Open access peer-reviewed chapter

Using Shifted Jacobi Polynomials to Handle Boundary Value Problems of Fractional Order

Written By

Kamal Shah, Eiman, Hammad Khalil, Rahmat Ali Khan and Thabet Abdeljawad

Reviewed: December 16th, 2021 Published: February 27th, 2022

DOI: 10.5772/intechopen.102054

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Abstract

This paper is concerned about the study of shifted Jacobi polynomials. By means of these polynomials, we construct some operational matrices of fractional order integration and differentiations. Based on these matrices, we develop a numerical scheme for the boundary value problems of fractional order differential equations. The construction of the procedure is new one for the coupled systems of fractional order boundary value problems. In the proposed scheme, we obtain a simple but highly accurate systems of algebraic equations. These systems are easily soluble by means of Matlab or using Mathematica. We provide some examples to which the procedure is applied to verify the applicability of our proposed method.

Keywords

  • Jacobi polynomials
  • operational matrices
  • fractional order differential equations
  • coupled system
  • boundary conditions

1. Introduction

Recently, fractional order differential equations have gained much attention from the researchers of mathematics, physics, computer science and engineers. This is due to the large numbers of applications of fractional order differential equations in various fields of science and applied nature. The applications of fractional order differential equations are found in physics, mechanics, viscoelasticity, photography, biology, chemistry, fluid mechanics, image and signal processing phenomenons, etc.; for more details, see [1, 2, 3, 4]. The researchers are giving more attention to fractional order differential equations, because in most cases the fractional order models are more accurate and reliable than classical order models. On the other hand, fractional order models have more degrees of freedom than integer order models. In most cases, the hereditary phenomena and memory description process are accurately modeled by means of fractional order derivatives and integrations as compare with integer order derivative and integrations. Due to these facts, researchers had given much attention to study the existence and uniqueness of positive solutions as well as multiplicity of solutions for fractional order differential equations and their systems. This area is very well explored, and large numbers of research articles can be found in literature; for details, see [5, 6, 7, 8, 9, 10]. It is very difficult to obtain exact solutions for such type of differential equations in all cases always. However, the area involving numerical solutions is in initial stage. Therefore, various numerical schemes have been developed in last few years by different researchers. The aim of these schemes was to achieve better accuracy. These schemes have their own merits and demerits. For example, in [11, 12, 13], the authors developed Adomain decomposition methods, Homotopy analysis methods, and Variational iterational methods to obtained good approximate solutions for certain fractional order differential equations. In recent years, spectral methods have got much attention as they were applied to solve some real-word problems of various fields of science and engineering. High accuracy was obtained for solving such problems [14]. Spectral method needs operational matrices for the numerical solutions, which have been constructed by using some polynomials, for example, in [15], the authors developed an operational matrix for shifted Legendre polynomials corresponding to fractional order derivative. In [16], the authors introduced operational matrix by using shifted Legendre polynomials corresponding to fractional order integrations. Similarly in [17, 18], authors constructed operational matrices for fractional order derivative by using Chebyshev and Jacobi polynomials. In all these cases, these matrices were applied to solve multiterms fractional order differential equations together with Tau-collocation method. In [19], Singh et al. derived an operational matrices for Berestein polynomials corresponding to fractional order integrations. Haar wavelet operational matrices were also developed, and some problems of fractional order differential equations were solved. In all these methods, collocation method was used together with these methods to obtain numerical solutions for fractional order differential equations. They only solved single problems.

Most of the physical and biological phenomena can be modeled in the form of coupled systems. For example, when two masses and two spring systems are modeled, it led to a coupled system of differential equations.

DαX+λm1Dν1Xμm1Dν2Y+κ1m1Xκ2m1Y=fx,1<α2,0<ν1,ν21,DβYλm2Dω1X+λ+μm2Dω2Yκ1m2X+κ1+κ2m2Y=gx,1<β2,0<ω1,ω21,

subject to the boundary conditions

X0=0,X1=0Y0=1,Y1=1.

Many coupled systems can be found in physics and in biological model, chemistry, fluid dynamics, etc.; see [16, 20, 21, 22, 23, 24, 25, 26, 27]. In fractional calculus, we generalize these systems by replacing the integer order by any fractional values that lie in certain range. All the above numerical methods were applied to solve singles problems, and to the best of our knowledge, very few articles are devoted to the numerical solutions of coupled systems or models. In all these cases, the authors have solved initial value problems of coupled systems of fractional order differential and partial differential equations; for details, see [15, 28] and the references therein. For example, in [15], authors have solved some initial value problems of coupled systems of fractional differential equations (FDEs) by using shifted Jacobi polynomials operational matrix method. However, the scheme is not properly applied yet for solving coupled systems of boundary value problems of fractional order differential equations. In this paper, we discuss the shifted Jacobi polynomial operational matrices methods to solve boundary value problems for a coupled systems of fractional order differential equations. In this scheme, we introduced a new matrix corresponding to boundary conditions, which is required for the approximate solutions. First, we solve single boundary value problems given by

Dαxt+ADνxt+BDωxt+Cxt=ft,t0T,A,B,CR,E1

subject the boundary conditions

x0=x0,xT=x1,

where ftC0Tis a source term and 1<α2,0<ν,ω<1.Then, we extend our technique to solve coupled system of fractional differential equations (FDEs) given as

Dαxt+A1Dν1xt+B1Dν2yt+C1xt+D1yt=ft,t0T,Dβyt+A2Dω1xt+B2Dω2yt+C2xt+D2yt=gt,t0T,E2

subject to the boundary conditions given by

x0=x0,xT=x1,y0=y0,yT=y1,E3

where 1<α,β2,0<νi,ωi<1i=12.Further, ft,gtC0Tare the source term of the system (42), (43) and x0,y0,x1,y1are real constants, Ai,Bi,Ci,Dii=12are any constants. We convert the differential equations/systems of differential equations to a coupled systems of simple algebraic equations of Selvester or Laypunove type, which are easily soluble for unknown matrix required for approximations. In our proposed method, we do no need collation method, which is required for abovementioned methods. We applied our technique to some practical problems to verify the applicability of the proposed method.

Our article is organized as follows: Some fundamental concepts and results of fractional calculus and Jacobi polynomials needed are provided in sections 2. Section 3 contains some operational matrices for shifted Jacobi polynomials corresponding to fractional order derivative and integrations. Also in this section, we provide new operational matrix corresponding to the boundary conditions. In Section 4, we derive main procedure for the general class of boundary value problems of fractional order differential equations as well as for coupled system of FDEs. In Section 5, we apply the proposed method to some examples. Also we plot the images of these examples. In Section 6, we provide a short conclusion of the papers.

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2. Preliminaries

In this section, we recall some basic definitions and results of fractional calculus, also we will give fundamental properties, definitions related to Jacobi polynomials as in [1, 4, 6, 7, 8, 14, 17, 18, 28, 29, 30, 31].

Definition 2.1.The fractional integral of order α>0of a function z:0Ris defined by

Iαzt=1Γα0tzsts1αds,E4

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

Iαzδ=Γδ+1Γδ+α+1zα+δ.E5

Definition 2.2.The Caputo fractional derivative of orderα>0of a functionzCn01is defined by

D0+αzt=1Γnα0ttsnα1dndtnzsds,n1<αn,t>0,wheren=α+1,E6

provided that the right side is pointwise defined on 0. Also one of the important properties of fractional order derivative is given by

Dαtk=Γk+1Γkα+1tkα.Also foranyconstantC,we haveDαC=0.E7

The following results are needed in the sequel.

Lemma 2.2.1.[6], Let α>0then

IαDαyt=yt+C0+C1t++Cn1tαn,E8

for arbitrary

CiR,i=0,1,2,,n1,n=α+1.

2.1 The shifted Jacobi polynomials and its fundamental properties

In this section, we provide basic properties of shifted Jacobi polynomials. The famous Jacobi polynomials Piabzare defined over the interval 11as given by

Piabz=a+b+2i1a2b2+za+b+2i2a+b+2i22ia+b+ia+b+2i2Pi1abza+i1b+i1a+b+2iia+b+ia+b+2i2Pi2abz,i=2,3,,whereP0abz=1,P1abz=a+b+22z+ab2.E9

By means of the substitution z=2tξ1, we get a new version of the above polynomials defined in (9), which is called the shifted Jacobi polynomials over the interval 0ξ. The analytical form of this shifted Jacobi polynomials is given by the relations

Pξ,iabt=k=0i1ikΓi+b+1Γi+k+a+b+1Γk+b+1Γi+a+b+1Γik+1Γk+1ξktk,wherePξ,iab0=1iΓi+b+1Γb+1Γi+1.E10

The orthogonality conditions of the shifted Jacobi polynomials are given by

0ξPξ,jabtPξ,iabtWξabtdt=Ωξ,jabδji,whereδji=1,ifi=j,other wiseδji=0,and the weight function is givenbyWξabt=ξtatb,E11

and

Ωξ,jabt=ξa+b+1Γj+a+1Γj+b+12j+a+b+1Γj+1Γj+a+b+1.E12

Any function ftsuch that f2tis integrable over 0ξ,can be approximated in terms of shifted Jacobi polynomials as given by

ftk=0mBjPξ,kabt=KMTΦMt,E13

where the shifted Jacobi coefficient vector is denoted by KMand ΦMtis Mterms function vector. Also M=m+1,when mthe approximation converges to the exact value of the function. The coefficient bkcan be calculated by using (11)(13) as

Bj=1Ωξ,jab0ξWξabtftPξ,iabtdt,i=0,1,.E14
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3. Operational matrices

In this section, we provide some operational matrices for fractional order differentiation as well as for fractional order integration. Further, in this section, we give construction of new matrix for the boundary conditions.

Theorem 3.1.Let ΦMtbe the function vector, then the operational matrix of fractional order derivative αis given by

DαΦMtHM×MαΦMt,E15

where HM×Mαis the operational matrix of fractional order derivative αgiven by

HM×Mα=ϒ0,0,kϒ0,1,kϒ0,j,kϒ0,m,kϒ2,1,kϒ2,2,kϒ2,r,kϒ1,m,kϒi,0,kϒi,1,kϒi,j,kϒi,m,kϒm,0,kϒm,1,kϒm,j,kϒm,m,k,E16

where

ϒi,j,k=k=0i1ikΓi+b+1Γi+k+a+b+1Γk+1Γk+b+1Γi+a+b+1Γik+1Γk+1Γk+1αξk×l=0j1jl2j+a+b+1Γj+1Γj+l+a+b+1Γkalpha+l+b+1Γa+1ξαΓj+a+1Γl+b+1Γjl+1Γl+1Γkα+l+b+a+2.Ifϒi,j,k=0,fori<α.E17

Proof.The proof of this theorem is same as given in [14, 17] in lemma 3.2and 3.4, respectively.□

Theorem 3.2.LetΦMtbe the function vector corresponding to the shifted Jacobi polynomials andα>0, then the operational matrix corresponding to the fractional order integration is given by

IαΦMtHM×MαΦMt,E18

where HM×Mαis the operational matrix of fractional order integration αand is given by

HM×Mα=ϒ0,0,kϒ0,1,kϒ0,j,kϒ0,m,kϒ2,1,kϒ2,2,kϒ2,r,kϒ1,m,kϒi,0,kϒi,1,kϒi,j,kϒi,m,kϒm,0,kϒm,1,kϒm,j,kϒm,m,k,E19

where

ϒi,j,k=k=0i1ikΓi+b+1Γi+k+a+b+1Γk+1Γk+b+1Γi+a+b+1Γik+1Γk+1Γk+1+αξk×l=0j1jl2j+a+b+1Γj+1Γj+l+a+b+1Γk+α+l+b+1Γa+1ξαΓj+a+1Γl+b+1Γjl+1Γl+1Γk+α+l+b+a+2.E20

Proof.The proof of this theorem is available in [14]. Therefore, we omit the proof.

Theorem 3.3LetΦMtbe a function vector, and letϕtbe any function defined asϕt=ctn,n=0,1,2,,cRandut=KMTΦMt. Then

ϕt0Iξαut=KMTQM×McϕαΦMt,E21

where QM×Mcϕαis an operational matrix corresponding to some boundary value and is given by

QM×Mcϕα=ϒ0,0,kϒ0,1,kϒ0,j,kϒ0,m,kϒ2,1,kϒ2,2,kϒ2,r,kϒ1,m,kϒi,0,kϒi,1,kϒi,j,kϒi,m,kϒm,0,kϒm,1,kϒm,j,kϒm,m,k,E22

where

ϒi,j,k=j=0MΔi,k,αBjPξ,jabt,E23

where

Δi,k,α=k=0i1iki+b+1Γi+a+b+1cξαΓk+b+1Γi+a+b+1Γik+1Γk+αE24

and

Bj=l=0j1jl2j+a+b+1Γj+1Γj+l+a+b+1Γn+b+1Γa+1ξnlΓj+a+1Γl+b+1Γjl+1Γl+1Γn+a+b+1.E25

Proof.Consider the general term of ΦMtas

0IξαPξ,iabt=1Γα0ξξsα1Pξ,iabsds=1Γα0ξξsα1k=0i1iki+b+1Γi+k+a+b+1skΓk+b+1Γi+a+b+1Γik+1Γk+1ξkds=1Γαk=0i1iki+b+1Γi+k+a+b+1Γk+b+1Γi+a+b+1Γik+1Γk+1ξk0ξξsα1skds.E26

Now by applying Laplace transform, we get

L0ξξsα1skds=ΓαΓk+1sk+α+10ξξsα1skds=Γk+1Γαξk+αΓα+1.E27

Now putting (27) in (26), we get

0IξαPξ,iabt=k=0i1iki+b+1Γi+a+b+k+1ξαΓk+b+1Γi+a+b+1Γik+1Γk+α.E28

Now if ϕt=tn, then

t0IξαPξ,iabt=k=0i1iki+b+1Γi+a+b+k+1ξαctnΓk+b+1Γi+a+b+1Γik+1Γk+α.E29

Representing tnin terms of shifted Jacobi polynomials as

tn=j=0MBjPξ,jabt.E30

By the use of orthogonality relation (11), we have

Bj=1Ωξ,jab0ξtnPξ,jabtWξabξdt=1Ωξ,jabl=0j1jlΓj+b+1Γj+l+a+b+1Γl+b+1Γj+a+b+1Γjl+1Γl+1ξl0ξtn+bξtadt.E31

Now by means of Laplace transform, we have

L0ξtn+bξtadt=Γn+b+1Γa+1ξn+b+a+20ξtn+bξtadt=Γn+b+1Γa+1Γn+a+b+1ξn+a+b+1.E32

Thus putting (32) in (31) and using (12), we get

Bj=l=0j1jl2j+a+b+1Γj+1Γj+l+a+b+1Γn+b+1Γa+1ξnlΓj+a+1Γl+b+1Γjl+1Γl+1Γn+a+b+1.E33

Now (29) implies that

ϕt0IξαPξ,iabt=j=0Mk=0i1iki+b+1Γi+a+b+k+1ξαcΓk+b+1Γi+a+b+1Γik+1Γk+αBjPξ,jabt=j=0MΔi,k,αBjPξ,jabt=j=0Mϒi,j,kPξ,jabt,E34

where

ϒi,j,k=j=0MΔi,k,αBj,Δi,k,α=k=0i1iki+b+1Γi+a+b+k+1ξαcΓk+b+1Γi+a+b+1Γi+k+1Γk+α.

Evaluating the result (34) for different values of jwe get the required result.

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4. Applications of operational matrices

In this section, we show fundamental importance of operational matrices of fractional order derivative and integration. We apply them to solve some multiterms fractional order boundary value problems of fractional differential equations. Consider the following general fractional differential equation with constant coefficient and given boundary conditions

Dαxt+ADνxt+BDωxt+Cxt=ft,t0T,A,B,CR,E35

subject the boundary conditions

x0=x0,xT=x1,

where ftC0Tis a source terms and 1<α2,0<ν,ω<1..

To obtain the solutions of the (35) in terms of shifted Jacobi polynomials, we proceed as

Dαxt=KMTΦMt.E36

Applying Iα, definition (7), and lemma (2.2.1), then we get

xt=KMTHM×MαΦMt+c0+c1t.E37

By means of boundary conditions we get

xt=KMTHM×MαΦMt+x0+tx1x0TtTKMTHM×MαΦMtt=TE38

using the approximation x0+tx1x0TFM1ΦMt,tT=ϕt,also using tKMTHM×MαΦMtt=T=KMTQM×MαϕΦMt,then (38) implies that

xt=KMTHM×MαΦMt+FM1ΦttKMTHM×MαΦMtt=T=KMTHM×MαΦMtKMTQM×MαϕΦMt+FM1ΦMt=KMTHM×MαQM×MαϕΦMt+FM1ΦMt.E39

Now from (39) we have

Dνxt=KMTHM×MαQM×MαϕGM×MνΦMt+FM1GM×MνΦMtDωxt=KMTHM×MαQM×MαϕGM×MωΦMt+FM1GM×MωΦMtand approximating the source termasftFM2ΦMt.E40

Putting (36)(40) in (35), which yields

KMTΦMt+AKMTHM×MαQM×MαϕGM×MνΦMt+FM1GM×MνΦMt+BKMTHM×MαQM×MαϕGM×MωΦMt+FM1GM×MωΦMt+CKMTHM×MαQM×MαϕΦMt+FM1ΦMtFM2ΦMt=0KMTΦMt+KMTHM×MαQM×MαϕAGM×Mν+BGM×Mω+CIΦMt+FM1AGM×Mν+BGM×Mω+CFM1FM2ΦMt=0KMT+KMTHM×MαQM×MαϕAGM×Mν+BGM×Mω+CI+FM1AGM×Mν+BGM×Mω+CIFM2=0.E41

Which is a simple algebraic equation. Solving this equation for KMand putting it in (39) we get the required approximate solution of the corresponding boundary value problem.

4.1 Coupled system of boundary value problems for fractional order differential equations

In this subsection, we use operational matrices to derive procedure for the numerical solutions of coupled system. We consider the following general coupled system of FDEs as

Dαxt+A1Dν1xt+B1Dν2yt+C1xt+D1yt=ft,t0T,Dβyt+A2Dω1xt+B2Dω2yt+C2xt+D2yt=gt,t0T,E42

subject to the boundary conditions given by

x0=x0,xT=x1,y0=y0,yT=y1,E43

where 1<α,β2,0<νi,ωi<1i=12.Further, ft,gtC0Tare the source term of the system (42), (43) and x0,y0,x1,y1are real constants, Ai,Bi,Ci,Dii=12are any constants. To approximate the above system in terms of shifted Jacobi polynomials, we take

Dαxt=KMTΦMtE44

and

Dβyt=LMTΦMtE45

Applying Iα,Iβand lemma (1) and boundary conditions, the above Eqs. (44), (45) imply that

xt=KMTHM×MαQM×MαϕΦMt+FM1ΦMtE46
yt=LTMHM×MβQM×MβϕΦMt+FM2ΦMt.E47

Now taking fractional order derivative of (46), (47), we have

Dν1xt=KMTHM×MαQM×MαϕGM×Mν1ΦMt+FM1GM×Mν1ΦMtDω1xt=KMTHM×MαQM×MαϕGM×Mω1ΦMt+FM1GM×Mω1ΦMtE48

and

Dν2yt=LMTHM×MβQM×MβϕGM×Mν2ΦMt+FM2GM×Mν2ΦMtDω2yt=LMTHM×MβQM×MβϕGM×Mω2ΦMt+FM2GM×Mω2ΦMt.E49

Putting (45)(49) together with the use of approximation ftFM3ΦMt,gtFM4ΦMtin (42), we obtain

KMTΦMt+A1KMTHM×MαQM×MαϕGM×Mν1ΦMt+FM1GM×Mν1ΦMt+B1LMTHM×MβQM×MβϕGM×Mν2ΦMt+FM2GM×Mν2ΦMt+C1KMTHM×MαQM×MαϕΦMt+FM1ΦMt+D1LMTHM×MβQM×MβϕΦMt+FM2ΦMtFM3ΦMt=0E50
LMTΦMt+A2KMTHM×MαQM×MαϕGM×Mω1ΦMt+FM1GM×Mω1ΦMt+B2LMTHM×MβQM×MβϕGM×Mω2ΦMt+FM2GM×Mω2ΦMt+C2KMTHM×MαQM×MαϕΦMt+FM1ΦMt+D2LMTHM×MβQM×MβϕΦMt+FM2ΦMtFM4ΦMt=0.

Rearranging the terms in above system (50) and using the notation for simplicity

S=FM1A1Gν1+C1I+FM2B1Gν2+D1IFM3R=FM1A2Gω1+C2I+FM2B2Gω2+D2IFM4HαQαϕ=P,HβQβϕ=QE50a

then witting in matrix form, we have

KMTLMTΦMt00ΦMt+KMTLMTPA1Gν1+C1IQB1Gν2+D1IPA2Gω1+C2IQB2Gω2+C2IΦMt00ΦMt+SRΦMt00ΦMt=0KMTLMT+KMTLMTPA1Gν1+C1IQB1Gν2+D1IPA2Gω1+C2IQB2Gω2+C2I+SR=0E51

which is an algebraic equation and can easily be solved for unknown matrices KM,LM, then putting it in (38) and (39), we get required approximations for xt,yt.

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5. Numerical examples

In this section, we will apply our proposed scheme to some practical problems.

Example 1.Consider the problems of Buckling of a thin vertical column such that when a constant compressive forceFis applied to the said thin column of lengthLhanged at both ends whose general differential equations in the absence of source term are given by

ELdαydxα+Fy=ft,1<α2,subject to the boundary conditionsy0=y1,yL=y2,E52

where E,Iare Young modulus and moment of inertia, respectively. Where E=2.6×107lb/in,I=0.25π2R4,where Ris the radius of the column. Taking L=1in,R=1inthe exact deflection at α=2is yx=sinπx. We apply our technique introduced in section 4for various choices of a,b,M. The observations are given in Table 1 for the absolute error yappyexact.

Mabyappyexactabyappyexact
2008×102114×102
4006×103113×103
6221.2×103112×103
80.50.58×1041.51.52×104
8001×1050.50.52×105
10202.3×105024×105

Table 1.

Maximum absolute error for T=1,α=1.8,ν1=ν2=0for example, (1).

In Table 2, we have shown the maximum absolute error for various choices of a,b,M. Also in Figures 1 and 2, we have shown the images of their comparison between exact and approximate values. Further, we observed that as the α2, the error is reducing and one can check that at α=2, the absolute error is below 1.8×1013as given in the next table. From above tables, we observe that as order α2the approximation converges to the value at α=2. This phenomenon shows the accuracy of the approximate solutions.

Mαν1ν2YappYexact
42111×103
62117×1017
1.80.90.82×104
81.710.82×104
101.90.9990.9993×105

Table 2.

Maximum absolute error at a=b=0,and for different α,ν1,ν2for example, (2).

Figure 1.

Subplot (a) represents comparison of approximate and exactyxof example 1. Stars and dots represent exact solution while the red curve represents approximate solution. Settingα=1.8,M=4,T=1,a=b=1.the subplot represents their absolute error atM=8.

Figure 2.

Subplot (a) represents comparison of approximate and exactyxof example 1.Red dots represent exact solution while the blue curve represents approximate solution. Settingα=1.8,a=b=0,M=8,T=1.the subplot (b) represents their absolute error atM=8.

Example 2.Consider the following general problem

DαY+ADν1Y+BDν2Y+CY=fx,1<α2,0<ν1,ν21,subject to the boundary conditionsY0=1,Y1=5.E53

Let the exact solution Yx=x4+x3+x2+x+1,and A=1,B=2,C=3,the source terms

fx=3x4+3x3+15x2+9x+5+24Γ215x165+6Γ165x115+2Γ115x65+1Γ65x15+48Γ235x185+12Γ185x135+4Γ135x85+2Γ85x35.E54

We approximate the solutions of this problem with proposed method by setting α=2,ν1=0.8,ν2=0.4,a=b=1,M=8,T=1.and observed that the scheme provides much more accurate approximations for very small-scale level even at scale level M=8maximum absolute error is 9×1017as shown in Figure 3.

Figure 3.

Subplot (a) represents comparison of approximate and exactYxof example 2. Blue curve represents exact solution while the red square dots represent approximate solution. The subplot (b) represents their absolute error atM=8.

The absolute error for different choices of a,band Mand for α=2,ν1=0.8,ν2=0.4is given in Table 3.

Mabαyappyexact
4001.74×103
61.84×105
111.91.5×104
23.5×1013
60021.8×1013

Table 3.

Maximum absolute error at various values of α,a,band Mfor example, (1).

From Table 4, we observed that as the orders of derivatives tend to their corresponding integer values, the approximate solutions tend to its exact value, which demonstrate the accuracy of numerical solutions obtained in our proposed method.

MabYappYexact
4001×103
60.50.51.6×1015
8119×1017
80.50.52.5×1015
10024×1016

Table 4.

Maximum absolute error at α=2,ν1=0.8,ν2=0.4for example, (2).

In the following examples, we solve some coupled systems under the given conditions by our proposed method.

Example 3.Consider the coupled system given by

DαX+A1Dν1X+B1Dν2Y+C1X+D1Y=fx,1<α2,0<ν1,ν21,DβY+A2Dω1X+B2Dω2Y+C2X+D2Y=gx,1<β2,0<ω1,ω21,E55

subject to the boundary conditions

X0=0.5,X1=0.5Y0=0.6,Y1=0.6.E56

We solve this problem under the given parameters

A1=2,A2=2,B1=2,B2=2,C1=3,C2=4,D1=5,D2=5,α=β=2,ν1=ω2=1,ν1=ω1=1,a=b=0.

Let the exact solutions at α=β=2,νi=ωi=1i=12are given by

Xx=x2Tx2+0.5,Yx=xTx3+0.6,E57

the source terms are given by

fx=4x2x26xx125xx13+2x122x13+3x2x12+3172393274221175x7550x285x+3433495522228568064+2x2+92E58
gx=2x22x22xx125xx136x122x133x2x2+3x2x12+92E59

We approximate the coupled systems at scale level M=6for the given parameters. We see from Figures 4 and 5 that our method works very well, and the absolute error is about 3×1016for Xxand for Yxthis amount is 7×1016.Which is very small numbers, which prove the applicability of our methods. Absolute error for various a,b,Mis given in next Table 5.

Figure 4.

Subplot (a) represents comparison of approximate and exactXxof example 3. Blue curve represents exact solution while the red circles represent approximate solution. The subplot (b) represents their absolute error atM=6.

Figure 5.

Subplot (c) represents comparison of approximate and exactYxof example 3. Blue dots represent exact solution while the red curve represents approximate solution. The subplot (d) represents their absolute error atM=6.

MabXappXexactYappYexact
4001×1031×103
63×10167×1016
6114.5×10167×1016
80.514.5×10166×1016
810.53.5×10161.2×1015

Table 5.

Maximum absolute error at various values of a,band Mfor example, (3).

Absolute error for specific a,band for various α,β,ν1,ν2,ω1,ω2is given in Table 6.

Mαβν1ν2ω1ω2XappXexactYappYexact
41.51.50.50.50.50.52.5×1031×103
61.61.60.60.60.60.63×1032×103
61.71.70.70.70.70.76×1041.2×103
81.81.80.80.80.80.84.5×1041×104
81.91.90.90.90.90.99×1052×105
102211114.5×10166×1016

Table 6.

Maximum absolute error at specific values of a=b=0for example, (3).

From Table 5, we see the effect of a,bat various scale level M. While Table 6 indicates the effect of various choices of α,β,ν1,ν2,ω1,ω2at different scale level. As the scale level increases, values of α,β,ν1,ν2,ω1,ω2tend to their integer values. The absolute error is reducing and the solutions are tending to their exact value, which demonstrate the applicability of the proposed method.

Example 4.Consider the mathematical models of fractionally damped coupled system of spring masses system whose model is given by

DαX+λm1Dν1Xμm1Dν2Y+κ1m1Xκ2m1Y=fx,1<α2,0<ν1,ν21,DβYλm2Dω1X+λ+μm2Dω2Yκ1m2X+κ1+κ2m2Y=gx,1<β2,0<ω1,ω21,E60

subject to the boundary conditions

X0=0,X1=0Y0=1,Y1=1.E61

Where the source terms are given by

fx=cosπxsinπxπcosπxπsinπxπ2sinπxgx=sinπx2cosπx+πcosπx+2πsinπxπ2cosπx.E62

Where λ,μare damping parameters and κ1,κ2are spring constants and m1,m2are masses of objects and springs are hanged from both ends. We solve this problem under the following values:

λ=2,μ=2,κ1=2,κ2=2,m1=m2=2,then calculating the coefficient of the problemsasA1=1,A2=1,B1=1,B2=2,C1=1,C2=1,D1=1,D2=1.

Let the exact solutions at α=β=2,νi=ωi=1i=12are given by

Xx=sinπx,Yx=cosπxE63

The above model is approximated for the solutions by our proposed methods (Figures 6 and 7). We observed that our method provides best approximate solutions to the problems for small-scale level M=5.We also find numerical solutions for fractional values of α,β,νi,ωii=12. We observe that as these orders tend to their integer values, the solutions are tending to the exact, which prove the applicability of our method in the following tables.

Figure 6.

Subplot (a) represents comparison of approximate and exactXxatM=5. The subplot (b) represents comparison of approximate and exactYxatM=5for example,(4). Blue curve represents exact solution while the red curve represents approximate solution fora=b=0.

Figure 7.

Subplot (c) represents absolute errorXappXexact, and subplot (d) represents their absolute errorYappYexactatM=6for example,(4)fora=b=0.

Absolute error for various a,b,Mand α=β=2,νi=ωi=1i=12is given in Table 7.

MabXappXexactYappYexact
4001×1031×103
510×1046×103
69×1079×106
6114×1037×103
80.5.52×1031×103
810.53.5×1031.2×103

Table 7.

Maximum absolute error at various values of a,band Mfor example, (4).

We observe from Table 7 that values of a,bplay important role in the approximate solution. By giving integral value to a,b,methods give best approximate solution for the given problem. Similarly for a=b=0and for various choices of α,β,ν1,ν2,ω1,ω2, the absolute errors are given in Table 8.

Mαβν1ν2ω1ω2XappXexactYappYexact
41.51.50.50.50.50.52.5×1021×102
61.61.60.60.60.60.64×1032×103
61.71.70.70.70.70.71×1031×103
81.81.80.80.80.80.84.5×1041×104
81.91.90.90.90.90.91.2×1052×105
102211119×1078×106

Table 8.

Maximum absolute error at a=b=0and for various α,β,ν1,ν2,ω1,ω2for example, (4).

From Table 8, it is obvious that when the orders of the derivatives approach to their integral values, the error is reducing and the approximate solutions are converging to the exact solutions. This phenomenon indicates that the proposed method is highly accurate.

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6. Conclusion and discussion

In this article, we have studied shifted Jacobi polynomials. Based on these polynomials, we recalled some already existing matrices of fractional order derivative and integrations from the literature as well as we constructed a new operational matrix corresponding to boundary conditions. By means of these operational matrices, we converted the system of fractional differential equations to simple and easily soluble system of algebraic equations. There is no need of Tau-collocation method. The simple algebraic equations were easily solved, and the result were plotted. From the plot, we observe that our proposed method is highly accurate and can be applied to a variety of problems of fractional order ordinary as well as partial differential equations. We also compared our results to the exact solutions and observed that our method gave satisfactory results. The proposed method can easily and accurately can be applied to a variety of problems of applied mathematics, engineering, and physics, etc.

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Acknowledgments

The authors Aziz Khan, Kamal Shah and Thabet Abdeljawad would like to thank Prince Sultan university for the support through TAS research lab.

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Mathematics Subject Classification:

26A33; 34A08; 34K37.

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Written By

Kamal Shah, Eiman, Hammad Khalil, Rahmat Ali Khan and Thabet Abdeljawad

Reviewed: December 16th, 2021 Published: February 27th, 2022