Open access peer-reviewed chapter

Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre Polynomials

By Adnan Khan, Kamal Shah and Danfeng Luo

Submitted: November 15th 2019Reviewed: December 4th 2019Published: January 2nd 2020

DOI: 10.5772/intechopen.90754

Abstract

This article is devoted to compute numerical solutions of some classes and families of fractional order differential equations (FODEs). For the required numerical analysis, we utilize Laguerre polynomials and establish some operational matrices regarding to fractional order derivatives and integrals without discretizing the data. Further corresponding to boundary value problems (BVPs), we establish a new operational matrix which is used to compute numerical solutions of boundary value problems (BVPs) of FODEs. Based on these operational matrices (OMs), we convert the proposed (FODEs) or their system to corresponding algebraic equation of Sylvester type or system of Sylvester type. The resulting algebraic equations are solved by MATLAB® using Gauss elimination method for the unknown coefficient matrix. To demonstrate the suggested scheme for numerical solution, many suitable examples are provided.

Keywords

• FODEs
• numerical solution
• Laguerre polynomials
• operational matrices

1. Introduction

The theory of integrals as well as derivatives of arbitrary order is known by the special name “fractional calculus.” It has an old history just like classical calculus. The chronicle of fractional calculus and encyclopedic book can be studied in [1, 2]. Researchers have now necessitated the use of fractional calculus due to its diverse applications in different fields, specially in electrical networks, signal and image processing and optics, etc. For conspicuous work on FODEs in the fields of dynamical systems, electrochemistry, advanced techniques of microorganisms culturing, weather forecasting, as well as statistics, we refer to peruse [3, 4]. Fractional derivatives show valid results in most cases where ordinary derivatives do not. Also annotating that fractional order derivatives as well as fractional integrals are global operators, while ordinary derivatives are local operators. Fractional order derivative provides greater degree of freedom. Therefore from different aspects, the aforesaid areas were investigated. For instance, many researchers have provide understanding to existence and uniqueness results about FODEs, for few results, we refer [5, 6, 7], and many others have actualized the instinctive framework of fractional differential equations in various problems [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] with many references included in them.

Often it is very difficult to obtain the exact solution due to global nature of fractional derivatives in differential equations. Contrarily approximate solutions are obtained by numerical methods assorted in [20, 21, 22]. Various new numerical methods have been developed, among them is one famous method called “spectral method” which is used to solve problems in various realms [23]. In this method operational matrices are obtained by using orthogonal polynomials [24]. Many authors have successfully developed operational matrices by using Legender, Jacobi, and various other polynomials [25, 26]. For delay differential and various other related equations, Laguerre spectral methods have been used [27, 28, 29, 30, 31, 32]. Bernstein polynomials and various classes of other polynomials were also used to obtain operational matrices corresponding to fractional integrals and derivatives [33, 34, 35, 36, 37, 38, 39, 40]. Apart from them, operational matrices were also developed with the collocation method (see Refs. [41, 42, 43]). Since spectral methods are powerful tools to compute numerical solutions of both ODEs and FODEs. Therefore, we bring out numerical analysis via using Laguerre polynomials of some families and coupled systems of FODEs under initial as well as boundary conditions. In this regard we investigate the numerical solutions to the given families under initial conditions

0cDtγzt±zt=0,0<γ1,z0=z0,z0R,E1

and subject to boundary conditions

0cDtγzt±zt=0,1<γ2,z0=z0,z1=z1,z0,z1R.E2

By similar numerical techniques, we also investigate the numerical solutions to the following systems with fractional order derivatives under initial and boundary conditions as

0cDtγzt+azt+byt=ft,0cDtγyt+cyt+dzt=gt,z0=z0,y0=y0E3

for 0<γ1and

0cDtγztazt+byt=ft,0cDtγyt+cyt+dzt=gt,z0=z0,y0=y0,z1=z1,y1=y1,E4

for 1<γ2where f,g:01×R2Rand z0,y0,z1,y1R.We first obtain OMs for fractional derivatives and integrals by using Laguerre polynomials. Also corresponding to boundary conditions, we construct an operational matrix which is needed in numerical analysis of BVPs. With the help of the OMs we convert the considered problem of FODEs under initial/boundary conditions to Sylvester-type algebraic equations. Solving the mentioned matrix equations by using MATLAB®, we compute the numerical solutions of the considered problems.

2. Preliminaries

Here we recall some basic definition results that are needed in this work onward, keeping in mind that throughout the paper we use fractional derivative in Caputo sense.

Definition 1.The fractional integral of order γ>0of a function z:0Ris defined by

0Itγzt=1Γγ0tzsts1γds,

provided the integral converges at the right sides. Further a simple and important property of 0Itγis given by

0Itγtδ=Γδ+1Γδ+γ+1tγ+δ.

Definition 2.Caputo fractional derivative is defined as

0cDtγft=1Γnγ0ttsnγ1fnsds,

where nis a positive integer with the property that n1<γn.For example, if 0<γ1, then Caputo fractional derivative becomes

0cDtγft=1Γ1γ0ttsγ1f'sds.

Theorem 1.The FODE given by

0cDtγft=0

has a unique solution, such that

ft=d0+d1t+d2t2++dn1tn1,n=γ+1.

Lemma 1.Therefore in view of this result, if hLn0T,then the unique solution of nonhomogenous FODE

0cDtγft=ht,n1<γn

is written as

ft=d0+d1t+d2t2++dn1tn1+0Itγht,

where difor i=0,1,2,3n1are real constants.

The above lemma is also stated as

ft=0Itγht+i=0n1fi0i!ti.

Definition 3.The famous Laguerre polynomials are represented by Liγtand defined as

Liγt=k=0i1kΓi+γ+1Γk+1+γΓik+1Γk+1tk.

They are orthogonal on 0.If Liγtand Ljγtare Laguerre polynomials, then the orthogonality condition is given as

0LiγtLjγtWγtdt=δi,jUk,

where

Wγt=tγet,

is the weight function and

Uk=Γ1+γ+kΓ1+k,i=j0ij.

Now let Ztbe any function, defined on the interval 0.We express the function in terms of Laguerre polynomials as

Zt=i=0nciLiγt.=c0L0γt+c1L1γt++cNLNγt=c0c1cNL0γtLnγt.E5

We set the above two vectors into their inner product and represent the column matrix by Ψt,so that

Zt=ctΨt.

Again as

Zt=i=0nciLiγt,0LZtWγtLjγtdt=0Li=0nciLiγtLjγtWγtdt,

which is written as

i=0nci0LLiγtLjγtWγtdt.

We call hito the general term of integration

0LZtWγtLjγtdt=i=0ncihi.

Hence the coefficient ciis

ci=1hi0LZtWγtLjγtdt.

In vector form we can write Eq. (5) as

Zt=cMtΨMt.

where M= m+1,cMis the Mterms coefficient vector and ΨMtis the Mterms function vector.

2.1 Representation of Laguerre polynomial with Caputo fractional order derivative

If the Caputo fractional order derivative is applied to Laguerre polynomial, by considering whole function constant except tk.We use the definition of Caputo fractional order derivative for tkto obtain (6) as

0cDtγLiγt=k=0itkγ1kΓi+γ+1Γk+1+γΓik+1Γ1+kγ.E6

2.2 Error analysis

The proof of the following results can be found with details in [20].

Lemma 2.Let Liβtbe given; then

0cDtγLiβt=0,i=0,1,2,,β1,γ>0.

Theorem 2.For error analysis, we state the theorem such that, abe any integer and 0sa, and then

PM,azztAαs,ΛcMsa2ztAαa,Λ,ztϵAαaΛ,

where Aαa={z/zis measurable on Λand zAαa,Λ<}and

zAαa,Λ=paz+a,Λ,zAαa,Λ=k=0azAαa,Λ212.

Now let Λ=ϱ/0<ϱ<with χϱbe a weight function. Then

Lχ2Λ={κ/κis measurable on Λand uLχ2,Λ<}.

with the following inner product and norm

uvχ,Λ=Λuϱvϱdϱ,vχ,Λ=uvχ,Λ.

3. Operational matrices corresponding to fractional derivatives and integrals

Here in this section, we provide the required OMs via Laguerre polynomials of fractional derivatives and integrals.

Lemma 3.Let ΨMtbe a function vector; the fractional integral of order γfor the function ΨMtcan be generalized as

0ItγΨMtGN×NγΨMt,

where GN×Nγis the OM of integration of fractional order γand given by

0,0,k,rγ0,1,k,rγ0,j,k,rγ0,m,k,rγ1,0,k,rγ1,i,k,rγ1,j,k,rγ1,m,k,rγi,0,k,rγi,1,k,rγi,j,k,rγi,m,k,rγm,0,k,rγm,1,k,rγm,j,k,rγm,m,k,rγ,

where

i,j,k,rγ=k=0ir=0i1k+rΓj+1Γi+γ+1Γk+γ+α+r+1Γjr+1Γik+1Γr+1Γk+γ+1Γk+α+1Γγ+r+1.

Proof. We apply the fractional order integral of order γto the Laguerre polynomials

0cItγLiγt=k=0iΓi+γ+1Γik+1Γk+γ+1Γk+10cItγtk.E7

Since from (7), we have

0cItγtk=Γk+1Γ1+k+αtk+γ.

Therefore Eq. (7) implies that

0cItγLiγt=k=0itk+γΓi+γ+1Γik+1Γk+γ+1Γk+1Γk+1Γ1+k+α,

which is equal to

0cItγLiγt=k=0i1kΓi+γ+1Γik+1Γk+γ+1Γ1+kγtk+γ.E8

We approximate tk+γin (8) with Laguerre polynomials, i.e.

tk+γj=0nHjLjγt.

By using the relation of orthogonality, we can find coefficients

Hj=r=0j1kΓj+1Γk+α+r+γ+1Γ1+jrΓ1+rΓ1+r+γ.

So Eq. (8) implies

0cItγLiγt=k=0i1kΓi+γ+1Γik+1Γk+γ+1Γ1+kγ×r=0j1rΓj+1Γk+α+r+γ+1Γjr+1Γr+1Γr+γ+1.
0cItγLiγt=k=0ir=0j1k+rΓj+1Γi+γ+1Γk+α+r+γ+1Γ1k+iΓjγ+1Γγ+1Γk+γ+1Γk+α+1Γγ+r+1.

which is the desired result.

Lemma 4.Let ΨMtbe a function vector; then the fractional derivative of order γfor ΨMtis generalized as

0cDtγΨMtWM×MγΨMt,

where WM×Mγis the OM of derivative of order γ, defined as in (9)

WM×Mγ=00000Θγ,0,k,αγΘγ,1,k,αγΘγ,j,k,αγΘγ,n,k,αγΘi,0,k,αγΘi,1,k,αγΘi,j,k,αγΘi,n,k,αγΘn,0,k,αγΘn,1,k,αγΘn,j,k,αγΘn,n,k,αγ,E9

where

Θi,j,k,αγ=k=γir=0i1γ+kΓj+1Γi+α+1Γk+αr+γ+1Γjr+1Γik+1Γr+1Γk+α+1Γkγ+1Γα+γ+1.

Proof. Leaving the proof as it is very similar to the proof of the above lemma.

Lemma 5.We consider a function Ztdefined on 0and yt=KMΨMTt; then

Zt0Itγyt=KMQM×MγΨMt,

where QM×Mγis the operational matrix, given by

C0,0,C0,1C0,jC0,mC1,0C1,1C1,jC1,mCi,0Ci,1Ci,jCi,mCm,0Cm,1Cm,jCm,m,

where

Ci,j=1hi01Δi,γ,kZtLjγtdt,

with

wi=k=0i1i+1Γi+1+γΓk+γ+1Γ1k+iΓk+γ.

Proof. By considering the general term of ΨMt

0I1γLit=1Γγ011sγ1Lisds.0I1γLit=1Γγ011sγ1k=0isk1kΓi+1+γΓk+1+iΓk+1+γΓ1+kds.0I1γLit=k=0i1kΓi+1+γΓγΓk+1+iΓk+1+γΓ1+k011sγ1skds.E10

Using the famous Laplace transform, we have from (10)

£(011sγ1skds=ΓγΓk+1Γγ+k.0I1γLit=k=0i1kΓi+1+γΓγΓk+1+iΓk+1+γΓ1+kΓγΓk+1Γγ+k.k=0i1kΓi+1+γΓk+1+iΓk+γ+1Γ1+k=Δi,γ,k.

Now using Laguerre polynomials, we have

Δi,γ,kzt=j=0mCi,jLit,

where Ci,jis calculated by using orthogonality as

Ci,j=1hi01Δi,γ,kztLjγtdt.E11

To get the desired result, we evaluate the above (11) relation for i=0,1,,mand j=0,1,,m.

4. Main result

In this section, we discuss some cases of FODEs with initial condition as well as boundary conditions. The approximate solution obtained through desired method is compared with the exact solution. Similarly we investigate numerical solutions to various coupled systems under some initial conditions as well as boundary conditions.

4.1 Treatment of FODEs under initial and boundary conditions

Here we discuss different cases.

Case 1.In the first case, we consider the fractional order differential equation

0cDtγzt±zt=0,0<γ1,z0=z0,z0RE12

we see that

0cDtγzt=ŁMψMTt.

and applying 0Itγby the Lemma 1, on (12) we write

zt=e0+0ItγŁMψMTt,

Using the initial condition to get e0=z0and approximate z0as z0FMψMTt, Eq. (12) implies

ŁMψMTt+ŁMGM×MγψMTt+FMψMTt=0.

Finally the Sylvester-type algebraic equation is obtained as

ŁM+ŁMGM×MγψMTt+FM=0.

Solving the Sylvester matrix for ŁM,we get the numerical value for zt.

Example 1.

0cDtγzt±zt=0,0<γ1,z0=1,z0R.

Since the exact solution is given by

zt=Eγtγ,

where Eγis the Mittag-Leffler representation, and at γ=1,zt=et.

Approximating the solution through the proposed method and plotting the exact as well as numerical solution by using scale M=8corresponding to γ=1in Figure 1, we see that the proposed method works very well.

Case 2.

0cDtγzt+zt=0,1<γ2,z0=z0,z1=z1,z0,z1R.E13

We take

0cDtγzt=KMψMTt.E14

Applying Lemma 1 to Eq. (14), we get

zt=e0+e1t+0ItγKMψMTt.E15

Using the conditions by putting t=0and t=1to get e0=z0and

e1=z1z0KM0I1γψMTt/t=1.

Equation (15) implies

zt=z0+z1z0ttKM0I1γψMTt/t=1+0ItγKMψMTt,

where z0+z1z0tis the smooth function of tand constants; we approximate it as

z0+z1z0tGM×MγψMTt

and

tKM0I1γψMT1KMQM×MγψMTt.

Hence

zt=GM×MγψMTtKMQM×MγψMTt+KMGM×MγψMTt

So Eq. (13) implies

KMψMTt+GM×MγψMTtKMQM×MγψMTt+KMGM×MγψMTt=0

which is further solved for KMto get the required numerical solution.

For Case 2, we give the following example.

Example 2.

0cDtγzt+zt=0,0<γ2,z0=1,z1=1.E16

At γ=2,we get the exact solution as of (16) as given by (17)

zt=114.58sinxcosxE17

Upon using the suggested method, we see from the subplot at the left of Figure 2 that exact and numerical solutions are very close to each other for very low scale level. Also, the absolute error is given in subplot at the right of Figure 2.

4.2 Coupled systems of linear FODEs under initial and boundary conditions

In this subsection, we consider different forms of coupled systems of FODEs with the initials as well as boundary conditions.

Case 1.First we take the coupled system of FODEs as

0cDtγzt+azt+byt=ft0cDtγyt+cyt+dzt=gt,E18

with the conditions

z0=z0,y0=y0,z0,y0R.E19

Let

0cDtγzt=ŁMψMTt,0cDtγyt=KMψMTt.E20

Applying Lemma 1 to Eq. (20), we get

zt=e0+ŁMGM×MγψMTt,yt=d0+KMGM×MγψMTt.E21

Using the initial conditions given in Eq. (19), from Eq. (21), we get

zt=FM1ψMTt+ŁMGM×MγψMTt,yt=y0FM2ψMTt+KMGM×MγψMTt.E22

We take approximation as

z0FM1ψMTt,

and

y0FM2ψMTt,

while source functions are approximated as

ftFM3ΨMTt,

and

gtFM4ΨMTt.

Therefore the consider system on using (19)(22), (18) becomes

ŁMψMT+aFM1ψMTt+ŁMGM×MγψMTt+b(FM2ψMTt+KMGM×MγψMTt=FM3ψMTt.KMψMT+cFM2ψMTt+KMGM×MγψMTt+d(FM1ψMTt+ŁMGM×MγψMTt=FM4ψMTt.

On further rearrangement we have

ŁM+aFM1+ŁMGM×Mγ+b(FM2+KMGM×Mγ=FM3KM+cFM2+KMGM×Mγ+d(FM1+ŁMGM×Mγ=FM4.

which further can be written as

ŁMIM×M+aGM×Mγ+KMbGM×Mγ+aFM1+bFM2FM3=0KMIM×M+cGM×Mγ+ŁMdGM×Mγ+cFM2+dFM1FM4=0.

In matrix form we write as

ŁMKMIM×M+aGM×Mγ00IM×M+cGM×Mγ+ŁMKM0dGM×MγbGM×Mγ0+aFM1+bFM2FM3cFM2+dFM1FM4=0.

We solve this system of matrix equation for ŁMKMby using Gaussian’s elimination method. The considered system is in the form of XA¯+XB¯+C¯=0,.

where X=ŁMKMA¯=IM×M+aGM×Mγ00IM×M+cGM×Mγ,.

B¯=0dGM×MγbGM×Mγ0and C¯=aFM1+bFM2FM3cFM2+dFM1FM4..

Upon computation of matrices ŁM,KMby using MATLAB®, we put these matrices in Eq. (22) to find zappand yapp, respectively.

Example 3.We now provide its example by considering the system of FODEs:

0cDtγzt+zt+yt=ft0cDtγyt+yt+zt=gt,z0=2,y0=1.

By taking γ=1,the exact solution is obtained as

zt=cost+et,y=sint+et,

where the external source functions are given by ft=cost+et+2etand gt=et+sint+2cost.The exact solution zex,yexcan be computed by any method of ODEs. Approximating the problem by the considered method, we see that the computed numerical and exact solutions have close agreement at very small-scale level. The corresponding accuracy has been recorded in Table 1. Further the comparison between exact and numerical solution and the results about absolute error have been demonstrated in Figures 3 and 4, respectively. In Figure 3 we are given the comparison between exact solution and approximate solutions by using proposed method. Similarly the absolute errors have been described in Figure 4.

tCPU time (s)Absolute error zappzexAbsolute error yappyexCPU time (s)
030.50.000030.00000632.5
0.1532.70.0000160.00003433.3
o.3535.80.0000130.0000333.9
0.6533.60.0000120.0000335.6
0.8734.80.0000180.00003636.5
135.90.000030.00000636.8

Table 1.

Absolute error at M=5,γ=0.9, for different values of tin Example 3.

By comparing the exact and numerical solution through the proposed method, we observe that our numerical solution does not show any disagreement with the exact solution as can be seen in Figure 3. The absolute errors zappzexand yappyexplotted at the scale M=5are very low as given in Figure 4, which describes the efficiency of the proposed method.

Case 2.Similarly for the coupled system of FODEs with boundary conditions, we consider

0cDtγzt+azt+byt=ft,0cDtγyt+cyt+dzt=gt,z0=z0,y0=y0,z1=z1,y1=y1.E23

Let us assume

0cDtγzt=ŁMψMTt,0cDtγyt=KMψMTt.E24

Applying Lemma 1 to Eq. (24), we get

zt=e0+e1t+ŁMGM×MγΨMTtyt=d0+d1t+KMGM×MγΨMTt,E25

where d0,d1,e0,e1R.Using the initial conditions in Eq. (25), we have e0=z0,d0=y0.On using boundary conditions, we have from Eq. (25)

z1=z0+e1+ŁMGM×MγΨMTtt=1,z1z0ŁMGM×MγΨMTtt=1=e1.

Similarly

y1=y0+d1+KMGM×MγΨMTtt=1,y1y0KMGM×MγΨMTtt=1=d1.

Equation (25) implies that

zt=z0+tz1z0t(LMGM×MγΨMTtt=1)+LMGM×MγΨMTtyt=y0+ty1y0t(KMGM×MγΨMTtt=1)+KMGM×MγΨMTt.E26

Let z0+tz1z0FM1ΨMTtand y0+ty1y0FM2ψMTt,with

ŁMGM×MγΨMTt=ŁMQM×Mγ,zΨMTttKMGM×MγΨMTt=KMQM×Mγ,yΨMTt.E27

Hence Eq. (26) implies

zt=FM1ΨMTtLMQM×Mγ,zΨMTt+LMGM×MγΨMTtyt=FM2ΨMTtKMQM×Mγ,yΨMTt+KMGM×MγΨMTt.E28

approximating ftand gtsuch that

ftFM3ΨMTtgtFM4ΨMTt.E29

On using (24)(29), system (23) can be written as

LMΨMTt+aFM1ΨMTtLMQM×Mγ,zΨMTt+LMGM×MγΨMTt+bFM2ΨMTtKMQM×MγΨMTt+KMGM×MγΨMTtFM3ΨMTt=0KMΨMTt+cFM2ΨMTtKMQM×Mγ,yΨMTt+KMGM×MγΨMTt+dFM1ΨMTtŁMQM×Mγ,zΨMTt+ŁMGM×MγΨMTtFM4ΨMTt=0.

On rearrangement of terms, the above equations give

ŁMIM×MaQM×Mγ,z+aGM×Mγ+KMIM×MbQM×Mγ,y+bGM×Mγ+aFM1+bFM2FM3=0KMIM×McQM×Mγ,y+cGM×Mγ+ŁMIM×MdQM×Mγ,z+dGM×Mγ+cFM2+dFM1FM4=0.

In matrix form, we can write

ŁMKMIM×MaQM×Mγ,z+aGM×Mγ00IM×McQM×Mγ,y+cGM×Mγ+LMKM0IM×MdQM×Mγ,z+dGM×MγIM×MbQM×Mγ,y+bGM×Mγ0+aFM1+bFM2FM3cFM2+dFM1FM4=0.

We convert the system to algebraic equation by considering

L¯=IM×MaQM×Mγ,z+aGM×Mγ00IM×McQM×Mγ,y+cGM×MγM¯=0IM×MdQM×Mγ,z+dGM×MγIM×MbQM×Mγ,y+bGM×Mγ0andN¯=aFM1+bFM2FM3cFM2+dFM1FM4.

so that the system is of the form

XL¯+XM¯+N¯=0,

and solving the given equation for the unknown matrix X=LMKM,we get the required solution.

Example 4.As an example, we consider the Caputo fractional differential equation for the coupled system with the boundary conditions as

0cDtγzt+2zt2ytft=0,0cDtγyt3yt+2ztgt=0,z0=4z1=4,y0=2,y1=2.

At γ=2,the exact solutions are

zt=t6+t5+t4t3+t+1,yt=t7t6+t5+t4+t3t2t+1.

where the source functions are given by

ft=2t7+4t6+30t4+16t3+12t22t+2gt=3t7+12t6+35t527t419t3+20t2+9t4.

We approximate the solution at the considered method by taking scale level M=5.One can see that numerical plot and exact solution plot coincide very well as shown in Figure 5. Similarly the absolute error has been plotted at the given scale M=5in Figure 6, which is very low. The lowest value of absolute error zappzexand yappyexindicates efficiency of the proposed method. The table shows the comparison of errors for exact and approximate solutions for fixed scale level M=5and order γ=1.9.Further the absolute error has been recorded at different values of space variable in Table 2 which provides the information about efficiency of the proposed method.

tAbsolute error zappzexCPU time (s)Absolute error yappyexCPU time (s)
00.01149.40.01050.0
0.150.006250.30.005252.5
0.350.005851.20.004754.6
0.650.00651.50.00555.5
0.850.007552.60.00756.4
10.01153.80.01056.2

Table 2.

Absolute error at different values of t for Example 4.

5. Conclusion

We have successfully used the class of orthogonal polynomials of Laguerre polynomials to establish a numerical method to compute the numerical solution of FODEs and their coupled systems under some initial and boundary conditions. By using these polynomials, we have obtained some operational matrices corresponding to fractional order derivatives and integration. Also we have computed a new matrix corresponding to boundary conditions for boundary value problems of FODEs. Using the aforementioned matrices, we have converted the considered problem of FODEs to Sylvester-type algebraic equations. To obtain the numerical solution, we easily solved the desired algebraic equations by taking help from MATLAB®. Corresponding to the established procedure, we have provided numbers of examples to demonstrate our results. Also some error analyses have been provided along with graphical representations. By increasing the scale level, the accuracy is increased and vice versa. On the other hand, when the fractional order is approaching to integer value, the solutions tend to the exact solutions of the considered FODE. Therefore in each example, we have compared the exact and approximate solution and found that both the solutions were in closure contact with each other. Hence the established method can be very helpful in solving many classes and systems of FODEs under both initial and boundary conditions. In future the shifted Laguerre polynomials can be used to compute numerical solutions of partial differential equations of fractional order.

Competing interests

We declare that no competing interests exist regarding this manuscript.

Author contribution

All authors equally contributed this paper and approved the final version.

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Adnan Khan, Kamal Shah and Danfeng Luo (January 2nd 2020). Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre Polynomials, Nonlinear Systems -Theoretical Aspects and Recent Applications, Walter Legnani and Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.90754. Available from:

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