Theory and Modern Applications

# The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications

## Abstract

In this paper, the modified generalized Laguerre operational matrix (MGLOM) of Caputo fractional derivatives is constructed and implemented in combination with the spectral tau method for solving linear multi-term FDEs on the half-line. In this approach, truncated modified generalized Laguerre polynomials (MGLP) together with the modified generalized Laguerre operational matrix of Caputo fractional derivatives are analyzed and applied for numerical integration of such equations subject to initial conditions. The modified generalized Laguerre pseudo-spectral approximation based on the modified generalized Laguerre operational matrix is investigated to reduce the nonlinear multi-term FDEs and their initial conditions to a nonlinear algebraic system. Through some numerical experiments, we evaluate the accuracy and efficiency of the proposed methods. The methods are easy to implement and yield very accurate results.

## 1 Introduction

Fractional calculus has been used to develop accurate models of many phenomena of science, engineering, economics, and applied mathematics. These models are found to be best described by FDEs . More recently, Butkovskii et al. [7, 8] used fractional calculus for description of the dynamics of various systems and control processes.

Several numerical and analytical methods have been proposed in the literature for approximating FDEs (see ). In this direction, Atabakzadeh et al.  used the operational matrix of Caputo fractional-order derivatives for Chebyshev polynomials, which was derived in  to solve a system of FDEs. In  Tripathi et al. presented an approximate solution of multi-term FDEs using the operational matrix of fractional integration of the generalized hat basis functions. The authors of  presented a direct solution technique for approximating the linear multi-order FDEs on semi-infinite interval based on the tau method and proposed the modified generalized Laguerre collocation methods for solving the nonlinear FDEs. Recently, Kazem et al.  investigated the operational matrices of the fractional-order Legendre functions and used them together with the spectral tau approximation to solve linear initial value problems of fractional orders. The Taylor matrix method, in which the power is fractional number, was developed for approximating the relaxation-oscillation equation in . A Jacobi operational matrix was proved and applied for approximating the multi-term FDEs in  and a linear fuzzy fractional differential equation in . Ahmadian et al.  derived the operational matrix of Legendre polynomials and applied it for solving a fuzzy fractional differential equation.

The authors of  presented a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. More recently, the authors of  proposed the operational matrices of Riemann-Liouville fractional integration of Chebyshev, Laguerre, modified generalized Laguerre polynomials which are employed in combination with tau approximation for the numerical solution of linear FDEs on finite and semi-infinite intervals. We refer also to the recent papers  in which the authors studied the existence and uniqueness of several classes of FDEs; for application of such equations, see  and the references therein.

In , the authors derived the operational matrix of Riemann-Liouville fractional integration of MGLP and applied it for approximating the linear FDEs on the half-line. In this article, we aim to derive the operational matrix of Caputo fractional derivatives of MGLP. Furthermore, the MGLOM will be used in combination with the spectral tau scheme for approximating linear FDEs, and with the pseudo-spectral scheme for approximating nonlinear FDEs on the half-line. It is worth mentioning here that the Laguerre operational matrix and the generalized Laguerre operational matrix  can be obtained as special cases of MGLP. Finally, the accuracy of the proposed algorithms are demonstrated by test problems.

This paper is organized as follows. In Section 2, we present some properties of MGLP. The MGLOM of fractional derivative is constructed and proved in detail in Section 3. Section 4 is devoted to implementing the tau and collocation spectral schemes in combination with the MGLOM of fractional derivative for approximating two classes of multi-term FDEs. Some numerical experiments are presented in Section 5. The results are summarized in the final section.

## 2 Some basic preliminaries

We give some definitions and properties of fractional derivatives and modified generalized Laguerre polynomials (see [2, 42]).

Definition 2.1 The Riemann-Liouville fractional integral operator of order ν ($\nu >0$) is defined as

$\begin{array}{r}{J}^{\nu }f\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\nu \right)}{\int }_{0}^{x}{\left(x-t\right)}^{\nu -1}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}\nu >0,x>0,\\ {J}^{0}f\left(x\right)=f\left(x\right).\end{array}$
(1)

Definition 2.2 The Caputo fractional derivatives of order ν are defined as

$\begin{array}{r}{D}^{\nu }f\left(x\right)={J}^{m-\nu }{D}^{m}f\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(m-\nu \right)}{\int }_{0}^{x}{\left(x-t\right)}^{m-\nu -1}\frac{{d}^{m}}{d{t}^{m}}f\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\\ \phantom{\rule{1em}{0ex}}m-1<\nu 0,\end{array}$
(2)

where ${D}^{m}$ is the classical differential operator of order m.

For the Caputo derivative, we have

(3)
(4)

where $⌈\nu ⌉$ and $⌊\nu ⌋$ are the ceiling and floor functions respectively, while $N=\left\{1,2,\dots \right\}$ and ${N}_{0}=\left\{0,1,2,\dots \right\}$.

The Caputo fractional differentiation is a linear operation similar to the integer-order differentiation

${D}^{\nu }\left(\lambda f\left(x\right)+\mu g\left(x\right)\right)=\lambda {D}^{\nu }f\left(x\right)+\mu {D}^{\nu }g\left(x\right),$
(5)

where λ and μ are constants.

We recall below some relevant properties of the modified generalized Laguerre polynomials (see ). Now, let $\mathrm{\Lambda }=\left(0,\mathrm{\infty }\right)$ and ${w}^{\left(\alpha ,\beta \right)}\left(x\right)={x}^{\alpha }{e}^{-\beta x}$ be a weight function on Λ in the usual sense. Define

equipped with the following inner product and norm

${\left(u,v\right)}_{{w}^{\left(\alpha ,\beta \right)}}={\int }_{\mathrm{\Lambda }}u\left(x\right)v\left(x\right){w}^{\left(\alpha ,\beta \right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{2em}{0ex}}{\parallel v\parallel }_{{w}^{\left(\alpha ,\beta \right)}}={\left(v,v\right)}_{{w}^{\left(\alpha ,\beta \right)}}^{\frac{1}{2}}.$

Next, let ${L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)$ be the MGLP of degree i for $\alpha >-1$, and $\beta >0$ is defined by

${L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{1}{i!}{x}^{-\alpha }{e}^{\beta x}{\partial }_{x}^{i}\left({x}^{i+\alpha }{e}^{-\beta x}\right),\phantom{\rule{1em}{0ex}}i=1,2,\dots .$

For $\alpha >-1$ and $\beta >0$, we have

$\begin{array}{c}{\partial }_{x}{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)=-\beta {L}_{i-1}^{\left(\alpha +1,\beta \right)}\left(x\right),\hfill \\ {L}_{i+1}^{\left(\alpha ,\beta \right)}\left(x\right)=\frac{1}{i+1}\left[\left(2i+\alpha +1-\beta x\right){L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)-\left(i+\alpha \right){L}_{i-1}^{\left(\alpha ,\beta \right)}\left(x\right)\right],\phantom{\rule{1em}{0ex}}i=1,2,\dots ,\hfill \end{array}$

where ${L}_{0}^{\left(\alpha ,\beta \right)}\left(x\right)=1$ and ${L}_{1}^{\left(\alpha ,\beta \right)}\left(x\right)=-\beta x+\frac{\mathrm{\Gamma }\left(\alpha +2\right)}{\mathrm{\Gamma }\left(\alpha +1\right)}$.

The set of MGLP is the ${L}_{{w}^{\left(\alpha ,\beta \right)}}^{2}\left(\mathrm{\Lambda }\right)$-orthogonal system, namely

${\int }_{0}^{\mathrm{\infty }}{L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right){L}_{k}^{\left(\alpha ,\beta \right)}\left(x\right){w}^{\left(\alpha ,\beta \right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={h}_{k}{\delta }_{jk},$
(6)

where ${\delta }_{jk}$ is the Kronecker function and ${h}_{k}=\frac{\mathrm{\Gamma }\left(k+\alpha +1\right)}{{\beta }^{\alpha +1}k!}$.

The modified generalized Laguerre polynomial of degree i on the interval Λ is given by

${L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)=\sum _{k=0}^{i}{\left(-1\right)}^{k}\frac{\mathrm{\Gamma }\left(i+\alpha +1\right){\beta }^{k}}{\mathrm{\Gamma }\left(k+\alpha +1\right)\left(i-k\right)!k!}{x}^{k},\phantom{\rule{1em}{0ex}}i=0,1,\dots ,$
(7)

where ${L}_{i}^{\left(\alpha ,\beta \right)}\left(0\right)=\frac{\mathrm{\Gamma }\left(i+\alpha +1\right)}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(i+1\right)}$.

The special value

${D}^{q}{L}_{i}^{\left(\alpha ,\beta \right)}\left(0\right)=\frac{{\left(-1\right)}^{q}{\beta }^{q}\mathrm{\Gamma }\left(i+\alpha +1\right)}{\left(i-q\right)!\mathrm{\Gamma }\left(q+\alpha +1\right)},\phantom{\rule{1em}{0ex}}i⩾q,$
(8)

will be of important use later.

## 3 MGLOM of Caputo fractional derivatives

Let $u\left(x\right)\in {L}_{{w}^{\left(\alpha ,\beta \right)}}^{2}\left(\mathrm{\Lambda }\right)$, then $u\left(x\right)$ may be expressed in terms of MGLP as

$u\left(x\right)=\sum _{j=0}^{\mathrm{\infty }}{a}_{j}{L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right),\phantom{\rule{2em}{0ex}}{a}_{j}=\frac{1}{{h}_{k}}{\int }_{0}^{\mathrm{\infty }}u\left(x\right){L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right){w}^{\left(\alpha ,\beta \right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}j=0,1,2,\dots .$
(9)

In particular applications, the MGLP up to degree $N+1$ are considered. Then we have

${u}_{N}\left(x\right)=\sum _{j=0}^{N}{a}_{j}{L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right)={C}^{T}\varphi \left(x\right),$
(10)

where the modified generalized Laguerre coefficient vector C and the modified generalized Laguerre vector $\varphi \left(x\right)$ are given by

${C}^{T}=\left[{c}_{0},{c}_{1},\dots ,{c}_{N}\right],\phantom{\rule{2em}{0ex}}\varphi \left(x\right)={\left[{L}_{0}^{\left(\alpha ,\beta \right)}\left(x\right),{L}_{1}^{\left(\alpha ,\beta \right)}\left(x\right),\dots ,{L}_{N}^{\left(\alpha ,\beta \right)}\left(x\right)\right]}^{T},$
(11)

then the derivative of the vector $\varphi \left(x\right)$ can be expressed by

$\frac{d\varphi \left(x\right)}{dx}={\mathbf{D}}^{\left(1\right)}\varphi \left(x\right),$
(12)

where ${\mathbf{D}}^{\left(1\right)}$ is the $\left(N+1\right)×\left(N+1\right)$ operational matrix of derivative given by

${\mathbf{D}}^{\left(1\right)}=-\beta \left(\begin{array}{cccccccc}0& 0& 0& 0& 0& \cdots & 0& 0\\ 1& 0& 0& 0& 0& \cdots & 0& 0\\ 1& 1& 0& 0& 0& \cdots & 0& 0\\ 1& 1& 1& 0& 0& \cdots & 0& 0\\ 1& 1& 1& 1& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 1& 1& 1& 1& 1& \cdots & 1& 0\end{array}\right).$

By using Eq. (12), it is clear that

$\frac{{d}^{n}\varphi \left(x\right)}{d{x}^{n}}={\left({\mathbf{D}}^{\left(1\right)}\right)}^{n}\varphi \left(x\right),$
(13)

where $n\in N$ and the superscript in ${\mathbf{D}}^{\left(1\right)}$ denotes matrix powers. Thus

${\mathbf{D}}^{\left(n\right)}={\left({\mathbf{D}}^{\left(1\right)}\right)}^{n},\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
(14)

Lemma 3.1 Let ${L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)$ be a modified generalized Laguerre polynomial, then

${D}^{\nu }{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)=0,\phantom{\rule{1em}{0ex}}i=0,1,\dots ,⌈\nu ⌉-1,\nu >0.$
(15)

Proof Immediately, if we use Eqs. (4) and (5) in Eq. (7), the lemma can be proved. □

In what follows, we derive the operational matrix of Caputo fractional derivatives for MGLP.

Theorem 3.2 Let $\varphi \left(x\right)$ be a modified generalized Laguerre vector defined in Eq. (11) and consider $\nu >0$, then

${D}^{\nu }\varphi \left(x\right)\simeq {\mathbf{D}}^{\left(\nu \right)}\varphi \left(x\right),$
(16)

where ${\mathbf{D}}^{\left(\nu \right)}$ is the $\left(N+1\right)×\left(N+1\right)$ operational matrix of Caputo fractional derivatives of order ν and is given by

${\mathbf{D}}^{\left(\nu \right)}=\left(\begin{array}{ccccc}0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ 0& 0& 0& \cdots & 0\\ {\mathrm{\Omega }}_{\nu }\left(⌈\nu ⌉,0\right)& {\mathrm{\Omega }}_{\nu }\left(⌈\nu ⌉,1\right)& {\mathrm{\Omega }}_{\nu }\left(⌈\nu ⌉,2\right)& \cdots & {\mathrm{\Omega }}_{\nu }\left(⌈\nu ⌉,N\right)\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathrm{\Omega }}_{\nu }\left(i,0\right)& {\mathrm{\Omega }}_{\nu }\left(i,1\right)& {\mathrm{\Omega }}_{\nu }\left(i,2\right)& \cdots & {\mathrm{\Omega }}_{\nu }\left(i,N\right)\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathrm{\Omega }}_{\nu }\left(N,0\right)& {\mathrm{\Omega }}_{\nu }\left(N,1\right)& {\mathrm{\Omega }}_{\nu }\left(N,2\right)& \cdots & {\mathrm{\Omega }}_{\nu }\left(N,N\right)\end{array}\right),$
(17)

where

${\mathrm{\Omega }}_{\nu }\left(i,j\right)=\sum _{k=⌈\nu ⌉}^{i}\sum _{\ell =0}^{j}\frac{{\left(-1\right)}^{k+\ell }\phantom{\rule{0.25em}{0ex}}{\beta }^{\nu }j!\mathrm{\Gamma }\left(i+\alpha +1\right)\mathrm{\Gamma }\left(k-\nu +\alpha +\ell +1\right)}{\left(i-k\right)!\left(j-\ell \right)!\ell !\mathrm{\Gamma }\left(k-\nu +1\right)\mathrm{\Gamma }\left(k+\alpha +1\right)\mathrm{\Gamma }\left(\alpha +\ell +1\right)}.$

Note that in ${\mathbf{D}}^{\left(\nu \right)}$, the first $⌈\nu ⌉$ rows are all zero.

Proof The analytic form of the MGLP ${L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)$ of degree i is obtained by (7). Making use of (4), (5) and (7), we get

$\begin{array}{rl}{D}^{\nu }{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)& =\sum _{k=0}^{i}{\left(-1\right)}^{k}\frac{{\beta }^{k}\mathrm{\Gamma }\left(i+\alpha +1\right)}{\left(i-k\right)!k!\mathrm{\Gamma }\left(k+\alpha +1\right)}{D}^{\nu }{x}^{k}\\ =\sum _{k=⌈\nu ⌉}^{i}{\left(-1\right)}^{k}\frac{{\beta }^{k}\mathrm{\Gamma }\left(i+\alpha +1\right)}{\left(i-k\right)!\mathrm{\Gamma }\left(k-\nu +1\right)\mathrm{\Gamma }\left(k+\alpha +1\right)}{x}^{k-\nu },\phantom{\rule{1em}{0ex}}i=⌈\nu ⌉,\dots ,N.\end{array}$
(18)

Now, approximating ${x}^{k-\nu }$ by $N+1$ terms of modified generalized Laguerre series, we have

${x}^{k-\nu }=\sum _{j=0}^{N}{b}_{j}{L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right),$
(19)

where ${b}_{j}$ is given from (9) with $u\left(x\right)={x}^{k-\nu }$, and

${b}_{j}=\sum _{\ell =0}^{j}{\left(-1\right)}^{\ell }\frac{{\beta }^{-k+\nu }j!\phantom{\rule{0.25em}{0ex}}\mathrm{\Gamma }\left(k-\nu +\alpha +\ell +1\right)}{\left(j-\ell \right)!\left(\ell \right)!\mathrm{\Gamma }\left(\ell +\alpha +1\right)}.$
(20)

Employing Eqs. (18)-(20), we get

${D}^{\nu }{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)=\sum _{j=0}^{N}{\mathrm{\Omega }}_{\nu }\left(i,j\right){L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right),\phantom{\rule{1em}{0ex}}i=⌈\nu ⌉,\dots ,N,$
(21)

where

${\mathrm{\Omega }}_{ijk}=\sum _{k=⌈\nu ⌉}^{i}\sum _{\ell =0}^{j}\frac{{\left(-1\right)}^{k+\ell }\phantom{\rule{0.25em}{0ex}}{\beta }^{\nu }j!\mathrm{\Gamma }\left(i+\alpha +1\right)\mathrm{\Gamma }\left(k-\nu +\alpha +\ell +1\right)}{\left(i-k\right)!\left(j-\ell \right)!\ell !\mathrm{\Gamma }\left(k-\nu +1\right)\mathrm{\Gamma }\left(k+\alpha +1\right)\mathrm{\Gamma }\left(\alpha +\ell +1\right)}.$
(22)

Accordingly, Eq. (21) can be written in a vector form as follows:

${D}^{\nu }{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)\simeq \left[{\mathrm{\Omega }}_{\nu }\left(i,0\right),{\mathrm{\Omega }}_{\nu }\left(i,1\right),{\mathrm{\Omega }}_{\nu }\left(i,2\right),\dots ,{\mathrm{\Omega }}_{\nu }\left(i,N\right)\right]\varphi \left(x\right),\phantom{\rule{1em}{0ex}}i=⌈\nu ⌉,\dots ,N.$
(23)

Also, according to Lemma 3.1, we can write

${D}^{\nu }{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)\simeq \left[0,0,0,\dots ,0\right]\varphi \left(x\right),\phantom{\rule{1em}{0ex}}i=0,1,\dots ,⌈\nu ⌉-1.$
(24)

A combination of Eqs. (23) and (24) leads to the desired result. □

Remark In the case of $\nu =n\in N$, Theorem 3.2 gives the same result as Eq. (13).

## 4 Applications of the MGLOM for FDEs

The main aim of this section is to approximate linear and nonlinear multi-term FDEs using the MGLOM based on modified generalized Laguerre tau and pseudo-spectral approximations respectively.

### 4.1 Linear multi-term FDEs

Consider the linear FDE

(25)

with initial conditions

${u}^{\left(i\right)}\left(0\right)={d}_{i},\phantom{\rule{1em}{0ex}}i=0,\dots ,m-1,$
(26)

where ${\gamma }_{j}$ ($j=1,\dots ,k+1$) are real constant coefficients, $m-1<\nu \le m$, $0<{\zeta }_{1}<{\zeta }_{2}<\cdots <{\zeta }_{k}<\nu$ and ${d}_{i}$ ($i=0,\dots ,m-1$) are the initial values of $u\left(x\right)$.

To solve the fractional initial value problem, (25)-(26), we approximate $u\left(x\right)$ and $g\left(x\right)$ by modified generalized Laguerre polynomials as

$u\left(x\right)\simeq \sum _{i=0}^{N}{c}_{i}{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)={C}^{T}\varphi \left(x\right),$
(27)
$g\left(x\right)\simeq \sum _{i=0}^{N}{g}_{i}{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)={G}^{T}\varphi \left(x\right),$
(28)

where a vector $G={\left[{g}_{0},\dots ,{g}_{N}\right]}^{T}$ is known and $C={\left[{c}_{0},\dots ,{c}_{N}\right]}^{T}$ is an unknown vector.

By using Theorem 3.2 (Eqs. (16) and (27)), we have

${D}^{\nu }u\left(x\right)\simeq {C}^{T}{D}^{\nu }\varphi \left(x\right)={C}^{T}{D}^{\left(\nu \right)}\varphi \left(x\right),$
(29)
${D}^{{\zeta }_{j}}u\left(x\right)\simeq {C}^{T}{D}^{{\zeta }_{j}}\varphi \left(x\right)={C}^{T}{D}^{\left({\zeta }_{j}\right)}\varphi \left(x\right),\phantom{\rule{1em}{0ex}}j=1,\dots ,k.$
(30)

If we use (27)-(30), then the residual ${R}_{N}\left(x\right)$ for Eq. (25) can be given by

${R}_{N}\left(x\right)=\left({C}^{T}{D}^{\left(\nu \right)}-{C}^{T}\sum _{j=1}^{k}{\gamma }_{j}{\mathbf{D}}^{\left({\zeta }_{j}\right)}-{\gamma }_{k+1}{C}^{T}-{G}^{T}\right)\varphi \left(x\right).$
(31)

As in a typical tau method, see [20, 30, 46], we generate $\left(N-m+1\right)$ linear equations by applying

$〈{R}_{N}\left(x\right),{L}_{j}^{\left(\alpha ,\beta \right)}\left(x\right)〉=0,\phantom{\rule{1em}{0ex}}j=0,1,\dots ,N-m.$
(32)

Also, by substituting Eqs. (13) and (27) into Eq (26), we get

${u}^{\left(i\right)}\left(0\right)={C}^{T}{\mathbf{D}}^{\left(i\right)}\varphi \left(0\right)={d}_{i},\phantom{\rule{1em}{0ex}}i=0,1,\dots ,m-1.$
(33)

Eqs. (32) and (33) generate $\left(N-m+1\right)$ and m sets of linear equations, respectively. These linear equations can be solved for unknown coefficients of the vector C. Consequently, $u\left(x\right)$ given in Eq. (27) can be calculated, which gives the solution of the initial value problem in Eqs. (25) and (26).

### 4.2 Nonlinear multi-term FDEs

In this section, in order to show the high importance of the MGLOM of Caputo fractional derivatives, we apply it to approximate nonlinear multi-term FDEs. Regarding such problems on the interval Λ, we investigate the modified generalized Laguerre pseudo-spectral scheme based on the MGLOM to obtain the numerical solution ${u}_{N}\left(x\right)$. The nonlinear FDE is collocated at $\left(N-m+1\right)$ nodes of the modified generalized Laguerre-Gauss interpolation on the interval Λ, and then the problem reduces to the $\left(N+1\right)$ system of algebraic equations.

Consider the nonlinear FDE

(34)

with initial conditions (26), where F can be nonlinear in general.

In order to use the MGLOM for approximating (34), we expand $u\left(x\right)$, ${D}^{\nu }u\left(x\right)$ and ${D}^{{\beta }_{j}}u\left(x\right)$ for $j=1,\dots ,k$ as (27), (29) and (30) respectively, and then (34) becomes

${C}^{T}{\mathbf{D}}^{\left(\nu \right)}\varphi \left(x\right)=F\left(x,{C}^{T}\varphi \left(x\right),{C}^{T}{\mathbf{D}}^{\left({\beta }_{1}\right)}\varphi \left(x\right),\dots ,{C}^{T}{\mathbf{D}}^{\left({\beta }_{k}\right)}\varphi \left(x\right)\right).$
(35)

Substituting (27) and (13) in (34) gives

${u}^{\left(i\right)}\left(0\right)={C}^{T}{\mathbf{D}}^{\left(i\right)}\varphi \left(0\right)={d}_{i},\phantom{\rule{1em}{0ex}}i=0,1,\dots ,m-1.$
(36)

Collocating (35) at $\left(N-m+1\right)$ nodes of modified generalized Laguerre-Gauss interpolation yields

$\begin{array}{r}{C}^{T}{\mathbf{D}}^{\left(\nu \right)}\varphi \left({x}_{i}^{\left(\alpha ,\beta \right)}\right)=F\left({x}_{i}^{\left(\alpha ,\beta \right)},{C}^{T}\varphi \left({x}_{i}^{\left(\alpha ,\beta \right)}\right),{C}^{T}{\mathbf{D}}^{\left({\beta }_{1}\right)}\varphi \left({x}_{i}^{\left(\alpha ,\beta \right)}\right),\dots ,{C}^{T}{\mathbf{D}}^{\left({\beta }_{k}\right)}\varphi \left({x}_{i}^{\left(\alpha ,\beta \right)}\right)\right),\\ \phantom{\rule{1em}{0ex}}i=0,1,\dots ,N-m+1.\end{array}$
(37)

For every two special choices of the modified generalized Laguerre parameters α and β, the above-mentioned system of algebraic equations together with (36) gives the $\left(N+1\right)$ nonlinear system of equations which may be solved by Newton’s iterative method.

## 5 Illustrative examples

To illustrate the effectiveness of the proposed algorithms, six illustrative examples are implemented by the two methods proposed in the previous section. The results obtained by the proposed algorithms reveal that these algorithms are very effective and convenient for approximating the linear and nonlinear FDEs on a semi-infinite interval.

Example 1 Consider the FDE

$\begin{array}{r}{D}^{2}u\left(x\right)+{D}^{\frac{3}{2}}u\left(x\right)+u\left(x\right)={x}^{2}+2+\frac{2.6666666667}{\mathrm{\Gamma }\left(0.5\right)}{x}^{1.5},\phantom{\rule{2em}{0ex}}u\left(0\right)=0,\phantom{\rule{2em}{0ex}}{u}^{\prime }\left(0\right)=0,\\ \phantom{\rule{1em}{0ex}}x\in \mathrm{\Lambda }.\end{array}$
(38)

The exact solution is $u\left(x\right)={x}^{2}$.

Now, we implement the tau technique based on the MGLOM of fractional derivative with $N=2$, then the approximate solution may be written as

$u\left(x\right)={c}_{0}{L}_{0}^{\left(\alpha ,\beta \right)}\left(x\right)+{c}_{1}{L}_{1}^{\left(\alpha ,\beta \right)}\left(x\right)+{c}_{2}{L}_{2}^{\left(\alpha ,\beta \right)}\left(x\right)={C}^{T}\varphi \left(x\right).$

Here, we get the following operational matrices:

$\begin{array}{c}{\mathbf{D}}^{\left(1\right)}=-\beta \left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 1& 1& 0\end{array}\right),\phantom{\rule{2em}{0ex}}{\mathbf{D}}^{\left(2\right)}={\beta }^{2}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right),\hfill \\ {\mathbf{D}}^{\left(\frac{3}{2}\right)}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ {\mathrm{\Omega }}_{\frac{3}{2}}\left(2,0\right)& {\mathrm{\Omega }}_{\frac{3}{2}}\left(2,1\right)& {\mathrm{\Omega }}_{\frac{3}{2}}\left(2,2\right)\end{array}\right),\phantom{\rule{2em}{0ex}}G=\left(\begin{array}{c}{g}_{0}\\ {g}_{1}\\ {g}_{2}\end{array}\right),\hfill \end{array}$

where ${g}_{j}$ and ${\mathrm{\Omega }}_{\nu }\left(i,j\right)$ are evaluated from (9) and (17), respectively.

Firstly, applying the tau method to (38) (see relation (32)) yields

${c}_{0}+\left[{\mathrm{\Omega }}_{\frac{3}{2}}\left(2,0\right)+{\beta }^{2}\right]{c}_{2}-{g}_{o}=0.$
(39)

Secondly, applying (33) to the initial conditions yields

${c}_{0}+\left(\alpha +1\right){c}_{1}+\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}{c}_{2}=0,$
(40)
$-\beta {c}_{1}-\beta \left(\alpha +2\right){c}_{2}=0.$
(41)

Finally, solving (39)-(41), we have the three unknown coefficients with various choices of α and β which are given in Table 1. Then, we have

${c}_{0}=\frac{{\alpha }^{2}+3\alpha +2}{{\beta }^{2}},\phantom{\rule{2em}{0ex}}{c}_{1}=\frac{-2\alpha -4}{{\beta }^{2}},\phantom{\rule{2em}{0ex}}{c}_{2}=\frac{2}{{\beta }^{2}}.$

Consequentially, we can write

$u\left(x\right)=\left(\begin{array}{lll}{c}_{0}& {c}_{1}& {c}_{2}\end{array}\right)\left(\begin{array}{c}{L}_{0}^{\left(\alpha ,\beta \right)}\left(x\right)\\ {L}_{1}^{\left(\alpha ,\beta \right)}\left(x\right)\\ {L}_{2}^{\left(\alpha ,\beta \right)}\left(x\right)\end{array}\right)={x}^{2},$

which is the exact solution.

Example 2 Consider the equation

$\begin{array}{r}{D}^{2}u\left(x\right)-2{D}^{\frac{5}{3}}u\left(x\right)+{D}^{\frac{2}{3}}u\left(x\right)+u\left(x\right)={x}^{3}+6x-\frac{12}{\mathrm{\Gamma }\left(\frac{7}{3}\right)}{x}^{\frac{4}{3}}+\frac{6}{\mathrm{\Gamma }\left(\frac{10}{3}\right)}{x}^{\frac{7}{3}},\\ u\left(0\right)=0,\phantom{\rule{2em}{0ex}}{u}^{\prime }\left(0\right)=0,\phantom{\rule{1em}{0ex}}x\in \mathrm{\Lambda },\end{array}$
(42)

whose exact solution is given by $u\left(x\right)={x}^{3}$.

By applying the technique described in Section 4.1 with $N=3$ and $x\in \mathrm{\Lambda }$, we approximate the solution as

$u\left(x\right)=\sum _{i=0}^{3}{c}_{i}{L}_{i}^{\left(\alpha ,\beta \right)}\left(x\right)={C}^{T}\varphi \left(x\right).$

Here, we have

$\begin{array}{c}{\mathbf{D}}^{\left(2\right)}={\beta }^{2}\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 2& 1& 0& 0\end{array}\right),\phantom{\rule{2em}{0ex}}{\mathbf{D}}^{\left(\frac{5}{3}\right)}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ {\mathrm{\Omega }}_{\frac{5}{3}}\left(2,0\right)& {\mathrm{\Omega }}_{\frac{5}{3}}\left(2,1\right)& {\mathrm{\Omega }}_{\frac{5}{3}}\left(2,2\right)& {\mathrm{\Omega }}_{\frac{5}{3}}\left(2,3\right)\\ {\mathrm{\Omega }}_{\frac{5}{3}}\left(3,0\right)& {\mathrm{\Omega }}_{\frac{5}{3}}\left(3,1\right)& {\mathrm{\Omega }}_{\frac{5}{3}}\left(3,2\right)& {\mathrm{\Omega }}_{\frac{5}{3}}\left(3,3\right)\end{array}\right),\hfill \\ {\mathbf{D}}^{\left(\frac{2}{3}\right)}=\left(\begin{array}{cccc}0& 0& 0& 0\\ {\mathrm{\Omega }}_{\frac{2}{3}}\left(1,0\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(1,1\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(1,2\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(1,3\right)\\ {\mathrm{\Omega }}_{\frac{2}{3}}\left(2,0\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(2,1\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(2,2\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(2,3\right)\\ {\mathrm{\Omega }}_{\frac{2}{3}}\left(3,0\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(3,1\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(3,2\right)& {\mathrm{\Omega }}_{\frac{2}{3}}\left(3,3\right)\end{array}\right),\phantom{\rule{2em}{0ex}}G=\left(\begin{array}{c}{g}_{0}\\ {g}_{1}\\ {g}_{2}\\ {g}_{3}\end{array}\right),\hfill \end{array}$

where ${g}_{j}$ and ${\mathrm{\Omega }}_{\nu }\left(i,j\right)$ are computed from (9) and (17), respectively.

Therefore using (32), we obtain

$\begin{array}{r}{c}_{0}+\left[1+{\mathrm{\Omega }}_{\frac{2}{3}}\left(1,0\right)\right]{c}_{1}+\left[1-2{\mathrm{\Omega }}_{\frac{5}{3}}\left(2,0\right)+{\mathrm{\Omega }}_{\frac{2}{3}}\left(2,0\right)\right]{c}_{2}\\ \phantom{\rule{1em}{0ex}}+\left[2-2{\mathrm{\Omega }}_{\frac{5}{3}}\left(3,0\right)\right]{c}_{3}-{g}_{0}=0,\end{array}$
(43)
$\begin{array}{r}\left[1+{\mathrm{\Omega }}_{\frac{2}{3}}\left(1,1\right)\right]{c}_{1}+\left[{\mathrm{\Omega }}_{\frac{2}{3}}\left(2,1\right)-2{\mathrm{\Omega }}_{\frac{5}{3}}\left(2,1\right)\right]{c}_{2}\\ \phantom{\rule{1em}{0ex}}+\left[1-2{\mathrm{\Omega }}_{\frac{5}{3}}\left(3,1\right)+{\mathrm{\Omega }}_{\frac{2}{3}}\left(3,1\right)\right]{c}_{3}-{g}_{1}=0.\end{array}$
(44)

Now, by applying (33), we have

$\begin{array}{r}{C}^{T}\varphi \left(0\right)={c}_{0}+\left(\alpha +1\right){c}_{1}+\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}{c}_{2}+\frac{\left(\alpha +1\right)\left(\alpha +2\right)\left(\alpha +3\right)}{6}{c}_{3}=0,\\ {C}^{T}{\mathbf{D}}^{\left(1\right)}\varphi \left(0\right)=-\beta {c}_{1}-\beta \left(\alpha +2\right){c}_{2}-\frac{\beta \left(\alpha +3\right)\left(\alpha +2\right)}{2}{c}_{3}=0.\end{array}$
(45)

Finally, by solving (43)-(45), we have the four unknown coefficients with various choices of α and β which are displayed in Table 2. Then we get

${c}_{0}=\frac{{\alpha }^{3}+6{\alpha }^{2}+11\alpha +6}{{\beta }^{3}},\phantom{\rule{2em}{0ex}}{c}_{1}=\frac{-3{\alpha }^{2}-15\alpha -18}{{\beta }^{3}},\phantom{\rule{2em}{0ex}}{c}_{2}=\frac{6\alpha +18}{{\beta }^{3}},\phantom{\rule{2em}{0ex}}{c}_{3}=\frac{-6}{{\beta }^{3}},$

and

$u\left(x\right)=\left(\begin{array}{llll}{c}_{0}& {c}_{1}& {c}_{2}& {c}_{3}\end{array}\right)\left(\begin{array}{c}{L}_{0}^{\left(\alpha ,\beta \right)}\left(x\right)\\ {L}_{1}^{\left(\alpha ,\beta \right)}\left(x\right)\\ {L}_{2}^{\left(\alpha ,\beta \right)}\left(x\right)\\ {L}_{3}^{\left(\alpha ,\beta \right)}\left(x\right)\end{array}\right)={x}^{3}.$

Example 3 Consider the FDE

${D}^{2}u\left(x\right)+{D}^{\frac{3}{2}}u\left(x\right)+3u\left(x\right)=g\left(x\right),\phantom{\rule{2em}{0ex}}u\left(0\right)=1,\phantom{\rule{2em}{0ex}}{u}^{\prime }\left(0\right)=0,\phantom{\rule{1em}{0ex}}x\in \left[0,20\right],$
(46)

where

$g\left(x\right)=\left(3-{\gamma }^{2}\right)cos\left(\gamma x\right)+\frac{1}{\mathrm{\Gamma }\left(-\frac{3}{2}\right)}{\int }_{0}^{x}{\left(x-t\right)}^{-\frac{3}{2}-1}u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$

and the exact solution is given by $u\left(x\right)=cos\left(\gamma x\right)$.

The maximum absolute errors using the MGLOM together with the tau approximation, for $\gamma =0.1$ and various choices of N, α and β are shown in Table 3. Moreover, the approximate solution obtained by this method for $\alpha =2$, $\beta =4$ and two choices of N and γ is shown in Figures 1 and 2. Also, maximum absolute error for $N=10$ and different values of γ, α, and β are displayed in Table 4.

Example 4 Consider the nonlinear FDE

$\begin{array}{r}{D}^{2}u\left(x\right)+\left[{D}^{\frac{4}{3}}u\left(x\right){D}^{\frac{4}{3}}u\left(x\right)\right]+u\left(x\right)={\gamma }^{2}{e}^{\gamma x}+{\gamma }^{\frac{8}{3}}{e}^{\gamma 2x}+{e}^{\gamma x},\\ u\left(0\right)=1,\phantom{\rule{2em}{0ex}}{u}^{\prime }\left(0\right)=\gamma ,\phantom{\rule{1em}{0ex}}x\in \left[0,10\right],\end{array}$
(47)

where $u\left(x\right)={e}^{\gamma x}$ is the exact solution.

The solution of this problem is obtained by applying the modified generalized Laguerre collocation method. The approximate solution obtained by the proposed method for $\gamma =\left\{0.1,0.01\right\}$, $\alpha =2$, $\beta =3$ and two choices of N are shown in Figures 3 and 4 to make it easier to compare with the analytic solution. Moreover, the absolute errors for $\gamma =\left\{0.1,0.01\right\}$, $\alpha =2$, $\beta =3$ and $N=5$ are given in Figures 5 and 6.

Example 5 Consider the nonlinear FDE

$\begin{array}{r}{D}^{2}u\left(x\right)+{D}^{\frac{5}{3}}u\left(x\right)+{u}^{2}\left(x\right)=g\left(x\right),\\ u\left(0\right)=0,{u}^{\prime }\left(0\right)=\gamma ,\phantom{\rule{1em}{0ex}}x\in \left[0,10\right],\end{array}$
(48)

where

$g\left(x\right)=-{\gamma }^{2}sin\left(\gamma x\right)+{sin}^{2}\left(\gamma x\right)+\frac{1}{\mathrm{\Gamma }\left(-\frac{5}{3}\right)}{\int }_{0}^{x}{\left(x-t\right)}^{-\frac{5}{3}-1}u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,$

and the exact solution is given by $u\left(x\right)=sin\left(\gamma x\right)$.

The solution of this problem is implemented by the modified generalized Laguerre collocation method. The maximum absolute error obtained by the proposed method for various choices of N, α and β, where $\gamma =0.1$, is given in Figure 7. Moreover, the approximate solution obtained by the proposed method for $\alpha =5$, $\beta =3$ and two choices of N is shown in Figure 8.

Example 6 Consider the following nonlinear initial value problem:

${D}^{\nu }u\left(x\right)+{u}^{2}\left(x\right)=\mathrm{\Gamma }\left(\nu +2\right)x+\frac{6{x}^{3-\nu }}{\mathrm{\Gamma }\left(4-\nu \right)}+{\left({x}^{\nu +1}+{x}^{3}\right)}^{2},\phantom{\rule{1em}{0ex}}0<\nu \le 2,$

whose exact solution is given by $u\left(x\right)={x}^{\nu +1}+{x}^{3}$.

Comparison between the curves of exact solutions and the approximate solutions at $\alpha =3$ and $\beta =5$ of the proposed problem subject to $u\left(0\right)={u}^{\prime }\left(0\right)=0$ in case of $N=12$ and four different fractional orders $\nu =1.2,1.4,1.6,1.8$ are shown in Figures 9 and 10. The maximum absolute errors at $\nu =1.5$ for various choices of α, β and N in the interval $\left[0,1\right]$ are shown in Table 5. Moreover, Figures 11 and 12 display a comparison between the curves of exact solutions and the approximate solutions at $\alpha =3$ and $\beta =5$ of the proposed problem subject to $u\left(0\right)=0$ in case of $N=12$ and four different fractional orders $\nu =0.2,0.4,0.6,0.8$. From Figures 9-12, it can be seen that the numerical solutions are in complete agreement with the exact solutions for all values of ν. Also, from the numerical results implemented in this example and the previous ones, the classical Laguerre polynomial ($\alpha =0$, $\beta =1$), which is used most frequently in practice, is not the best one, especially when we are approximating the solution of FDEs.

## 6 Conclusions

In this paper, we have proposed two efficient spectral methods based on modified generalized Laguerre polynomials for tackling linear and nonlinear FDEs on the half-line. In these methods, the problem is reduced to the solution of a system of algebraic equations in the expansion coefficient of the solution. Illustrative examples were implemented to demonstrate the applicability of the proposed algorithms. The computational results show that the proposed methods can be effectively used in numerical solution of a time-dependent fractional partial differential equation and other problems on the half-line.

## References

1. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Singapore; 2012.

2. Miller K, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.

3. Ortigueira M: Introduction to fraction linear systems. Part 1: continuous-time case. 147. IEE Proceedings Vision, Image, Signal Processing 2000, 6-70.

4. Herzallah MAE, Baleanu D: Fractional Euler-Lagrange equations revisited. Nonlinear Dyn. 2012, 69: 977-982. 10.1007/s11071-011-0319-5

5. Baleanu D: New applications of fractional variational principles. Rep. Math. Phys. 2008, 61: 199-206. 10.1016/S0034-4877(08)80007-9

6. Baleanu D, Muslih S: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 2005, 72: 119-121. 10.1238/Physica.Regular.072a00119

7. Butkovskii AG, Postnov SS, Postnova EA: Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation. Autom. Remote Control 2013, 74: 543-574. 10.1134/S0005117913040012

8. Butkovskii AG, Postnov SS, Postnova EA: Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: modeling and hardware implementation. Autom. Remote Control 2013, 74: 725-749. 10.1134/S0005117913050019

9. Doha EH, Bhrawy AH, Ezz-Eldien SS: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 2011, 35: 5662-5672. 10.1016/j.apm.2011.05.011

10. Song J, Yin F, Cao X, Lu F: Fractional variational iteration method versus Adomian’s decomposition method in some fractional partial differential equations. J. Appl. Math. 2013., 2013: Article ID 10

11. El-Ajou A, Abu Arqub O, Momani S: Solving fractional two-point boundary value problems using continuous analytic method. Ain Shams Engineering Journal 2013. 10.1016/j.asej.2012.11.010

12. Saadatmandi A: Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Model. 2013. 10.1016/j.apm.2013.08.007

13. Bhrawy AH, Alshomrani M: A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Differ. Equ. 2012., 2012: Article ID 8

14. Baleanu D, Alipour M, Jafari H: The Bernstein operational matrices for solving the fractional quadratic Riccati differential equations with the Riemann-Liouville derivative. Abstr. Appl. Anal. 2013., 2013: Article ID 461970

15. Jankowski T: Initial value problems for neutral fractional differential equations involving a Riemann-Liouville derivative. Appl. Math. Comput. 2013, 219: 7772-7776. 10.1016/j.amc.2013.02.001

16. Bhrawy AH, Alghamdi MA: A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals. Bound. Value Probl. 2012., 2012: Article ID 62

17. Ma X, Huang C: Spectral collocation method for linear fractional integro-differential equations. Appl. Math. Model. 2013. 10.1016/j.apm.2013.08.013

18. Ghandehari MAM, Ranjbar M: A numerical method for solving a fractional partial differential equation through converting it into an NLP problem. Comput. Math. Appl. 2013, 65: 975-982. 10.1016/j.camwa.2013.01.003

19. Atabakzadeh MH, Akrami MH, Erjaee GH: Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Appl. Math. Model. 2013. 10.1016/j.apm.2013.04.019

20. Doha EH, Bhrawy AH, Ezz-Eldien SS: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 2011, 62: 2364-2373. 10.1016/j.camwa.2011.07.024

21. Tripathi MP, Baranwal VK, Pandey RK, Singh OP: A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 1327-1340. 10.1016/j.cnsns.2012.10.014

22. Baleanu D, Bhrawy AH, Taha TM: A modified generalized Laguerre spectral method for fractional differential equations on the half line. Abstr. Appl. Anal. 2013., 2013: Article ID 413529

23. Kazem S, Abbasbandy S, Kumar S: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 2013, 37: 5498-5510. 10.1016/j.apm.2012.10.026

24. Gulsu M, Ozturk Y, Anapali A: Numerical approach for solving fractional relaxation-oscillation equation. Appl. Math. Model. 2013, 37: 5927-5937. 10.1016/j.apm.2012.12.015

25. Doha EH, Bhrawy AH, Ezz-Eldien SS: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 2012, 36: 4931-4943. 10.1016/j.apm.2011.12.031

26. Ahmadian A, Suleiman M, Salahshour S, Baleanu D: A Jacobi operational matrix for solving a fuzzy linear fractional differential equation. Adv. Differ. Equ. 2013., 2013: Article ID 104

27. Ahmadian A, Suleiman M, Salahshour S: An operational matrix based on Legendre polynomials for solving fuzzy fractional-order differential equations. Abstr. Appl. Anal. 2013., 2013: Article ID 505903

28. Esmaeili S, Shamsi M, Dehghan M: Numerical solution of fractional differential equations via a Volterra integral equation approach. Cent. Eur. J. Phys. 2013. 10.2478/s11534-013-0212-6

29. Bhrawy AH, Alofi AS: The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 2013, 26: 25-31. 10.1016/j.aml.2012.01.027

30. Bhrawy AH, Taha TM: An operational matrix of fractional integration of the Laguerre polynomials and its application on a semi-infinite interval. Math. Sci. 2012. 10.1186/2251-7456-6-41

31. Bhrawy AH, Alghamdi MM, Taha TM: A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line. Adv. Differ. Equ. 2012. 10.1186/1687-1847-2012-179

32. Balachandran K, Kiruthika S: Existence of solutions of nonlinear fractional pantograph equations. Acta Math. Sci. 2013, 33: 712-720. 10.1016/S0252-9602(13)60032-6

33. Alikhani R, Bahrami F: Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 2007-2017. 10.1016/j.cnsns.2012.12.026

34. Baleanu D, Mustafa OG, Agarwal RP: An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 2010, 23: 1129-1132. 10.1016/j.aml.2010.04.049

35. Salahshour S, Allahviranloo T, Abbasbandy S, Baleanu D: Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Differ. Equ. 2012., 2012: Article ID 112

36. Javidi M, Nyamoradi N: Dynamic analysis of a fractional order prey-predator interaction with harvesting. Appl. Math. Model. 2013. 10.1016/j.apm.2013.04.024

37. Ezzat MA, El-Karamany AS, El-Bary AA, Fayik MA: Fractional calculus in one-dimensional isotropic thermo-viscoelasticity. C. R., Méc. 2013. 10.1016/j.crme.2013.04.001

38. Allahviranloo T, Salahshour S, Ebadi MJ, Avazpour L, Baleanu D: Fuzzy fractional Ostrowski inequality with Caputo differentiability. J. Inequal. Appl. 2013., 2013: Article ID 50

39. Yang X, Baleanu D, Tenreiro Machado JA: Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis. Bound. Value Probl. 2013, 2013: 131-150. 10.1186/1687-2770-2013-131

40. Tenreiro Machado JA: Optimal tuning of fractional controllers using genetic algorithms. Nonlinear Dyn. 2010, 62: 447-452. 10.1007/s11071-010-9731-5

41. Baleanu D, Bhrawy AH, Taha TM: Two efficient generalized Laguerre spectral algorithms for fractional initial value problems. Abstr. Appl. Anal. 2013., 2013: Article ID 546502

42. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

43. Guo B-Y, Zhang X: A new generalized Laguerre spectral approximation and its applications. J. Comput. Appl. Math. 2005, 181: 342-363. 10.1016/j.cam.2004.12.008

44. Szegö G Am. Math. Soc. Colloq. Pub. 23. Orthogonal Polynomials 1985.

45. Gulsu M, Gurbuz B, Ozturk Y, Sezer M: Laguerre polynomial approach for solving linear delay difference equations. Appl. Math. Comput. 2011, 217: 6765-6776. 10.1016/j.amc.2011.01.112

46. Canuto C, Hussaini MY, Quarteroni A, Zang TA: Spectral Methods Fundamentals in Single Domains. Springer, Berlin; 2006.

## Acknowledgements

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the article.

## Author information

Authors

### Corresponding author

Correspondence to Ali H Bhrawy.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors have equal contributions to each part of this paper. All the authors read and approved the final manuscript.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

Reprints and Permissions

Bhrawy, A.H., Alghamdi, M.A. The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications. Adv Differ Equ 2013, 307 (2013). https://doi.org/10.1186/1687-1847-2013-307

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1847-2013-307

### Keywords

• nonlinear fractional differential equations
• operational matrix
• tau method
• pseudo-spectral approximation
• modified generalized Laguerre polynomials
• Caputo derivative 