral tau (SCT) method given in . As we see from Table 7, it is clear that the result obtained by the present method for each choice of the parameter α is superior to that obtained by SCT method. Fig 3 shows the absolute error function at N = 10, α = 0 and ν = λ = 0.5. The obtained results of this example show that the present method is very accurate by selecting a few number of fractional-order generalized Laguerre generalized functions.
We have defined new orthogonal functions namely FGLFs. The fractional operational matrices of Caputo fractional derivatives and Riemann-Liouville fractional integration were established for these functions. Two efficient spectral tau techniques were proposed based on these fractional operational matrices for solving linear FDEs of order ν (0 < ν < 1) on the half line.
In addition, we have developed the fractional-order generalized Laguerre pseudo-spectral approximation for solving the nonlinear initial value problem of fractional order ν. This technique was extended to solve systems of FDEs. The results of the proposed spectral schemes based on FGLFs were compared with other methods. Several numerical examples were implemented for FDEs and systems of FDEs including linear and nonlinear terms to demonstrate the high accuracy and the efficiency of the proposed techniques. The main idea and techniques developed in this work provide an efficient framework for the collocation method of various nonlinear FDEs on the half line. We also assert that the proposed technique can be extended to solve the one- and two-dimensional space/time fractional partial equations on the half line, (see [49–53]).
This paper was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (14-135-35-RG). The authors, therefore, acknowledge with thanks DSR technical and financial support.
Conceived and designed the experiments: AHB TMT DB EOA AAA. Performed the experiments: AHB TMT DB EOA AAA. Analyzed the data: AHB TMT DB EOA AAA. Contributed reagents/materials/analysis tools: AHB TMT DB EOA AAA. Wrote the paper: AHB TMT DB EOA AAA.
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