# Numerical Simulation of Coupled Fractional Differential-Integral Equations Utilizing the Second Kind Chebyshev Wavelets

Authors: Wei Shi; Jiaquan Xie; Tao Wang; Yanjie Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to solve coupled fractional differential-integral equations more effectively and to deal with the problem that the huge algebraic equations lead to considerable computational complexity and large data storage requirements in the calculation process, this paper approximates the function of the unknown solution based on the Chebyshev wavelet of the second kind and then combines the collocation method to solve the numerical solution of nonlinear fractional Fredholm integral-differential equations. By using the proposed method, the original problem can be reduced to a system of linear algebraic equations, which can be easily solved by some mathematical techniques. In addition, the convergence analysis of the system based on the second kind of Chebyshev wavelet is studied. Several numerical test problems are presented, and the absolute error values under different fractional orders are given, which proves the superiority and effectiveness of the proposed method. It provides support for improving the precision and reliability of the system.


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## Body

## 1. Introduction

Fractional differential-integral belongs to the field of applied mathematics and can handle integrals and derivatives of any order. It has the advantage of being a practical tool for reasonably explaining the memory and genetic quality of various materials and processes. With the development of natural science and social economy, it is found that integer order differential equation cannot be used to solve complex problems, but these problems can be solved by fractional calculus equation [1]. Therefore, differential-integral equation plays a very important role in various problems of natural science and related fields of engineering technology [2]. There are different types of integral equations, such as Fredholm integral equations, Volterra integral equations, and Volterra-Fredholm integral equations, depending on the structure of the integral [3–5]. Nonlinear integral equations have been widely studied in many different disciplines, such as vehicular traffic, biology, optimal control theory, economics. [6, 7]. In this paper, numerical calculations are carried out for a class of system of coupled fractional Fredholm differential-integral equations. The general form of the research question is as follows:(1)DtαY1t=λ11∫01k11x,tY1xp1dx+λ12∫01k12x,tY2xp2dx+g1tDtβY2t=λ21∫01k21x,tY1xq1dx+λ22∫01k22x,tY2xq2dx+g2t,where p1,p2,q1,q2>1, and the initial conditions are given as follows:(2)Y1i0=δi,i=0,1,…,r−1,∍r−1<α<r,r∈N∗,Y2j0=γj,j=0,1,…,m−1,∍m−1<β<m,m∈N∗,where g1,g2∈L20,1,k11,k12,k21,k22∈L20,1×0,1 are given functions, Y1t,Y2t are the solutions to be determined, Dtα,Dtβ refer to fractional derivatives in the Caputo sense, and p,q are positive integers.In recent years, different types of numerical methods have been proposed for solving fractional differential and integral equations. These methods mainly include the variation of parameters method [8], ADM [9, 10], VIM and HPM [11], CAS wavelet [12], and Taylor expansion method [13]. The wavelet methods stand out among these methods and have been adopted in various scientific or engineering applications. The wavelets method makes the accurate representation of different functions and operators and the establishment of the connection with fast numerical algorithms possible [14]. Wavelet methods, including CAS wavelet [15], Legendre wavelet [16], Haar wavelet [17], Chebyshev wavelet [18], and others, are widely used in solving linear or nonlinear differential equations, integral equations, and integro-differential equations. In [12], the numerical method of CAS wavelet is applied to solve the fractional integral and differential equations, and the numerical results are compared with the analytical results. In [19], the second kind of Chebyshev wavelet method is utilized to obtain the numerical solutions of fractional nonlinear Fredholm integral-differential equations. In recent years, wavelet methods develop rapidly [20–22]. At present, some scholars have done in-depth research on the time delay and the existence and controllability of equations for fractional differential-integral system [23–27]. In view of the problem, the huge algebraic equations will lead to considerable computational complexity and large data storage requirements in the calculation process. Because of the structural redundancy of the second kind of Chebyshev wavelet, the computational complexity of the algebraic system can be reduced. In order to more effectively solve the coupled fractional differential-integral equations [28, 29], the second kind of Chebyshev wavelet method is introduced in this paper, and the original problem can be reduced to a system of linear algebraic equations, which can be easily solved by some mathematical techniques. In the study, the main system of fractional Fredholm integral-differential equations is discussed in detail by the second kind Chebyshev wavelet, and the convergence analysis of the system is investigated.The main structure of the article is as follows: in Section2, some necessary definitions of the fractional integral-differential are introduced. In Section 3, the second kind Chebyshev wavelet and its operational matrix of the fractional integration are derived. In Section 4, the main computing steps are described in detail. In Section 5, the convergence analysis of the system is investigated. In Section 6, the effectiveness of the proposed method is verified by several test problems. Finally, a conclusion is drawn in Section 7.


## 2. Fractional Calculus

In the section, some necessary definitions and mathematical preliminaries about the fractional integral-differential theory are given [30, 31].Definition 1.
Riemann-Liouville fractional integral operatorℐα of order α defined in the region a,b is given by(3)ℐαYt=1Γα∫0tt−τα−1Yτdτ,α>0,ℐ0Yt=Yt.Definition 2.
Caputo definition of a fractional differential operator is given by(4)DαYt=1Γn−α∫0tt−τn−α−1Ynτdτ,n−1<α≤n,n∈N∗.
Caputo fractional differential operator has some basic properties:(5)DαℐαYt=Yt,and(6)ℐαDαYt=Yt−∑k=0n−1Yk0+t−akk!,t>0.


## 3. The Second Kind Chebyshev Wavelet

### 3.1. Definition

The second kind Chebyshev waveletψnmt=ψk,n,m,t has four arguments, n=1,2,…,2k−1,k∈N∗. t is the normalized time. They are defined on the interval 0,1 as follows [32, 33]:(7)ψnmt=2k/2U˜m2kt−2n+1,≤t<n2k−1,0,o.w.with(8)U˜mt=2πUmt,m=0,1,2,…,M−1.Here,Umt is the second kind Chebyshev polynomials with the weight function wnt=1−t2 and satisfies the following recursive formula U0t=1,U1t=2t,Um+1t=2tUmt−Um−1t,m=1,2,…,. A function Yt defined over Lwn20,1 may be expanded as(9)Yt≃∑n=12k−1∑m=0M−1ynmψnmt=YTΨt=Y˜t,where(10)ynm=Yt,ψnmtωn=∫01ωntYtψnmtdt,and the weight function wnt=w2kt−2n+1. Moreover, Y and Ψt are m^=2k−1M column vectors given by(11)Y=y10,y11,…,y1M−1,y20,y21,…,y2M−1,…,y2k−10,…,y2k−1M−1TΨt=ψ10,ψ11,…,ψ1M−1,ψ20,ψ21,…,ψ2M−1,…,ψ2k−10,…,ψ2k−1M−1T.Taking the collocation points as follows:(12)ti=2i−12kM,i=1,2,…,2k−1M,m^=2k−1M.We define the second kind Chebyshev wavelet matrixΦm^×m^ as(13)Φm^×m^=Ψ12m^,Ψ32m^,…,Ψ2m^−12m^.Any functionkx,t∈Lωn20,1×0,1 can be approximated as(14)kijx,t=ΨxTKijΨt,i,j=1,2,where Kij is m^×m^ matrix with(15)Kij=ψix,kx,t,ψjt.


### 3.2. Fractional Integration Operational Matrix

In the interval0,1, a m^− set of block-pulse functions (BPFs) is defined(16)Bit=1,i/m^≤t<i+1/m^,0,o.w.i=0,1,2,…,m^−1.The functionsBit are disjoint and orthogonal(17)BitBjt=0,i≠jBit,i=j∫01BisBjsds=0,i≠j1/m^,i=j.The second kind Chebyshev wavelet vector can be expressed by anm^-term block-pulse functions as(18)Ψt=Φm^×m^Bm^t.In the previous study, the fractional integral operational matrix of block-pulse functions can be expressed as(19)ℐαBm^t≈FαBm^t,where(20)Bm^t=B0t,B1t,…,Bm^−1tT,and(21)Fα=1m^α1Γα+21ξ1ξ2ξ3⋯ξm^−101ξ1ξ2⋯ξm^−2001ξ1⋯ξm^−3⋮⋮⋱⋱⋮00⋯01ξ1000⋯01.The fractional integration of the second kind Chebyshev wavelet can be given as(22)ℐαΨt=Pm^×m^αΨt,where Pm^×m^α is called fractional integral operational matrix of the second kind Chebyshev wavelet, and it can be given by(23)Pm^×m^α=Φm^×m^FαΦm^×m^−1.For more details, see [32].


## 3.1. Definition

The second kind Chebyshev waveletψnmt=ψk,n,m,t has four arguments, n=1,2,…,2k−1,k∈N∗. t is the normalized time. They are defined on the interval 0,1 as follows [32, 33]:(7)ψnmt=2k/2U˜m2kt−2n+1,≤t<n2k−1,0,o.w.with(8)U˜mt=2πUmt,m=0,1,2,…,M−1.Here,Umt is the second kind Chebyshev polynomials with the weight function wnt=1−t2 and satisfies the following recursive formula U0t=1,U1t=2t,Um+1t=2tUmt−Um−1t,m=1,2,…,. A function Yt defined over Lwn20,1 may be expanded as(9)Yt≃∑n=12k−1∑m=0M−1ynmψnmt=YTΨt=Y˜t,where(10)ynm=Yt,ψnmtωn=∫01ωntYtψnmtdt,and the weight function wnt=w2kt−2n+1. Moreover, Y and Ψt are m^=2k−1M column vectors given by(11)Y=y10,y11,…,y1M−1,y20,y21,…,y2M−1,…,y2k−10,…,y2k−1M−1TΨt=ψ10,ψ11,…,ψ1M−1,ψ20,ψ21,…,ψ2M−1,…,ψ2k−10,…,ψ2k−1M−1T.Taking the collocation points as follows:(12)ti=2i−12kM,i=1,2,…,2k−1M,m^=2k−1M.We define the second kind Chebyshev wavelet matrixΦm^×m^ as(13)Φm^×m^=Ψ12m^,Ψ32m^,…,Ψ2m^−12m^.Any functionkx,t∈Lωn20,1×0,1 can be approximated as(14)kijx,t=ΨxTKijΨt,i,j=1,2,where Kij is m^×m^ matrix with(15)Kij=ψix,kx,t,ψjt.


## 3.2. Fractional Integration Operational Matrix

In the interval0,1, a m^− set of block-pulse functions (BPFs) is defined(16)Bit=1,i/m^≤t<i+1/m^,0,o.w.i=0,1,2,…,m^−1.The functionsBit are disjoint and orthogonal(17)BitBjt=0,i≠jBit,i=j∫01BisBjsds=0,i≠j1/m^,i=j.The second kind Chebyshev wavelet vector can be expressed by anm^-term block-pulse functions as(18)Ψt=Φm^×m^Bm^t.In the previous study, the fractional integral operational matrix of block-pulse functions can be expressed as(19)ℐαBm^t≈FαBm^t,where(20)Bm^t=B0t,B1t,…,Bm^−1tT,and(21)Fα=1m^α1Γα+21ξ1ξ2ξ3⋯ξm^−101ξ1ξ2⋯ξm^−2001ξ1⋯ξm^−3⋮⋮⋱⋱⋮00⋯01ξ1000⋯01.The fractional integration of the second kind Chebyshev wavelet can be given as(22)ℐαΨt=Pm^×m^αΨt,where Pm^×m^α is called fractional integral operational matrix of the second kind Chebyshev wavelet, and it can be given by(23)Pm^×m^α=Φm^×m^FαΦm^×m^−1.For more details, see [32].


## 4. Numerical Implementation

In the section, the second kind Chebyshev wavelet method is presented to numerically solve the system of nonlinear fractional Fredholm integral-differential [34–36] equation (1). Firstly, the functions k11x,t,k12x,t,k21x,t and k22x,t can be approximated by the second kind Chebyshev as(24)k11x,t≅∑n1=12k−1∑l1=0M−1∑m1=12k−1∑v1=0M−1kn1,l1,m1,v1ψn1,l1xψm1,v1x=ΨTxK11Ψtk12x,t≅∑n2=12k−1∑l2=0M−1∑m2=12k−1∑v2=0M−1kn2,l2,m2,v2ψn2,l2xψm2,v2x=ΨTxK12Ψtk21x,t≅∑n3=12k−1∑l3=0M−1∑m3=12k−1∑v3=0M−1kn3,l3,m3,v3ψn3,l3xψm3,v3x=ΨTxK21Ψtk22x,t≅∑n4=12k−1∑l4=0M−1∑m4=12k−1∑v4=0M−1kn4,l4,m4,v4ψn4,l4xψm4,v4x=ΨTxK22Ψt,where(25)K11=kn1,l1,m1,v1,K12=kn2,l2,m2,v2K21=kn3,l3,m3,v3,K22=kn4,l4,m4,v4,and(26)kn1,l1,m1,v1=ψn1,l1x,k1x,t,ψm1,v1tkn2,l2,m2,v2=ψn2,l2x,k2x,t,ψm2,v2tkn3,l3,m3,v3=ψn3,l3x,k3x,t,ψm3,v3tkn4,l4,m4,v4=ψn4,l4x,k4x,t,ψm4,v4t.Moreover,(27)g1t=G1TΨtg2t=G2TΨt.Suppose that(28)DtαY1t≅Y1TΨtDtαY2t≅Y2TΨt.For simplification, assumingδi=γj=0, then by equations (2), (7), (24), and (28), we can obtain(29)Y1t≅Y1TPm^×m^αΦm^×m^Bm^tY2t≅Y2TPm^×m^αΦm^×m^Bm^t.Let(30)r=r0,r1,…,rm′−1=Y1TPm^×m^αΦm^×m^s=s0,s1,…,sm′−1=Y2TPm^×m^αΦm^×m^,then Y1t≅rBm^t. According to the property of BPFs, we have(31)Y1t2≅rBm^t2=r0B0t+r1B1t+⋯+rm^−1Bm^−1t2=r02B0t+r12B1t+⋯+rm^−12Bm^−1t=r˜2Bm^t,where r˜2=r02,r12,…,rm^−12. Through induction, we can easily obtain(32)Y1tp1≅r0p1,r1p1,…,rm′−1p1Bm^t=r˜p1Bm^tY1tq1≅r0q1,r1q1,…,rm′−1q1Bm^t=r˜q1Bm^tY2tp2≅s0p2,s1p2,…,sm′−1p2Bm^t=s˜p2Bm^tY2tq2≅s0q2,s1q2,…,sm′−1q2Bm^t=s˜q2Bm^t,where(33)r˜p1=r0p1,r1p1,…,rm^−1p1r˜q1=r0q1,r1q1,…,rm^−1q1s˜p2=s0p2,s1p2,…,sm^−1p2s˜q2=s0q2,s1q2,…,sm^−1q2.Then, we have(34)∫01k11x,tY1xp1dt=∫01ΨTtK11ΨxBm′Txr˜p1Tdx=∫01ΨTtK11Ψm′×m′Bm′xBm′Txr˜p1Tdx=ΨTtK11Φm′×m′∫01Bm^xBm^Txr˜p1Tdx,∫01k12x,tY2xp2dx=∫01ΨTtK12ΨxBm^Txs˜p2Tdx=∫01ΨTtK12Φm^×m^Bm′xBm^Txs˜p2Tdx=ΨTtK12Φm^×m^∫01Bm^xBm^Txs˜p2Tdx,and(35)∫01k21x,tY1xq1dt=∫01ΨTtK21ΨxBm^Txr˜q1Tdx=∫01ΨTtK21Φm^×m^Bm^xBm^Txr˜q1Tdx=ΨTtK21Φm^×m^∫01Bm^xBm^Txr˜q1Tdx,where(36)∫01Bm^xBm^Txs˜p2Tdx=∫01B0xoB1x⋱oBm^−1xs0p2s1p2⋮sm^−1p2dx=∫01s0p2B0x,s1p2B1x,…,sm^−1p2Bm^−1xTdx=1m^s0p2,s1p2,…,sm^−1p2T,∫01Bm^xBm^Txs˜q2Tdx=∫01B0xoB1x⋱oBm^−1xs0q2s1q2⋮sm^−1q2dx=∫01s0q2B0x,s1q2B1x,…,sm^−1q2Bm^−1xTdx=1m^s0q2,s1q2,…,sm^−1q2T,∫01Bm^xBm^Txr˜p1Tdx=∫01B0xoB1x⋱oBm^−1xr0p1r1p1⋮rm^−1p1dx=∫01r0p1B0x,r1p1B1x,…,rm^−1p1Bm^−1xTdx=1m^r0p1,r1p1,…,rm^−1p1T,and(37)∫01Bm^xBm^Txr˜q1Tdx=∫01B0xoB1x⋱oBm^−1xr0q1r1q1⋮rm^−1q1dx=∫01r0q1B0x,r1q1B1x,…,rm^−1q1Bm^−1xTdx=1m^r0q1,r1q1,…,rm^−1q1T.Then,(38)∫01k11x,tY1xp1dx=1m^ΨTtK11Φm^×m^r˜p1∫01k21x,tY1xq1dx=1m^ΨTtK21Φm^×m^r˜q1∫01k12x,tY2xp2dx=1m^ΨTtK12Φm^×m^s˜p2∫01k22x,tY2xq2dx=1m^ΨTtK22Φm^×m^s˜q2.Substituting equations (28), (29), and (43) into (1), we have(39)ΨTtY1−λ111m^ΨTtK11Φm^×m^r˜p1−λ121m^ΨTtK12Φm^×m^s˜p2=ΨTtG1,ΨTtY2−λ211m^ΨTtK21Φm^×m^r˜q1−λ221m^ΨTtK22Φm^×m^s˜q2=ΨTtG2.Obviously, (39) is a system of linear algebraic equations. Dispersing the unknown variable t using Matlab software, we can obtain r0,r1,…,rm^−1 and s0,s1,…,sm^−1, and then, by (29), the solutions of (1) can be obtained.


## 5. Convergence Analysis

In this section, we focus on the convergence analysis of the system. In fact, by (9), Y˜t converge (in the mean) to Yt as k approaches to ∞. If the function Yt is several times differentiable, we can bound the error, as established by the following theorem.Theorem 1.
Suppose that the functionY:0,1⟶R is m times continuously differentiable and Y∈Cm0,1. Then, Y˜t approximate Yt with mean error bounded as follows:(40)Yt−Y˜t2≤22mk−14mm!supt∈0,1Ymt.Proof.
See [37].
Assume thatCJ,⋅ is the Banach space of all continuous functions on the interval J=0,1 with norm(41)Y1t∞=max∀t∈JY1t,Y2t∞=max∀t∈JY2t.
Let(42)Fp1Y1x=Y1xp1Gp2Y2x=Y2xp2Fq1Y1x=Y1xq1Gq2Y2x=Y2xq2,and(43)ξ11x,t≤M11ξ12x,t≤M12ξ21x,t≤M21ξ22x,t≤M22∀0<x,t≤1.
Suppose that the nonlinear termsFp1Y1x,Fq1Y1x,Gp2Y2x, and Gq2Y2x satisfy the Lipschitz conditions:(44)Fp1Y1x−Fp1Y˜1x≤L11Y1x−Y˜1xFq1Y1x−Fq1Y˜1x≤L21Y1x−Y˜1xGp2Y2x−Gp2Y˜2x≤L12Y2x−Y˜2xGq2Y2x−Gq2Y˜2x≤L22Y2x−Y˜2x,where Y1t,Y2t are the m times continuous differentiable analytical functions of the system, and Y˜1t,Y˜2t are the approximate functions of the system, respectively.
Then, we have(45)Y1t−Y˜1t∞≤==max∀t∈JY1t−Y˜1t≤max∀t∈JIαθ11∫01ξ11x,tFp1Y1x−Fp1Y˜1xdx+θ12∫01ξ12x,tGp2Y2x−Gp2Y˜2xdx≤θ11M11L11Γ1+αmaxt∈JY1t−Y˜1t+θ12M12L12Γ1+αmaxt∈JY2t−Y˜2t≤θ11M11L11Γ1+αY1t−Y˜1t2+θ12M12L12Γ1+αY2t−Y˜2t2≤θ11M11L11Γ1+α22mk−14mm!supt∈0,1Y1mt+θ12M12L12Γ1+α22mk−14mm!supt∈0,1Y2mt,and(46)Y2t−Y˜2t∞=max∀t∈JY2t−Y˜2t≤max∀t∈JIβθ21∫01ξ21x,tFp2Y1x−Fp2Y˜1xdx+θ22∫01ξ22x,tGq2Y2x−Gq2Y˜2xdx≤θ21M21L21Γ1+βmaxt∈JY1t−Y˜1t+θ22M22L22Γ1+βmaxt∈JY2t−Y˜2t≤θ21M21L21Γ1+βY1t−Y˜1t2+θ22M22L22Γ1+βY2t−Y˜2t2≤θ21M21L21Γ1+β22mk−14mm!supt∈0,1Y1mt+θ22M22L22Γ1+β22mk−14mm!supt∈0,1Y2mt.


## 6. Numerical Experiments

In order to verify the effectiveness of the proposed method via the following examples, we defined the 2-norm error as(47)Yit−Y˜it2=∫01Yit−Y˜it2dt1/2≅1N∑i=0NYitj−Y˜itj21/2j=1,2,…,N,i=1,2,where Yit is the analytical solution of this system, and Y˜it is the approximate solution of this system.Test Problem 1.
Considering the following nonlinear system of fractional-order Fredholm differential-integral equations(48)Dt2.5Y1t=∫01x+tY1x2dx+∫01xtY2x2dx+12t0.5/π−15t56−18Dt2.5Y2t=∫01x−tY1x2dx+∫01xt2Y2x2dx+12t0.5/π+t7−18−t28,with the zero initial conditions: Y10=Y20=Y1′0=Y2′0=Y1″0=Y2″0=0. The analytical solutions of the system are Y1t=t3 and Y2t=−t3.
The numerical results of the system with some value ofk,M are shown in Tables 1 and 2. From Tables 1 and 2, it can be concluded that the numerical solutions approach to the analytical solutions as k,M increases. Tables 3 and 4 show the absolute error tables of Y1 and Y2, respectively. It can be seen from the error that the error decreases with the increase of the discrete term.Table 1
The numerical and analytical solutions ofY1t when k=4,M=3,4,5.
tAnalytical solutionsM=3M=4M=50.10.0010000000.0012453540.0010452540.0010054520.30.0270000000.0272363630.0270334140.0270056360.50.1250000000.1258574360.1250646320.1250078680.70.3430000000.3437454250.3430867680.3430079650.90.7290000000.7313636370.7290125260.729007683Table 2
Absolute error ofY1t when k=4,M=3,4,5.
tM=3M=4M=50.12.4535 × 10−44.5254 × 10−55.4520 × 10−60.32.3636 × 10−43.3414 × 10−55.6360 × 10−60.58.5744 × 10−46.4632 × 10−57.8680 × 10−60.77.4543 × 10−48.6768 × 10−57.9650 × 10−60.92.3636 × 10−31.2526 × 10−57.6830 × 10−6Table 3
The numerical and analytical solutions ofY2t when k=4,M=3,4,5.
tAnalytical solutionsM=3M=4M=50.1−0.001000000−0.001143453−0.001046533−0.0010076530.3−0.027000000−0.027442879−0.020877439−0.0270046560.5−0.125000000−0.125879658−0.125026536−0.1250066730.7−0.343000000−0.343636322−0.343096420−0.3430078750.9−0.729000000−0.728576698−0.729064631−0.729005628Table 4
Absolute error ofY2t when k=4,M=3,4,5.
tM=3M=4M=50.11.4345 × 10−44.6533 × 10−57.6530 × 10−60.34.4288 × 10−46.1226 × 10−34.6560 × 10−60.58.7966 × 10−42.6536 × 10−56.6730 × 10−60.76.3632 × 10−49.6420 × 10−57.8750 × 10−60.94.2330 × 10−46.4631 × 10−55.6280 × 10−6Test Problem 2.
Considering the following nonlinear system of fractional-order Fredholm differential-integral equations(49)Dt4/3Y1t=5∫01xtY1x2dx+6∫01x2tY2x1.5dx+Γ5/2Γ7/6t1/6−2tDt5/3Y2t=3∫01txY1x2dx+6∫01x2t2Y2x1.5dx+2Γ4/3t1/3−t−t20≤x,t<1,with the initial conditions:(50)Y10=Y1′0=0,Y20=Y2′0=0.
The analytical solutions of the system areY1t=t3/2 and Y2t=t2. When M=4,k=3,4,5, the obtained numerical results are shown in Tables 5 and 6. From Tables 5 and 6, it can be concluded that the numerical solutions approach to the analytical solutions as k,M increases.Table 5
The numerical and analytical solutions ofY1t with k=3,4,5 for Test Problem 2.
tAnalytical solutionsk=3k=4k=50.10.03160.03080.03140.03160.20.08940.08800.08920.08940.30.16430.16210.16400.16430.40.25300.25030.25250.25300.50.35360.35070.35310.35360.60.46480.46120.46420.46480.70.58570.58230.58500.58570.80.71550.71190.71470.71550.90.85380.84900.85300.8538Table 6
The numerical and analytical solutions ofY2t with k=3,4,5 for Test Problem 2.
tAnalytical solutionsk=3k=4k=50.10.01000.00920.00980.01000.20.04000.03890.03970.04000.30.09000.08780.08950.09000.40.16000.15740.15950.16000.50.25000.24700.24940.25000.60.36000.35630.35930.36000.70.49000.48560.48920.49000.80.64000.63380.63930.64000.90.81000.80200.80910.8100Test Problem 3.
Considering the nonlinear system of first-order Fredholm differential-integral equations(51)Y1′t=∫01x2tY2x3dx+πcosπt−0.241213254623034tY2′t=∫01xt2Y1x3dx+cosht−2t23π,subject to the initial conditions Y10=Y20=0. The analytical solutions of the system are Y1t=sinπt and Y2t=sinht.
Whenm^=12k=M=3,m^=32k=M=4, and m^=80k=M=5, the 2-norm errors are listed in Table 7. Table 7 shows that the proposed method has a good convergence precision for solving these kinds of problems.
Through the analysis of the above two experimental examples, it can be concluded that the numerical results approach to the analytical results asm grows. The results show that the absolute error between the analytical and numerical results can achieve a high convergence precision.
Figures1–3 show the absolute error of Y1 and Y2 under different fractional orders. It can be seen from the figure that the error is increasing with the increase of the order, but it is not an absolute increase. Due to some calculation factors, the error will also fluctuate to a certain extent.Table 7
The 2-norm errorsY1t−Y˜1t2 and Y˜2t−Y˜2t2 with m^=12,32,80.
m^=12m^=32m^=80Y1t−Y˜1t22.3764 × 10−33.7629 × 10−45.0317 × 10−6Y2t−Y˜2t23.8206 × 10−34.1846 × 10−47.9300 × 10−6Figure 1
Absolute error diagram ofY1 and Y2 when k=4,M=4, (a) e1t=Y1t−Y˜1t, (b) e2t=Y2t−Y˜2t.
(a)(b)Figure 2
Absolute error diagram ofY1 and Y2 when k=3,M=5, (a) e1t=Y1t−Y˜1t, (b) e2t=Y2t−Y˜2t.
(a)(b)Figure 3
Absolute error diagram ofY1 and Y2 when k=M=5. (a) e1t=Y1t−Y˜1t, (b) e2t=Y2t−Y˜2t.
(a)(b)


## 7. Conclusions

In this paper, in view of the problem that the huge algebraic equations will lead to considerable computational complexity and large data storage requirements in the calculation process, the second kind Chebyshev wavelet method is applied to obtain the numerical solutions of nonlinear system of fractional Fredholm integral-differential equations. Using this method, the system of differential-integral equations has been reduced to a system of algebraic equations. The results show that the second Chebyshev wavelet method is used; the accuracy is improved with the increase ofK and M values. In addition, the convergence analysis of the system based on the second kind of Chebyshev wavelet is studied. The analysis is carried out by several numerical experiments, and the absolute error values under different fractional orders are given, which proves the superiority and effectiveness of the proposed method. It provides support for improving the precision and reliability of the system.



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