# Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

Authors: Guodong Shi; Yanlei Gong; Mingxu Yi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.


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## Body

## 1. Introduction

In this study, the operational alternative Legendre method is introduced and employed to solve a class of nonlinear fractional integro-differential equations with weakly singular kernel:(1)Dtαyt=Fyt+∫0tt−s−βGysds+ft,α>0,0≤β<1,t∈0,1,(2)yi0=y0i,i=0,1,2,…,α−1,where F:C0,1⟶ℝ and G:C0,1⟶ℝ (sufficiently smooth operators) are considered to be nonlinear. Dtα are the Caputo fractional derivative operators, α is the ceiling function of α, ft is a continuous function, and yt is an unknown function to be determined.These equations come from the mathematical modeling of various phenomena, such as radiative equilibrium, heat conduction problems, elasticity, and fracture mechanics (see [1–5]). Since the numerical solution of the nonlinear fractional equations is almost a new subject and because of having a singular kernel, there are many schemes for solving this kind of equations. Heydari et al. have utilized the Chebyshev wavelet method to solve systems of the linear and nonlinear singular fractional Volterra integro-differential equations (see [6]). Mohammadi has applied the block pulse functions for the linear and nonlinear singular fractional integro-differential equations (see [7]). Zhao et al. have proposed the piecewise polynomial collocation method for solving the fractional integro-differential equations with weakly singular kernels (see [8]). In [9, 10], the operational Tau method has been used to solve this kind of equations. A spectral method based on the Chebyshev polynomials of the second kind has been applied in [11]. Yi et al. have dealt with the CAS wavelets and Legendre wavelets method for solving the linear and nonlinear fractional weakly singular integro-differential equations (see [12, 13]), and so on (see [14–16]).In this paper, application of the alternative Legendre polynomials is extended to solve the nonlinear weakly singular fractional order integro-differential equations. For this purpose, the fractional operational matrices of integration and product are derived. Also, an operational matrix is derived to approximate the integral part with the singular kernel in equation (1). The matrices and approximations resulted are substituted into the given equation to convert it into the nonlinear system of algebraic equations. The nonlinear systems can be solved by the well-known Newton iteration method. Also, error analysis and convergence analysis of the proposed method are presented.


## 2. Definitions of Fractional Derivatives and Integrals

Definition 1.
Riemann–Liouville fractional integral of orderα,α≥0 (see [17]):(3)Itαyt=1Γα∫0tt−τα−1yτdτ,t>0,It0yt=yt,where Γα=∫0∞tα−1e−tdt is the gamma function.Definition 2.
Caputo’s fractional derivative of orderα,α≥0 (see [17]):(4)Dtαyt=1Γr−α∫0tyrτt−τα−r+1dτ,0≤r−1<α<r.
Specially, the operatorDtα satisfies the following properties (c is a constant) (see [18]):(5)Dtαc=0,Dtαtβ=0,β∈N0,β<α,Γβ+1Γβ+1−αtβ−α,β∈N0,β≥α;β∉N0,β>α,DtαItαyt=yt,ItαDtαyt=yt−∑k=0α−1yk0+tkk!,t>0.


## 3. Alternative Legendre Polynomials and Their Operational Matrices

The setPn=Pnktk=0n of alternative Legendre polynomials of degree n in 0,1 (see [19]) is given by(6)Pnkt=∑j=0n−k−1jn−kjn+k+j+1n−ktk+j,k=0,1,…,n.They are orthogonal over the interval0,1 with respect to the weight function ωt=1 and satisfy the property as follows:(7)∫01PnktPnltdt=1k+l+1,k=l0,k≠l,k,l=0,1,…,n.Equation (6) may be rewritten as Rodrigues’s type:(8)Pnkt=1n−k!1tk+1dn−kdtn−ktn+k+11−tn−k,k=0,1,…,n.Integrating equation (8) from 0 to 1, we have(9)∫01Pnktdt=1n+1,k=0,1,…,n.Suppose the alternative Legendre polynomials vector as(10)Φt=Pn0t,Pn1t,…,PnntT.Equation (10) can be rewritten as the form(11)Φt=Q⋅Tt,where(12)Tt=1,t,t2,…,tnT,and Q is the upper triangular matrix defined by(13)Q=qkjk,j=0n,qkj=0,0≤j<k,−1j−kn−kj−kn+j+1n−k,k≤j≤n.A square integrable functionyt∈L20,1 can be expressed in terms of the alternative Legendre polynomials basis as follows:(14)yt=∑k=0∞ckPnkt.In practice, only the firstn+1 term of alternative Legendre polynomials is considered. Hence, one has(15)yt≈∑k=0nckPnkt=CTΦt,where C=c0,c1,…,cnT, ck=y,Pnk/Pnk,Pnk=2k+1y,Pnk,k=0,1,…,n are called alternative Legendre polynomials coefficients, and y,y=∫01y2tdt.In implementing the operations on the alternative Legendre basis, we frequently encounter the integration and the product of the vectorΦt and also it is necessary to evaluate the integration, the product of the vector Φt. For this purpose, some operational matrices will be derived. To pursue, we need the following lemmas. The proofs of these lemmas are quite easy by using the definition.Lemma 1.
LetPnkt=∑r=0nprktr and Pnjt=∑r=0nprjtr be the kth and jth alternative Legendre polynomials, respectively. Then, the product of Pnkt and Pnjt can be written as(16)PnktPnjt=∑r=02nqrk,jtr,where qrk,j=∑l=0rplkpr−lj,r≤n∑l=r−nnplkpr−ljr>n.Lemma 2.
Letr be a positive integer, then we have(17)∫01trPnktdt=∑l=0n−k−1ln−kln+k+l+1n−kk+l+r+1,k=0,1,…,n.Lemma 3.
LetPnit, Pnjt, and Pnkt be ith, jth, and kth alternative Legendre polynomials, respectively. Then, we have(18)∫01PnitPnjtPnktdt=∑r=02nqrk,j∑l=0n−i−1ln−iln+i+l+1n−ki+l+r+1.
The alternative Legendre polynomials operational matrix of fractional integration is shown by the following theorem.Theorem 1.
LetΦt be the alternative Legendre polynomials vector obtained by equation (10), then we have the following:(19)IαΦt=J⋅Φt,where J=pkrk,r=0n is the alternative Legendre polynomials operational matrix of fractional integration of order n+1×n+1 in which(20)pkr=2r+1∑j=0n−k−1jn−kjn+k+j+1n−kΓα+k+j+1∑l=0n−r−1ln−rln+r+l+1n−rk+l+r+j+1+α.
The following theorem presents a general formula for finding the operational matrix of the product.Theorem 2.
LetC=c0,c1,…,cnT be an arbitrary vector, then we have(21)ΦtΦTtC=C^Φt,where C^=c^iki,k=0n is the alternative Legendre polynomials operational matrix of the product in which(22)c^ik=2k+1∑j=0ncjξijk,ξijk=∫01PnitPnjtPnktdt.
The values ofξijk are calculated by using Lemma 3.Proof.
(see [20]).Theorem 3.
Suppose thatyt∈C0,1,0<β<1, and yt=CTΦt, then(23)∫0tyst−sβds=CTBβΦt,where Bβ=bikβi,k=0n is a n+1×n+1 matrix and its elements are as follows:(24)bikβ=2r+1∑j=0n−k−1jn−kjn+k+j+1n−kΓk+j+1Γ1−βΓk+j+2−β×∑l=0n−r−1ln−rln+r+l+1n−rk+l+r+j+2−β.Proof.
According to the definition of the vectorΦt, one has(25)∫0tyst−sβds≈CT∫0tΦst−sβds,=CT∫0tPn0st−sβds,∫0tPn1st−sβds,…,∫0tPnnst−sβdsT.
Using equation (6) leads to(26)∫0tPnkst−sβds=∑j=0n−k−1jn−kjn+k+j+1n−k∫0tsk+jt−sβds,=∑j=0n−k−1jn−kjn+k+j+1n−kΓk+j+1Γ1−βΓk+j−β+2tk+j−β+1.
Using equation (15), we can approximate tk+j−β+1 in terms of alternative Legendre polynomials as follows:(27)tk+j−β+1=∑r=0nbkjrPnrt,where, by Lemma 2, one can obtain(28)bkjr=2r+1∫01tk+j−β+1Pnrtdt=2r+1∑l=0n−r−1ln−rln+r+l+1n−rk+l+r+j+2−β.
Therefore,(29)∫0tPnkst−sβds=∑r=0n2r+1∑j=0n−k−1jn−kjn+k+j+1n−kΓk+j+1Γ1−βΓk+j+2−β×∑l=0n−r−1ln−rln+r+l+1n−rk+l+r+j+2−βPnrt,=∑r=0nbkrβPnrt.
So, equation (25) can be written as follows:(30)∫0tΦst−sβds=∫0tPn0st−sβds,∫0tPn1st−sβds,…,∫0tPnnst−sβdsT,=b00βb01β…b0nβb10βb11β⋮b1nβ⋮⋮⋱⋮bn0βbn1β…bnnβPn0tPn1t⋮Pnnt.
This proof is completed.


## 4. Solution of Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

To explain the implementation process of the proposed method, the following case of equation (1) is considered:(31)Dtαyt=Fym1t+∫0tt−s−βym2sds+ft,α>0,0≤β<1,t∈0,1,where m1 and m2 are the positive integers.By applying the introduced matrices and the basis vector in the previous section, the terms of the equations under investigation may be approximated in terms of the alternative Legendre polynomials as follows.Dtαyt can be approximated as(32)Dtαyt≈CTΦt.By applying the fractional integral operator of orderα to equation (28), an approximation of unknown function yt results as follows:(33)yt≈CTJΦt+∑k=0α−1yk0+tkk!=CTJΦt+XTΦt=YTΦt,where XT is a known n+1 vector and YT=CTJ+XT.Using Theorem2, we can obtain(34)ym1t≈YTC^m1−1Φt,(35)ym2t≈YTC^m2−1Φt,where C^m1−1 and C^m2−1 are m1−1 and m2−1 order, the alternative Legendre polynomials operational matrix of the product.Using Theorem3 and equation (35), the integral parts in equation (31) are approximated as follows:(36)∫0tt−s−βym2sds≈YTC^m2−1∫0tt−s−βΦsds=YTC^m2−1BβΦt.After substituting approximations equations (32)–(36) into equation (31), the main equation is converted into the following nonlinear algebraic equation:(37)CTΦt≈FYTC^m1−1Φt+YTC^m2−1BβΦt+ft.Equation (37) is collocated at the n+1 Gauss–Chelyshkov nodes (see [21]). By solving the nonlinear systems of the generated algebraic equations, the unknown vector CT can be determined. The resultant nonlinear systems can be solved by the Newton iteration method.


## 5. Error Analysis and Convergence

### 5.1. Error Estimate

Assume thatCn+10,1 is the space of functions and y:0,1⟶R are the continuous derivatives,(38)yit=didtiyt,t∈0,1,for all i such that 0≤i≤n+1, then we have the following theorem.Theorem 4.
Suppose thatyt∈Cn+10,1 and ynt=CTΦt are yt expansion in terms of alternative Legendre polynomials. Then, for t∈0,1,(39)yt−ynt2≤Cn+1!22n+1,where C is a constant such that(40)yn+1t≤C.Proof.
Assuming thatqnt are the interpolating polynomials to yt at points tl (tl are the roots of the shifted Chebyshev polynomials of degree n+1), one has(41)yt−qnt=yn+1ηn+1!∏l=0nt−tl,η∈0,1.
The estimates for Chebyshev interpolation nodes is that (see [22])(42)yt−qnt≤Cn+1!22n+1,∀t∈0,1.
As discussed in [20], ynt is the unique best approximation of yt, then we get(43)yt−ynt22≤yt−qnt22,=∫01yt−qnt2dt=∫01Cn+1!22n+12dt=Cn+1!22n+12.
Therefore,(44)yt−ynt2≤Cn+1!22n+1.
This proof is completed.


### 5.2. Convergence Analysis

Before the convergence analysis, we firstly transform equation (1) into an equivalent nonlinear Volterra integral equation with the same initial condition (2).Theorem 5.
If the right-hand side of equation (1) is continuous, then equations (1) and (2) are equivalent nonlinear Volterra integral equations as follows:(45)yt=gt+∫0tk1t,τGyτdτ,where(46)gt=∑i=0α−1y0itii!+1Γα∫0tt−τα−1Fyτdτ+1Γα∫0tt−τα−1fτdτ.(47)k1t,τ=Γ1−βΓα−β+1t−τα−β.Proof.
Applying the Riemann–Liouville integral operatorItα to both sides of equation (1), we have(48)yt=gt+1Γα∫0tt−τα−1∫0ττ−s−βGysdsdτ,where gt is defined by equation (46).
Using the Dirichlet’s formula to the last part of the right-hand side of equation (48), we get(49)1Γα∫0tt−τα−1∫0ττ−s−βGysdsdτ=1Γα∫0t∫0τt−τα−1τ−s−βGysdsdτ,=1Γα∫0t∫τtt−sα−1s−τ−βGyτdsdτ.
Letw=s−τ/t−τ, then equation (49) will be(50)1Γα∫0t∫τtt−sα−1s−τ−βGyτdsdτ=1Γα∫0tt−τα−β∫011−wα−1w−βdwGyτdτ,=Bα,1−βΓα∫0tt−τα−βGyτdτ=ΓαΓ1−βΓα−β+1⋅1Γα∫0tt−τα−βGyτdτ=∫0tΓ1−βΓα−β+1t−τα−βGyτdτ=∫0tk1t,τGGyτdτ,where B is the beta function.
This proof is completed.
Letgnt be the approximation of gt and fnt be the alternative Legendre approximation of ft and suppose the nonlinear terms Fu and Gu are satisfied in Lipschitz condition such that(51)Fu−Fv≤L1u−v,(52)Gu−Gv≤L2u−v.Theorem 6.
The solution of the nonlinear fractional order integro-differential equation (45) by using alternative Legendre polynomials approximations is convergent when n⟶+∞.Proof.
(53)yt−ynt=gt−gnt+∫0tk1t,τGyτdτ−∫0tk1t,τGynτdτ,≤gt−gnt+∫0tk1t,τGyτ−Gynτdτ=I1t+I2t,where I1t=gt−gnt and I2t=∫0tk1t,τGyτ−Gynτdτ.
Using Theorem4 and equation (51), we obtain(54)I1t=gt−gnt=1Γα∫0tt−τα−1Fyτ−Fynτdτ+∫0tt−τα−1fτ−fnτdτ,≤1Γα∫0tt−τα−1Fyτ−Fynτdτ+1Γα∫0tt−τα−1fτ−fnτdτ≤L1Γα∫0tt−τα−1yτ−ynτdτ+1Γα∫0tt−τα−1fτ−fnτdτ≤L1CΓαn+1!22n+1∫0tt−τα−1dτ+CΓαn+1!22n+1∫0tt−τα−1dτ≤L1+1CΓα+1n+1!22n+1.
Combining Theorem4 and equation (52) together, one has(55)I2t=∫0tk1t,τGyτ−Gynτdτ,≤∫0tk1t,τ⋅Gyτ−Gynτdτ≤L2∫0tk1t,τ⋅yτ−ynτdτ≤L2Cn+1!22n+1∫0tk1t,τdτ≤L2CΓ1−βn+1!22n+1Γα−β+1∫0tt−τα−βdτ≤L2CΓ1−βΓα−β+2n+1!22n+1.
Combining equations (53)–(55), we can get the relationship between the convergence order and α,β as follows:(56)yt−ynt≤L1+1C/Γα+1+L2CΓ1−β/Γα−β+2n+1!22n+1.
From equation (56), one has yt−ynt⟶0 when n⟶+∞.
This proof is completed.
Note: the conditions of Theorem6 are the same as Theorems 4 and 5.
By combining Theorems5 and 6, we can easily obtain the following theorem.Theorem 7.
The solution of the nonlinear fractional order integro-differential equation (1) by using alternative Legendre polynomials approximations is convergent when n⟶+∞.


## 5.1. Error Estimate

Assume thatCn+10,1 is the space of functions and y:0,1⟶R are the continuous derivatives,(38)yit=didtiyt,t∈0,1,for all i such that 0≤i≤n+1, then we have the following theorem.Theorem 4.
Suppose thatyt∈Cn+10,1 and ynt=CTΦt are yt expansion in terms of alternative Legendre polynomials. Then, for t∈0,1,(39)yt−ynt2≤Cn+1!22n+1,where C is a constant such that(40)yn+1t≤C.Proof.
Assuming thatqnt are the interpolating polynomials to yt at points tl (tl are the roots of the shifted Chebyshev polynomials of degree n+1), one has(41)yt−qnt=yn+1ηn+1!∏l=0nt−tl,η∈0,1.
The estimates for Chebyshev interpolation nodes is that (see [22])(42)yt−qnt≤Cn+1!22n+1,∀t∈0,1.
As discussed in [20], ynt is the unique best approximation of yt, then we get(43)yt−ynt22≤yt−qnt22,=∫01yt−qnt2dt=∫01Cn+1!22n+12dt=Cn+1!22n+12.
Therefore,(44)yt−ynt2≤Cn+1!22n+1.
This proof is completed.


## 5.2. Convergence Analysis

Before the convergence analysis, we firstly transform equation (1) into an equivalent nonlinear Volterra integral equation with the same initial condition (2).Theorem 5.
If the right-hand side of equation (1) is continuous, then equations (1) and (2) are equivalent nonlinear Volterra integral equations as follows:(45)yt=gt+∫0tk1t,τGyτdτ,where(46)gt=∑i=0α−1y0itii!+1Γα∫0tt−τα−1Fyτdτ+1Γα∫0tt−τα−1fτdτ.(47)k1t,τ=Γ1−βΓα−β+1t−τα−β.Proof.
Applying the Riemann–Liouville integral operatorItα to both sides of equation (1), we have(48)yt=gt+1Γα∫0tt−τα−1∫0ττ−s−βGysdsdτ,where gt is defined by equation (46).
Using the Dirichlet’s formula to the last part of the right-hand side of equation (48), we get(49)1Γα∫0tt−τα−1∫0ττ−s−βGysdsdτ=1Γα∫0t∫0τt−τα−1τ−s−βGysdsdτ,=1Γα∫0t∫τtt−sα−1s−τ−βGyτdsdτ.
Letw=s−τ/t−τ, then equation (49) will be(50)1Γα∫0t∫τtt−sα−1s−τ−βGyτdsdτ=1Γα∫0tt−τα−β∫011−wα−1w−βdwGyτdτ,=Bα,1−βΓα∫0tt−τα−βGyτdτ=ΓαΓ1−βΓα−β+1⋅1Γα∫0tt−τα−βGyτdτ=∫0tΓ1−βΓα−β+1t−τα−βGyτdτ=∫0tk1t,τGGyτdτ,where B is the beta function.
This proof is completed.
Letgnt be the approximation of gt and fnt be the alternative Legendre approximation of ft and suppose the nonlinear terms Fu and Gu are satisfied in Lipschitz condition such that(51)Fu−Fv≤L1u−v,(52)Gu−Gv≤L2u−v.Theorem 6.
The solution of the nonlinear fractional order integro-differential equation (45) by using alternative Legendre polynomials approximations is convergent when n⟶+∞.Proof.
(53)yt−ynt=gt−gnt+∫0tk1t,τGyτdτ−∫0tk1t,τGynτdτ,≤gt−gnt+∫0tk1t,τGyτ−Gynτdτ=I1t+I2t,where I1t=gt−gnt and I2t=∫0tk1t,τGyτ−Gynτdτ.
Using Theorem4 and equation (51), we obtain(54)I1t=gt−gnt=1Γα∫0tt−τα−1Fyτ−Fynτdτ+∫0tt−τα−1fτ−fnτdτ,≤1Γα∫0tt−τα−1Fyτ−Fynτdτ+1Γα∫0tt−τα−1fτ−fnτdτ≤L1Γα∫0tt−τα−1yτ−ynτdτ+1Γα∫0tt−τα−1fτ−fnτdτ≤L1CΓαn+1!22n+1∫0tt−τα−1dτ+CΓαn+1!22n+1∫0tt−τα−1dτ≤L1+1CΓα+1n+1!22n+1.
Combining Theorem4 and equation (52) together, one has(55)I2t=∫0tk1t,τGyτ−Gynτdτ,≤∫0tk1t,τ⋅Gyτ−Gynτdτ≤L2∫0tk1t,τ⋅yτ−ynτdτ≤L2Cn+1!22n+1∫0tk1t,τdτ≤L2CΓ1−βn+1!22n+1Γα−β+1∫0tt−τα−βdτ≤L2CΓ1−βΓα−β+2n+1!22n+1.
Combining equations (53)–(55), we can get the relationship between the convergence order and α,β as follows:(56)yt−ynt≤L1+1C/Γα+1+L2CΓ1−β/Γα−β+2n+1!22n+1.
From equation (56), one has yt−ynt⟶0 when n⟶+∞.
This proof is completed.
Note: the conditions of Theorem6 are the same as Theorems 4 and 5.
By combining Theorems5 and 6, we can easily obtain the following theorem.Theorem 7.
The solution of the nonlinear fractional order integro-differential equation (1) by using alternative Legendre polynomials approximations is convergent when n⟶+∞.


## 6. Numerical Examples

In this section, some numerical examples are presented to illustrate the proposed alternative Legendre polynomials method. In order to demonstrate the error of the method, the notation is shown as(57)en=max0≤i≤nyti−ynti,where yt and ynt are the exact solution and the numerical solution, respectively, ti∈0,1. All algorithms are performed by Mathematica 10.0.Example 1.
Consider the following nonlinear fractional order integro-differential equation with weakly singular kernel (see [23]):(58)Dt2/3yt=gt+ptyt+∫0tt−s−1/2y2sds,t∈0,1,where(59)gt=3Γ1/24Γ11/6t5/6−t5/2−3235t7/2,pt=t,with initial value y0=0.
The exact solution of this equation isyt=t3/2. The absolute errors between the numerical solutions and the exact solution with different values of n are displayed in Table 1 and compared with the results obtained by MHFs (modification of hat functions) (see [23]). From Table 1, it can be seen that the absolute errors become smaller and smaller with n increasing. Satisfactory results would be acquired by a small number of alternative Legendre polynomials. Figure 1 shows the comparison of en obtained by our method and MHFs for different n on logarithmic scale. From Figure 1, we find that en are smaller than that, and the accuracy of our method is higher.Table 1
The absolute errors with differentn for Example 1.
tn=4n=5n=6n=7n=8 (see [23])00.000000.000000.000000.000000.000000.1252.82184e-031.28015e-036.59465e-043.98172e-041.59756e-030.251.27453e-035.94191e-044.86154e-044.02646e-045.69812e-040.3751.10784e-038.65911e-046.26588e-043.89888e-046.53198e-040.51.80480e-031.09916e-035.90305e-043.92223e-048.56779e-040.6252.46826e-031.12048e-037.02201e-045.51706e-048.88166e-040.752.88819e-031.45003e-031.07896e-037.13425e-041.42594e-030.8753.92612e-032.48974e-031.50493e-031.05219e-031.81991e-0317.69399e-033.50573e-032.68452e-031.64642e-033.30803e-03Figure 1
en on logarithmic scale for different methods (Example 1).Example 2.
Consider the nonlinear fractional integro-differential equation with weakly singular kernel as follows (see [23]):(60)Dtαyt=gt+∫0tt−s−1/2y2sds,t∈0,1,where gt=3t2−πΓ7/Γ15/2t23/2, with y0=0.
The exact solution of the nonlinear equation forα=1 is yt=t3. The numerical solutions and comparisons are given in Table 2 and Figure 2. The absolute errors for α=1 are listed in Table 2. It can be seen that the values of the absolute errors decay as n increases from 5 to 8. The comparisons of en obtained by our method and MHFs for different n on logarithmic scale are plotted in Figure 2. This figure shows good coincidence between the numerical and exact solutions.
According to Figure2, it can be deduced that the results obtained by the alternative Legendre polynomials method are more precise than those obtained by MHFs. Figure 3 displays the numerical results for α=0.7,0.8,0.9,1 and the exact result for α=1. It shows that the numerical solutions converge to the exact solution as α⟶1.Table 2
The absolute errors with differentn for Example 2α=1.
tn=5n=6n=7n=8n=16 (see [23])00.000000.000000.000000.000000.000000.1255.24499e-042.26218e-058.97476e-079.93225e-084.75615e-090.254.42941e-041.70091e-056.88273e-078.62125e-081.46318e-070.3753.55136e-041.66330e-057.81163e-079.54967e-087.47778e-070.53.81245e-041.94631e-058.00897e-078.92258e-082.25373e-060.6254.57360e-041.95593e-057.51114e-079.51004e-085.19464e-060.754.84405e-041.86419e-058.55455e-071.03849e-071.01464e-050.8754.77051e-042.25186e-059.59522e-071.08441e-071.76618e-0517.12625e-042.23668e-051.22798e-061.22564e-072.81379e-05Figure 2
en on logarithmic scale for different methods (Example 2).Figure 3
Numerical solutions (differentα) and exact solution α=1 with n=6 for Example 2.Example 3.
Consider this equation:(61)Dt4/5yt=gt+y2t+∫0tt−s−1/2y2sds,t∈0,1,where(62)gt=Γ4Γ16/5t11/5+Γ7/3Γ23/15t8/15−t6−2t13/3−t8/3−πΓ7Γ15/2t13/2+Γ11/3Γ25/6t19/6+2Γ16/3Γ35/6t29/6,with initial value y0=0.
The exact solution of this equation isyt=t3+t4/3. The figures of the numerical solutions and the exact solution for the problem with different n are shown in Figures 4–7. From Figures 4–7, we can see that the approximate solutions are getting close to the exact solutions with n increasing.Figure 4
Comparison of the numerical solutions with exact solution forn=6.Figure 5
Comparison of the numerical solutions with exact solution forn=8.Figure 6
Comparison of the numerical solutions with exact solution forn=10.Figure 7
Comparison of the numerical solutions with exact solution forn=12.Example 4.
Consider the following nonlinear fractional integro-differential equation with weakly singular kernel:(63)Dtαyt=gt+sint⋅yt+cost⋅y2t+∫0tt−s−1/2y2sds,t∈0,1,where gt=mtm−1−sint⋅tm−cost⋅t2m−πΓ2m+1/Γ2m+1.5t2m+0.5, such that y0=0.
The exact solution isyt=tm if α=1, and m is a positive integer. The graphs of en for different m on logarithmic scale are depicted in Figures 8–10. From Figures 8–10, one can find that for large n, the error norm becomes smaller and their logarithm behaves approximately as a linear function of n, which means that the error norm decreases linearly.Figure 8
en on logarithmic scale for m=3 (Example 4).Figure 9
en on logarithmic scale for m=4 (Example 4).Figure 10
en on logarithmic scale for m=5 (Example 4).Example 5.
As the last example, consider the nonlinear fractional integro-differential equation with weakly singular kernel as follows:(64)Dtαyt=gt+siny2t+∫0tt−s−1/2y2sds,t∈0,1,where gt=mtm−1−sintm−πΓ2m+1/Γ2m+1.5t2m+0.5, such that y0=0.
The exact solution of this equation is alsoyt=tm for α=1. The approximate solutions are plotted in Figure 11 for the values of α=0.7,0.8,0.9,1 and m=4,n=6. Figure 11 shows that the exact and approximate solutions are in good agreement when α=1. It also illustrates that the numerical solutions converge to the exact solution as α⟶1.Figure 11
Numerical solutions (differentα) and exact solution α=1 with m=4,n=6 for Example 5.


## 7. Conclusion

The alternative Legendre polynomials operational matrix method has been developed for solving the weakly singular nonlinear integro-differential equations with fractional derivatives. The method exploits the operational matrices of the fractional integration and the product of alternative Legendre polynomials as the primary underlying tool. A given nonlinear equation is converted into a nonlinear system of algebraic equations. The method has also been analyzed for error and convergence. A new operational matrix is obtained to approximate the integral parts with the singular kernels. To illustrate the reliability and efficiency of the suggested approach, the alternative Legendre operational matrices and the relevant approximations are employed to solve several examples. In the examples in which the order of the fractional derivatives is known, the numerical results show good agreement between the numerical and exact solutions. When the order of the fractional derivativeα is uncertain, the numerical solutions for the various values of α approach to the exact solutions as α⟶1. The results obtained by our method are also compared with those obtained by some existing methods. From the tables and figures, it can be concluded that the results obtained by the suggested method are more precise than those. The results confirm the ability of the alternative Legendre polynomials method to solve the nonlinear fractional integro-differential equations with weakly singular kernels.



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