# Bivariate Chebyshev Polynomials to Solve Time-Fractional Linear and Nonlinear KdV Equations

Authors: Azam Zahrani; Mashaallah Matinfar; Mostafa Eslami
Journal: Journal of Mathematics
 (2022)

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## Abstract

This work concerns the numerical solutions of a category of nonlinear and linear time-fractional partial differential equations (TFPDEs) that are called time-fractional inhomogeneous KdV and nonlinear time-fractional KdV equations, respectively. The fractional derivative operators are of the Caputo type. Two-variable second-kind Chebyshev wavelets (SKCWs) are constructed using one-variable ones; then, utilizing corresponding integral operational matrices leads to an approximate solution to the problem under study. Also, it is found that the perturbation term tends to zero even if a finite number of the basis functions is adopted. To exhibit the applicability and efficiency of the proposed scheme, two models of the KdV equations are given.


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## Body

## 1. Introduction

Many scientists and researchers are interested in fractional integral and derivative operators as mathematical tools for modeling diverse physical, chemical, and biological phenomena [1–5]. Fractional operators have the memory property, and this characteristic converts them into a powerful tool for studying real-world problems [6–8]. Different fractional derivative operators have been introduced by researchers for successfully and effectively modeling scientific phenomena. For example, the fractional pseudohyperbolic telegraph partial differential equation employing the Caputo fractional derivative was solved in [9] utilizing the explicit finite difference method. Generalized Caputo and Caputo fractional derivatives were studied in [10], and the nonlinear heat equation in the sense of the generalized Caputo derivative was solved by fractional Green’s functions, the generalized Laplace transform, and generalized Mellin transform. A type of the fractional diffusion equation in the sense of the Grunwald–Letnikov derivative was solved by Gorenflo and Abdel-Rehim in [11] using a difference scheme. The 2+1-dimensional fractional Ablowitz–Kanup–Newell–Segur equation in the sense of the conformable derivative was studied to extract general analytical wave solutions in [12] implementing the exp−ϕξ-expansion method. A modified definition of the conformable fractional derivative was presented in [13], and then, the exact solutions of linear and nonlinear time- and space-fractional mixed partial differential equations involving a new fractional derivative were obtained applying the invariant subspace method. Abu-Shady and Kaabar proposed the generalized fractional derivative (GFD) and showed that this operator coincides with the Caputo and Riemann–Liouville fractional derivatives [14, 15]. Therefore, this computational tool can be used to model different scientific phenomena. Nonlinear fractional partial differential equations (FPDEs) have attracted wide attention for describing many phenomena in engineering, physics, material science, and acoustics [7, 16–19]. Korteweg and de Vries introduced a class of nonlinear evolution equations, namely, KdV equations, for the first time in 1895, to describe the nonlinear shallow-water waves [20]. The KdV equations emerge in diverse phenomena of physics like the one-dimensional waves in shallow-water waves; the Ferma-Pasto-Ulam problem in the continuum limit; the evolution of long, ion-acoustic waves in a plasma, and so on. Time-fractional KdV equations are obtained by replacing the first-order time derivatives with fractional ones of the arbitrary orders. Many works have been done on the KdV and generalized KdV equations. For example, Bagheri and Khani used rational functions, trigonometric functions, and hyperbolic functions, to reach the exact solutions of a fractional model of the KdV equation [21]. A balance method was given to obtain some closed forms of solutions of the KdV equation in [22]. Authors in [23] applied the q-homotopy analysis transform method to study the modified coupled KdV equations. An extended tanh-function method was used in [24] to achieve soliton solutions of modified coupled KdV and generalized Hirota–Satsuma coupled KdV equations. Sahoo and Saha applied the G′/G-expansion method to solve the time-fractional KdV equation [25]. Kaya et al. applied radial basis functions to KdV and mKdV equations [26]. Momani et al. [27] utilized the variational iteration method for time-fractional KdV. The analytical traveling wave solutions of the nonlinear fractional KdV equation are obtained by introducing an approximate-analytical method in [28]. Authors in [29] dealt with obtaining exact solutions to the fractional KdV equation. In [30–34], the new iteration method, Adomian decomposition method, variational iteration method, and homotopy perturbation method were utilized to derive approximate solutions to different forms of the KdV equations.The target of the current work is to achieve approximate solutions for two models of the KdV by means of the second-kind Chebyshev wavelets. From a viewpoint of comparison, the proposed method has a less computational size compared to some existing methods. The orthogonal second-kind Chebyshev polynomials are utilized as basis functions in diverse methods to obtain approximate solutions of integrodifferential equations [35], integral equations [36, 37], ordinary differential equations [38, 39], and partial differential equations [40, 41]. In the present paper, an approach based on the second-kind Chebyshev polynomials is presented to work out time-fractional inhomogeneous KdV and nonlinear time-fractional KdV equations. Finding analytic solutions to linear and especially nonlinear equations is hard; hence, presenting or modifying computational methods to find an approximate solution to these problems is noteworthy.The main goal of this paper is to assess the numerical solutions of the linear inhomogeneous fractional KdV equation and nonlinear time-fractional KdV equations. An orthogonal collocation scheme is proposed based upon the SKCW functions. Two-dimensional integral operational matrices of fractional and integer orders are derived utilizing one-dimensional ones. Resultant matrices accompanied by the collocation method convert the main problem into an algebraic equation by collocating this algebraic equation at tensor pointsθi,ϑj,i=0,1,⋯,M1,j=0,1,⋯,M2 leading to a linear or nonlinear algebraic system. θi and ϑj are roots of the second-kind Chebyshev polynomials of degrees M1 and M2, respectively. By solving the resulted system, an approximate solution is achieved.The organization of this paper is as follows: the fractional operators, one- and two-variable second-kind Chebyshev wavelets are introduced, and then, operational matrices of the integral are derived in Section2. In Section 3, two models of the equations under study are presented. Then, it can be seen how using appropriate approximations results in a residual function. In Section 4, some error bounds of the resulted approximations are computed. The established approach is utilized for two equations in Section 5, and a conclusion is provided in the last section.


## 2. Fractional Operators and SKCWs

This section presented some definitions of the fractional calculus, the SKCWs are introduced, and their integral operational matrices of integer and fractional orders are gained.


### 2.1. Fractional Operators

Definition 1.
The Caputo fractional derivative operator ofgθ,ϑ∈CnΩ with the order μ∈ℝ is given as the following [42]:
(1)D0Cϑμgθ,ϑ=1Γn−μ∫0ϑϑ−ηn−μ−1∂ngθ,η∂ηndη,n−1<μ<n,n∈ℕ,∂ngθ,ϑ∂ϑn,μ=n∈ℕ.Definition 2.
The Riemann-Liouville fractional integral operator ofgθ,ϑ∈CΩ with the order μ∈ℝ is given as [42]
(2)J0RLϑμgθ,ϑ=1Γμ∫0ϑϑ−ημ−1gθ,ηdη,μ>0,J0RLϑ0gθ,ϑ=gθ,ϑ.Some features of the above-mentioned operators are as follows:(3)J0RLϑμ1J0RLϑμ2gθ,ϑ=J0RLϑμ2J0RLϑμ1gθ,ϑ=J0RLϑμ1+μ2gθ,ϑ,J0RLϑμtσ=Γσ+1Γσ+μ+1ϑσ+μ,σ>−1,D0Cϑμϑσ=0,μ>σ,Γσ+1Γσ−μ+1ϑσ−μ,σ≥μ,D0CϑμJ0RLϑμgθ,ϑ=gθ,ϑ,J0RLϑμD0Cϑμgθ,ϑ=gθ,ϑ−gθ,0,0<μ≤1.


### 2.2. SKCWs

The one-variable second-kind Chebyshev waveletψnmϑ is defined on the interval J=0,1 as
(4)ψnmϑ=2k/2U~m2kϑ−2n+1,n−12k−1<ϑ<n2k−1,0,otherwise,where U~mϑ=2/πUmϑ,m=0,1,⋯,M−1. Umϑ,m=0,1,⋯,M−1, are the Chebyshev polynomials of the second kind which are orthogonal with respect to the weight function ωϑ=1−ϑ21/2 on the interval −1,1; on the other hand,
(5)∫−11UmϑUkϑωϑdϑ=π2,m=k,0,m≠k.These polynomials are obtained from the following formula:(6)Um+1ϑ=2ϑUmϑ−Um−1ϑ,m=1,2,⋯,U0ϑ=1,U1ϑ=2ϑ.From (4), ψnmϑ involves four arguments, n=1,⋯,2k−1,k∈ℕ,m is the degree of the second-kind Chebyshev polynomials, and ϑ is the time variable. The SKCWs are orthogonal with respect to the weight functions ωnϑ=ω2kϑ−2n+1,n=1,2,⋯,2k−1, over the interval Jn=n−1/2k−1,n/2k−1.Every functiong∈Lωn2Jn can be expanded as
(7)gϑ=∑∞n=1∑∞m=0Gnmψnmϑ,where
(8)Gnm=∫01gϑψnmϑωnϑdϑ.Using a truncated form of the series in (7), an approximation to gϑ is gained as follows:
(9)gϑ≈gmϑ=∑2k−1n=1∑M−1m=0Gnmψnmϑ=G¯TΨ¯ϑ,where G¯ and Ψ¯ϑ are 2k−1M-order vectors as follows:
(10)G¯=G10,G11,⋯,G1M−1,G20,G21,⋯,G2M−1,⋯,G2k−10,G2k−11,⋯,G2k−1M−1T,Ψ¯ϑ=ψ10ϑ,ψ11ϑ,⋯,ψ1M−1ϑ,ψ20ϑ,ψ21ϑ,⋯,ψ2M−1ϑ,⋯,ψ2k−10ϑ,ψ2k−11ϑ,⋯,ψ2k−1M−1ϑT.The two-variable SKCWs can be defined on the intervalJ=0,1×0,1 using (4) as follows:
(11)ψn1m1n2m2θ,ϑ=2k1+k2/2U~m12k1θ−2n1+1U~m22k2ϑ−2n2+1,n1−12k1−1<θ<n12k1−1,n2−12k2−1<t<n22k2−1,0,otherwise,where ni=1,⋯,2ki−1,mi=0,1,⋯,Mi−1,ki∈ℕ,i=1,2. It is clear that ψn1m1n2m2θ,ϑ=ψn1m1θψn2m2ϑ. Every two-variable g∈LWn1n22J can be written as follows:
(12)gθ,ϑ=∑∞n1=1∑∞m1=0∑∞n2=1∑∞m2=0Gn1m1n2m2ψn1m1n2m2θ,ϑ,where the coefficients Gn1m1n2m2 are computed as
(13)Gn1m1n2m2=∫01∫01gθ,ϑψn1m1n2m2θ,ϑWn1n2θ,ϑdϑdθ,and Wn1n2θ,ϑ=ωn1θωn2ϑ. By considering the truncated series of the infinite series in (12), one gets the following approximation to gθ,ϑ:
(14)gθ,ϑ≈gM1M2θ,ϑ=∑2k1−1n1=1∑M1−1m1=0∑2k2−1n2=1∑M2−1m2=0Gn1m1n2m2ψn1m1n2m2θ,ϑ=GTΔθ,ϑ=GTΨθ⊗Ψϑ,where G and Δ are 2k1−1M12k2−1M2×1 vectors and ⊗ denotes the Kronecker product.


### 2.3. Operational Matrices of the Integration

The integration of the one-variable basis in (10) can be approximated as
(15)∫0ϑΨ¯ηdη≈P1Ψ¯ϑ,where P1 is the operational matrix of the integration, and its entries are calculated as
(16)P1i,j≔∫0ϑΨ¯iηdη,Ψ¯jϑωn,i,j=1,2,⋯,2k−1M.IfJ0RLϑμ is the fractional integral of order μ>0 [7], then the operational matrix of the integration of the fractional order μ, Pμ, is given as
(17)J0RLϑμΨ¯ϑ≈PμΨ¯ϑ,where
(18)J0RLϑμΨ¯ϑ=J0RLϑμψ10ϑ,⋯,J0RLϑμψ1M−1ϑ,⋯,J0RLϑμψ2k−10ϑ,⋯,J0RLϑμψ2k−1M−1ϑT,J0RLϑμψmnϑ=2k/2J0RLϑμU~m2kϑ−2n+1,n−12k−1<ϑ<n2k−1,0,otherwise.Now, the two-dimensional operational matrices of the integration are constructed usingP1 and Pμ:
(19)∫0θΔξ,ϑdξ≈PθΔθ,ϑ=P1⊗IΔθ,ϑ,∫0ϑΔθ,ηdη≈PϑΔθ,ϑ=I⊗P1Δθ,ϑ,J0RLϑμΔθ,η≈PϑμΔθ,ϑ=I⊗PμΔθ,ϑ,where Pθ, Pϑ, and Pϑμ are two-dimensional operational matrices regarding the classic and fractional integral operators, respectively, and I is the M×M identity matrix.


## 2.1. Fractional Operators

Definition 1.
The Caputo fractional derivative operator ofgθ,ϑ∈CnΩ with the order μ∈ℝ is given as the following [42]:
(1)D0Cϑμgθ,ϑ=1Γn−μ∫0ϑϑ−ηn−μ−1∂ngθ,η∂ηndη,n−1<μ<n,n∈ℕ,∂ngθ,ϑ∂ϑn,μ=n∈ℕ.Definition 2.
The Riemann-Liouville fractional integral operator ofgθ,ϑ∈CΩ with the order μ∈ℝ is given as [42]
(2)J0RLϑμgθ,ϑ=1Γμ∫0ϑϑ−ημ−1gθ,ηdη,μ>0,J0RLϑ0gθ,ϑ=gθ,ϑ.Some features of the above-mentioned operators are as follows:(3)J0RLϑμ1J0RLϑμ2gθ,ϑ=J0RLϑμ2J0RLϑμ1gθ,ϑ=J0RLϑμ1+μ2gθ,ϑ,J0RLϑμtσ=Γσ+1Γσ+μ+1ϑσ+μ,σ>−1,D0Cϑμϑσ=0,μ>σ,Γσ+1Γσ−μ+1ϑσ−μ,σ≥μ,D0CϑμJ0RLϑμgθ,ϑ=gθ,ϑ,J0RLϑμD0Cϑμgθ,ϑ=gθ,ϑ−gθ,0,0<μ≤1.


## 2.2. SKCWs

The one-variable second-kind Chebyshev waveletψnmϑ is defined on the interval J=0,1 as
(4)ψnmϑ=2k/2U~m2kϑ−2n+1,n−12k−1<ϑ<n2k−1,0,otherwise,where U~mϑ=2/πUmϑ,m=0,1,⋯,M−1. Umϑ,m=0,1,⋯,M−1, are the Chebyshev polynomials of the second kind which are orthogonal with respect to the weight function ωϑ=1−ϑ21/2 on the interval −1,1; on the other hand,
(5)∫−11UmϑUkϑωϑdϑ=π2,m=k,0,m≠k.These polynomials are obtained from the following formula:(6)Um+1ϑ=2ϑUmϑ−Um−1ϑ,m=1,2,⋯,U0ϑ=1,U1ϑ=2ϑ.From (4), ψnmϑ involves four arguments, n=1,⋯,2k−1,k∈ℕ,m is the degree of the second-kind Chebyshev polynomials, and ϑ is the time variable. The SKCWs are orthogonal with respect to the weight functions ωnϑ=ω2kϑ−2n+1,n=1,2,⋯,2k−1, over the interval Jn=n−1/2k−1,n/2k−1.Every functiong∈Lωn2Jn can be expanded as
(7)gϑ=∑∞n=1∑∞m=0Gnmψnmϑ,where
(8)Gnm=∫01gϑψnmϑωnϑdϑ.Using a truncated form of the series in (7), an approximation to gϑ is gained as follows:
(9)gϑ≈gmϑ=∑2k−1n=1∑M−1m=0Gnmψnmϑ=G¯TΨ¯ϑ,where G¯ and Ψ¯ϑ are 2k−1M-order vectors as follows:
(10)G¯=G10,G11,⋯,G1M−1,G20,G21,⋯,G2M−1,⋯,G2k−10,G2k−11,⋯,G2k−1M−1T,Ψ¯ϑ=ψ10ϑ,ψ11ϑ,⋯,ψ1M−1ϑ,ψ20ϑ,ψ21ϑ,⋯,ψ2M−1ϑ,⋯,ψ2k−10ϑ,ψ2k−11ϑ,⋯,ψ2k−1M−1ϑT.The two-variable SKCWs can be defined on the intervalJ=0,1×0,1 using (4) as follows:
(11)ψn1m1n2m2θ,ϑ=2k1+k2/2U~m12k1θ−2n1+1U~m22k2ϑ−2n2+1,n1−12k1−1<θ<n12k1−1,n2−12k2−1<t<n22k2−1,0,otherwise,where ni=1,⋯,2ki−1,mi=0,1,⋯,Mi−1,ki∈ℕ,i=1,2. It is clear that ψn1m1n2m2θ,ϑ=ψn1m1θψn2m2ϑ. Every two-variable g∈LWn1n22J can be written as follows:
(12)gθ,ϑ=∑∞n1=1∑∞m1=0∑∞n2=1∑∞m2=0Gn1m1n2m2ψn1m1n2m2θ,ϑ,where the coefficients Gn1m1n2m2 are computed as
(13)Gn1m1n2m2=∫01∫01gθ,ϑψn1m1n2m2θ,ϑWn1n2θ,ϑdϑdθ,and Wn1n2θ,ϑ=ωn1θωn2ϑ. By considering the truncated series of the infinite series in (12), one gets the following approximation to gθ,ϑ:
(14)gθ,ϑ≈gM1M2θ,ϑ=∑2k1−1n1=1∑M1−1m1=0∑2k2−1n2=1∑M2−1m2=0Gn1m1n2m2ψn1m1n2m2θ,ϑ=GTΔθ,ϑ=GTΨθ⊗Ψϑ,where G and Δ are 2k1−1M12k2−1M2×1 vectors and ⊗ denotes the Kronecker product.


## 2.3. Operational Matrices of the Integration

The integration of the one-variable basis in (10) can be approximated as
(15)∫0ϑΨ¯ηdη≈P1Ψ¯ϑ,where P1 is the operational matrix of the integration, and its entries are calculated as
(16)P1i,j≔∫0ϑΨ¯iηdη,Ψ¯jϑωn,i,j=1,2,⋯,2k−1M.IfJ0RLϑμ is the fractional integral of order μ>0 [7], then the operational matrix of the integration of the fractional order μ, Pμ, is given as
(17)J0RLϑμΨ¯ϑ≈PμΨ¯ϑ,where
(18)J0RLϑμΨ¯ϑ=J0RLϑμψ10ϑ,⋯,J0RLϑμψ1M−1ϑ,⋯,J0RLϑμψ2k−10ϑ,⋯,J0RLϑμψ2k−1M−1ϑT,J0RLϑμψmnϑ=2k/2J0RLϑμU~m2kϑ−2n+1,n−12k−1<ϑ<n2k−1,0,otherwise.Now, the two-dimensional operational matrices of the integration are constructed usingP1 and Pμ:
(19)∫0θΔξ,ϑdξ≈PθΔθ,ϑ=P1⊗IΔθ,ϑ,∫0ϑΔθ,ηdη≈PϑΔθ,ϑ=I⊗P1Δθ,ϑ,J0RLϑμΔθ,η≈PϑμΔθ,ϑ=I⊗PμΔθ,ϑ,where Pθ, Pϑ, and Pϑμ are two-dimensional operational matrices regarding the classic and fractional integral operators, respectively, and I is the M×M identity matrix.


## 3. Methodology

To show the applicability of the proposed scheme, the time-fractional inhomogeneous KdV equation and nonlinear time-fractional KdV equation are considered.


### 3.1. Time-Fractional Inhomogeneous KdV Equation

A form of this model is given as follows [27]:
(20)D0Cϑμuθ,ϑ+pθ,ϑ∂uθ,ϑ∂θ+qθ,ϑ∂3uθ,ϑ∂θ3=fθ,ϑ,θ,ϑ∈J,μ∈0,1,with
(21)uθ,0=ρ1θ,u0,ϑ=ϕ1ϑ,∂u0,ϑ∂θ=ϕ2ϑ,∂2u0,ϑ∂θ2=ϕ3ϑ,where functions ρ1,ϕ1,ϕ2,ϕ3 are known continuous ones. By considering the highest orders of derivative operators regarding θ and ϑ, the following approximation is given:
(22)∂4uθ,ϑ∂t∂θ3≈CTΔθ,ϑ.Triple integrating (22) regarding θ and using conditions (21) lead to the following approximations:
(23)∂3uθ,ϑ∂ϑ∂θ2≈CTPθΔθ,ϑ+∂3u0,ϑ∂ϑ∂θ2=CTPθΔθ,ϑ+ϕ3′ϑ≈CTPθΔθ,ϑ+F1TΔθ,ϑ,(24)∂2uθ,ϑ∂ϑ∂θ≈CTPθ2Δθ,ϑ+F1TPϑΔθ,ϑ+ϕ2′ϑ≈CTPθ2Δθ,ϑ+F1TPϑΔθ,ϑ+F2TΔθ,ϑ,(25)∂uθ,ϑ∂t≈CTPθ3Δθ,ϑ+F1TPθ2Δθ,ϑ+F2TPθΔθ,ϑ+ϕ1′ϑ≈CTPθ3Δθ,ϑ+F1TPθ2Δθ,ϑ+F2TPθΔθ,ϑ+F3TΔθ,ϑ.Now, by integrating (23) regarding ϑ, an approximation to uθ,ϑ is obtained:
(26)uθ,ϑ≈CTPθ3PϑΔθ,ϑ+F1TPθ2PϑΔθ,ϑ+F2TPθPϑΔθ,ϑ+F3TPϑΔθ,ϑ+ρ1θ≈CTPθ3PϑΔθ,ϑ+F1TPθ2PϑΔθ,ϑ+F2TPθPϑΔθ,ϑ+F3TPϑΔθ,ϑ+F4TΔθ,ϑ.Again, by integrating (22) regarding ϑ and θ, approximations to uθθ and uθ are obtained:
(27)∂3uθ,ϑ∂θ3≈CTPϑΔθ,ϑ+ρ1′′′θ≈CTPϑΔθ,ϑ+F5TΔθ,ϑ,(28)∂2uθ,ϑ∂θ2≈CTPϑPθΔθ,ϑ+F5TPϑΔθ,ϑ+ϕ3ϑ≈CTPϑPθΔθ,ϑ+F5TPθΔθ,ϑ+F6TΔθ,ϑ,(29)∂uθ,ϑ∂θ≈CTPϑPϑ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+ϕ2ϑ≈CTPϑPθ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+F7TΔθ,ϑ.Now, an approximation toD0Cϑμuθ,ϑ is computed using (23):
(30)D0Cϑμuθ,ϑJ0RLϑ1−μ∂uθ,ϑ∂ϑ≈J0RLϑ1−μCTPθ3Δθ,ϑ+F1TPθ2Δθ,ϑ+F2TPθΔθ,ϑ+F3TΔθ,ϑ≈CTPθ3Pϑ1−μΔθ,ϑ+F1TPθ2Pϑ1−μΔθ,ϑ+F2TPθPϑ1−μΔθ,ϑ+F3TPϑ1−μΔθ,ϑ.Substituting approximations (27)–(30) into (20) yields Rθ,ϑ as the residual function as follows:
(31)Rθ,ϑ=CTPθ3Pϑ1−μΔθ,ϑ+F1TPθ2Pϑ1−μΔθ,ϑ+F2TPθPϑ1−μΔθ,ϑ+F3TPϑ1−μΔθ,ϑ+pθ,ϑCTPϑPθ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+F7TΔθ,ϑ+qθ,ϑCTPϑΔθ,ϑ+F5TΔθ,ϑ−fθ,ϑ.


### 3.2. Time-Fractional Nonlinear KdV Equation

In this paper, the following class of time-fractional nonlinear KdV equations is studied:(32)D0Cϑμuθ,ϑ+6uθ,ϑ∂uθ,ϑ∂θ+∂3uθ,ϑ∂θ3=0,θ,ϑ∈J,μ∈0,1,with the conditions in (21). Substituting approximations (26)–(30) into (32) yields the following residual function:
(33)Rθ,ϑ=CTPθ3Pϑ1−μΔθ,ϑ+F1TPθ2Pϑ1−μΔθ,ϑ+F2TPθPϑ1−μΔθ,ϑ+F3TPθ1−μΔθ,ϑ+CTPθ3PϑΔθ,ϑ+F1TPθ2PϑΔθ,ϑ+F2TPθPϑΔθ,ϑ+F3TPθΔθ,ϑ+F4TΔθ,ϑ×CTPϑPθ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+F7TΔθ,ϑ+CTPϑΔθ,ϑ+F5TΔθ,ϑ.Collocating residual functions (31) and (33) at points θi,ϑj,i=1,2,⋯,2k1−1M1,j=1,2,⋯,2k2−1M2 results in a system of algebraic equations, where θi and ϑj are roots of U~2k1−1M1θ and U~2k2−1M2ϑ, respectively. This algebraic system can be handled by the Newton scheme. Therefore, an approximate solution is acquired from (26).Two models were solved by the variational iteration method in [27], and some figures of approximate solutions were depicted. The nonlinear time-fractional KdV equation (32) was solved by El-Wakil et al. in [43] using He’s variational iteration method and presented a second-order solution including some parameters. Authors in [44] obtained an approximate solution utilizing the iteration method after spending many algebraic computational costs. Inc et al. acquired new numerical solutions of fractional-time KdV equation by a technique of fictitious time integration and group preserving [45]. Authors in [46–48] used algebraic computational methods such as the modified extended tanh method, Sardar-subequation method, and He’s semi-inverse variation method and the ansatz method to construct some soliton solutions of the nonlinear time-fractional KdV equation.


## 3.1. Time-Fractional Inhomogeneous KdV Equation

A form of this model is given as follows [27]:
(20)D0Cϑμuθ,ϑ+pθ,ϑ∂uθ,ϑ∂θ+qθ,ϑ∂3uθ,ϑ∂θ3=fθ,ϑ,θ,ϑ∈J,μ∈0,1,with
(21)uθ,0=ρ1θ,u0,ϑ=ϕ1ϑ,∂u0,ϑ∂θ=ϕ2ϑ,∂2u0,ϑ∂θ2=ϕ3ϑ,where functions ρ1,ϕ1,ϕ2,ϕ3 are known continuous ones. By considering the highest orders of derivative operators regarding θ and ϑ, the following approximation is given:
(22)∂4uθ,ϑ∂t∂θ3≈CTΔθ,ϑ.Triple integrating (22) regarding θ and using conditions (21) lead to the following approximations:
(23)∂3uθ,ϑ∂ϑ∂θ2≈CTPθΔθ,ϑ+∂3u0,ϑ∂ϑ∂θ2=CTPθΔθ,ϑ+ϕ3′ϑ≈CTPθΔθ,ϑ+F1TΔθ,ϑ,(24)∂2uθ,ϑ∂ϑ∂θ≈CTPθ2Δθ,ϑ+F1TPϑΔθ,ϑ+ϕ2′ϑ≈CTPθ2Δθ,ϑ+F1TPϑΔθ,ϑ+F2TΔθ,ϑ,(25)∂uθ,ϑ∂t≈CTPθ3Δθ,ϑ+F1TPθ2Δθ,ϑ+F2TPθΔθ,ϑ+ϕ1′ϑ≈CTPθ3Δθ,ϑ+F1TPθ2Δθ,ϑ+F2TPθΔθ,ϑ+F3TΔθ,ϑ.Now, by integrating (23) regarding ϑ, an approximation to uθ,ϑ is obtained:
(26)uθ,ϑ≈CTPθ3PϑΔθ,ϑ+F1TPθ2PϑΔθ,ϑ+F2TPθPϑΔθ,ϑ+F3TPϑΔθ,ϑ+ρ1θ≈CTPθ3PϑΔθ,ϑ+F1TPθ2PϑΔθ,ϑ+F2TPθPϑΔθ,ϑ+F3TPϑΔθ,ϑ+F4TΔθ,ϑ.Again, by integrating (22) regarding ϑ and θ, approximations to uθθ and uθ are obtained:
(27)∂3uθ,ϑ∂θ3≈CTPϑΔθ,ϑ+ρ1′′′θ≈CTPϑΔθ,ϑ+F5TΔθ,ϑ,(28)∂2uθ,ϑ∂θ2≈CTPϑPθΔθ,ϑ+F5TPϑΔθ,ϑ+ϕ3ϑ≈CTPϑPθΔθ,ϑ+F5TPθΔθ,ϑ+F6TΔθ,ϑ,(29)∂uθ,ϑ∂θ≈CTPϑPϑ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+ϕ2ϑ≈CTPϑPθ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+F7TΔθ,ϑ.Now, an approximation toD0Cϑμuθ,ϑ is computed using (23):
(30)D0Cϑμuθ,ϑJ0RLϑ1−μ∂uθ,ϑ∂ϑ≈J0RLϑ1−μCTPθ3Δθ,ϑ+F1TPθ2Δθ,ϑ+F2TPθΔθ,ϑ+F3TΔθ,ϑ≈CTPθ3Pϑ1−μΔθ,ϑ+F1TPθ2Pϑ1−μΔθ,ϑ+F2TPθPϑ1−μΔθ,ϑ+F3TPϑ1−μΔθ,ϑ.Substituting approximations (27)–(30) into (20) yields Rθ,ϑ as the residual function as follows:
(31)Rθ,ϑ=CTPθ3Pϑ1−μΔθ,ϑ+F1TPθ2Pϑ1−μΔθ,ϑ+F2TPθPϑ1−μΔθ,ϑ+F3TPϑ1−μΔθ,ϑ+pθ,ϑCTPϑPθ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+F7TΔθ,ϑ+qθ,ϑCTPϑΔθ,ϑ+F5TΔθ,ϑ−fθ,ϑ.


## 3.2. Time-Fractional Nonlinear KdV Equation

In this paper, the following class of time-fractional nonlinear KdV equations is studied:(32)D0Cϑμuθ,ϑ+6uθ,ϑ∂uθ,ϑ∂θ+∂3uθ,ϑ∂θ3=0,θ,ϑ∈J,μ∈0,1,with the conditions in (21). Substituting approximations (26)–(30) into (32) yields the following residual function:
(33)Rθ,ϑ=CTPθ3Pϑ1−μΔθ,ϑ+F1TPθ2Pϑ1−μΔθ,ϑ+F2TPθPϑ1−μΔθ,ϑ+F3TPθ1−μΔθ,ϑ+CTPθ3PϑΔθ,ϑ+F1TPθ2PϑΔθ,ϑ+F2TPθPϑΔθ,ϑ+F3TPθΔθ,ϑ+F4TΔθ,ϑ×CTPϑPθ2Δθ,ϑ+F5TPθ2Δθ,ϑ+F6TPθΔθ,ϑ+F7TΔθ,ϑ+CTPϑΔθ,ϑ+F5TΔθ,ϑ.Collocating residual functions (31) and (33) at points θi,ϑj,i=1,2,⋯,2k1−1M1,j=1,2,⋯,2k2−1M2 results in a system of algebraic equations, where θi and ϑj are roots of U~2k1−1M1θ and U~2k2−1M2ϑ, respectively. This algebraic system can be handled by the Newton scheme. Therefore, an approximate solution is acquired from (26).Two models were solved by the variational iteration method in [27], and some figures of approximate solutions were depicted. The nonlinear time-fractional KdV equation (32) was solved by El-Wakil et al. in [43] using He’s variational iteration method and presented a second-order solution including some parameters. Authors in [44] obtained an approximate solution utilizing the iteration method after spending many algebraic computational costs. Inc et al. acquired new numerical solutions of fractional-time KdV equation by a technique of fictitious time integration and group preserving [45]. Authors in [46–48] used algebraic computational methods such as the modified extended tanh method, Sardar-subequation method, and He’s semi-inverse variation method and the ansatz method to construct some soliton solutions of the nonlinear time-fractional KdV equation.


## 4. Error Bound

In this section, error bounds are derived for the residual functions/perturbation terms for two given models in Section3. First, some error bounds are computed for approximation errors.


### 4.1. Time-Fractional Inhomogeneous KdV Equation

Consider Equation (20) and suppose that uM1M2θ,ϑ is its approximate solution obtained from the presented algorithm in Section 3. Thus, uM1M2θ,ϑ satisfies the following equations:
(34)D0CϑμuM1M2θ,ϑ+pθ,ϑ∂uM1M2θ,ϑ∂θ+qθ,ϑ∂3uM1M2θ,ϑ∂θ3=fθ,ϑ−RM1M2θ,ϑ,where RM1M2θ,ϑ is called the residual function/perturbation term. By subtracting Equation (34) from Equation (20), one gets
(35)RM1M2θ,ϑ=D0Cϑμuθ,ϑ−0CDϑμuM1M2θ,ϑ+pθ,ϑ∂uθ,ϑ∂θ−∂uM1M2θ,ϑ∂θ+qθ,ϑ∂3uθ,ϑ∂θ3−∂3uM1M2θ,ϑ∂θ3.Suppose thatpθ,ϑ,qθ,ϑ are continuous functions over J. By taking L2-norm on Equation (35), one has
(36)RM1M2L2≤D0Cϑμu−D0CϑμuM1M2L2+pL2∂u∂θ−∂uM1M2∂θL2+qL2∂3u∂θ3−∂3uM1M2∂θ3L2.First, error bounds are calculated for terms on the right-hand side in (36). Assume that TM1M2θ,ϑ is the Taylor series expansion of uθ,ϑ, Θ1=maxθ,ϑ∈JuM1+M2−μθ,ϑ, and Jn1n2=n1−1/2k1−1,n1/2k1−1×n2−1/2k2−1,n2/2k2−1. One has,
(37)D0Cϑμu−D0CϑμuM1M2L22=∫01∫01Dϑμuθ,ϑ−D0CϑμuM1M2θ,ϑ2Wθ,ϑdϑdθ=∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1Dϑμuθ,ϑ−D0CϑμuM1M2θ,ϑ2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1Dϑμuθ,ϑ−D0CϑμTM1M2θ,ϑ2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1maxξn1,ηn2∈Jn1n2uM1+M2−μξn1,ηn2M1!M2−μ!2M1k1−12M2k2−12Wn1n2θ,ϑdϑdθ≤Θ1M1!M2−μ!2M1k1−12M2k2−12∫01∫01Wθ,ϑ⏟ωθωϑdϑdθ=Θ1M1!M2−μ!2M1k1−12M2k2−12∫01θ1/21−θ1/2dθ∫01ϑ1/21−ϑ1/2dϑ=Θ1M1!M2−μ!2M1k1−12M2k2−12π82.So, one gets(38)D0Cϑμu−D0CϑμuM1M2L2≤πΘ1M1!M2−μ!2M1k1−12M2k2−123.In a similar way, ifΘ2,l=maxθ,ϑ∈JuM1+M2−lθ,ϑ,l=0,1,2,3, one has
(39)∂lu∂θl−∂luM1M2∂θlL22=∫01∫01∂luθ,ϑ∂θl−∂luM1M2θ,ϑ∂θl2Wθ,ϑdϑdθ=∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1∂luθ,ϑ∂θl−∂luM1M2θ,ϑ∂θl2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1∂luθ,ϑ∂θl−∂lTM1M2θ,ϑ∂θl2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1maxξn1,ηn2∈Jn1n2uM1+M2−lξn1,ηn2M1−l!M2!2M1k1−12M2k2−12Wn1n2θ,ϑdϑdθ≤Θ2,lM1−l!M2!2M1k1−12M2k2−12π82.Thus, one gets(40)∂lu∂θl−∂luM1M2∂θlL2≤πΘ2,lM1−l!M2!2M1k1−12M2k2−123,l=0,1,2,3.Therefore, a bound is obtained for inequality (36) using (37) and (39) as follows:
(41)RM1M2L2πΘ1M1!M2−μ!2M1k1−12M2k2−123+pL2πΘ2,1M1−1!M2!2M1k1−12M2k2−123+qL2πΘ2,3M1−3!M2!2M1k1−12M2k2−123.It is evident from the right-hand side of (40) that RM1M2L2⟶0 when M1,M2⟶∞.


### 4.2. Time-Fractional Nonlinear KdV Equation

Consider Equation (32) and suppose that uM1M2θ,ϑ is its approximate solution obtained from the proposed method. Thus, uM1M2θ,ϑ satisfies the following equation:
(42)D0CϑμuM1M2θ,ϑ+6uM1M2θ,ϑ∂uM1M2θ,ϑ∂θ+∂3uM1M2θ,ϑ∂θ3=−RM1M2θ,ϑ.Subtracting Equation (41) from (32) leads to the following equation:
(43)RM1M2θ,ϑ=D0Cϑμuθ,ϑ−D0CϑμuM1M2θ,ϑ+6uθ,ϑ∂uθ,ϑ∂θ−uM1M2θ,ϑ∂uM1M2θ,ϑ∂θ+∂3uθ,ϑ∂θ3−∂3uM1M2θ,ϑ∂θ3.The nonlinear termuuθ−uM1M2uθM1M2 can be written as
(44)uuθ−uM1M2uθM1M2=u−uM1M2uθ+uθ−uθM1M2uM1M2=u−uM1M2uθ−uθM1M2+uθM1M2+uθ−uθM1M2uM1M2=u−uM1M2uθ−uθM1M2+u−uM1M2uθM1M2+uθ−uθM1M2uM1M2.Now, using bounds obtained in the previous section, an error bound of (42) can be calculated as follows:
(45)RM1M2L2πΘ1M1!M2−μ!2M1k1−12M2k2−123+6πΘ2,0M1!M2!2M1k1−12M2k2−123×πΘ2,1M1−1!M2!2M1k1−12M2k2−123+uθM1M2L2+πΘ2,1uM1M2L2M1−1!M2!2M1k1−12M2k2−123+πΘ2,3uM1M2L2M1−3!M2!2M1k1−12M2k2−123.Obviously, the right-hand side of (44) tends to zero, when M1,M2 are sufficiently large.


## 4.1. Time-Fractional Inhomogeneous KdV Equation

Consider Equation (20) and suppose that uM1M2θ,ϑ is its approximate solution obtained from the presented algorithm in Section 3. Thus, uM1M2θ,ϑ satisfies the following equations:
(34)D0CϑμuM1M2θ,ϑ+pθ,ϑ∂uM1M2θ,ϑ∂θ+qθ,ϑ∂3uM1M2θ,ϑ∂θ3=fθ,ϑ−RM1M2θ,ϑ,where RM1M2θ,ϑ is called the residual function/perturbation term. By subtracting Equation (34) from Equation (20), one gets
(35)RM1M2θ,ϑ=D0Cϑμuθ,ϑ−0CDϑμuM1M2θ,ϑ+pθ,ϑ∂uθ,ϑ∂θ−∂uM1M2θ,ϑ∂θ+qθ,ϑ∂3uθ,ϑ∂θ3−∂3uM1M2θ,ϑ∂θ3.Suppose thatpθ,ϑ,qθ,ϑ are continuous functions over J. By taking L2-norm on Equation (35), one has
(36)RM1M2L2≤D0Cϑμu−D0CϑμuM1M2L2+pL2∂u∂θ−∂uM1M2∂θL2+qL2∂3u∂θ3−∂3uM1M2∂θ3L2.First, error bounds are calculated for terms on the right-hand side in (36). Assume that TM1M2θ,ϑ is the Taylor series expansion of uθ,ϑ, Θ1=maxθ,ϑ∈JuM1+M2−μθ,ϑ, and Jn1n2=n1−1/2k1−1,n1/2k1−1×n2−1/2k2−1,n2/2k2−1. One has,
(37)D0Cϑμu−D0CϑμuM1M2L22=∫01∫01Dϑμuθ,ϑ−D0CϑμuM1M2θ,ϑ2Wθ,ϑdϑdθ=∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1Dϑμuθ,ϑ−D0CϑμuM1M2θ,ϑ2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1Dϑμuθ,ϑ−D0CϑμTM1M2θ,ϑ2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1maxξn1,ηn2∈Jn1n2uM1+M2−μξn1,ηn2M1!M2−μ!2M1k1−12M2k2−12Wn1n2θ,ϑdϑdθ≤Θ1M1!M2−μ!2M1k1−12M2k2−12∫01∫01Wθ,ϑ⏟ωθωϑdϑdθ=Θ1M1!M2−μ!2M1k1−12M2k2−12∫01θ1/21−θ1/2dθ∫01ϑ1/21−ϑ1/2dϑ=Θ1M1!M2−μ!2M1k1−12M2k2−12π82.So, one gets(38)D0Cϑμu−D0CϑμuM1M2L2≤πΘ1M1!M2−μ!2M1k1−12M2k2−123.In a similar way, ifΘ2,l=maxθ,ϑ∈JuM1+M2−lθ,ϑ,l=0,1,2,3, one has
(39)∂lu∂θl−∂luM1M2∂θlL22=∫01∫01∂luθ,ϑ∂θl−∂luM1M2θ,ϑ∂θl2Wθ,ϑdϑdθ=∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1∂luθ,ϑ∂θl−∂luM1M2θ,ϑ∂θl2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1∂luθ,ϑ∂θl−∂lTM1M2θ,ϑ∂θl2Wn1n2θ,ϑdϑdθ≤∑2k1−1n1=1∑2k2−1n2=1∫n1−1/2k1−1n1/2k1−1∫n2−1/2k2−1n2/2k2−1maxξn1,ηn2∈Jn1n2uM1+M2−lξn1,ηn2M1−l!M2!2M1k1−12M2k2−12Wn1n2θ,ϑdϑdθ≤Θ2,lM1−l!M2!2M1k1−12M2k2−12π82.Thus, one gets(40)∂lu∂θl−∂luM1M2∂θlL2≤πΘ2,lM1−l!M2!2M1k1−12M2k2−123,l=0,1,2,3.Therefore, a bound is obtained for inequality (36) using (37) and (39) as follows:
(41)RM1M2L2πΘ1M1!M2−μ!2M1k1−12M2k2−123+pL2πΘ2,1M1−1!M2!2M1k1−12M2k2−123+qL2πΘ2,3M1−3!M2!2M1k1−12M2k2−123.It is evident from the right-hand side of (40) that RM1M2L2⟶0 when M1,M2⟶∞.


## 4.2. Time-Fractional Nonlinear KdV Equation

Consider Equation (32) and suppose that uM1M2θ,ϑ is its approximate solution obtained from the proposed method. Thus, uM1M2θ,ϑ satisfies the following equation:
(42)D0CϑμuM1M2θ,ϑ+6uM1M2θ,ϑ∂uM1M2θ,ϑ∂θ+∂3uM1M2θ,ϑ∂θ3=−RM1M2θ,ϑ.Subtracting Equation (41) from (32) leads to the following equation:
(43)RM1M2θ,ϑ=D0Cϑμuθ,ϑ−D0CϑμuM1M2θ,ϑ+6uθ,ϑ∂uθ,ϑ∂θ−uM1M2θ,ϑ∂uM1M2θ,ϑ∂θ+∂3uθ,ϑ∂θ3−∂3uM1M2θ,ϑ∂θ3.The nonlinear termuuθ−uM1M2uθM1M2 can be written as
(44)uuθ−uM1M2uθM1M2=u−uM1M2uθ+uθ−uθM1M2uM1M2=u−uM1M2uθ−uθM1M2+uθM1M2+uθ−uθM1M2uM1M2=u−uM1M2uθ−uθM1M2+u−uM1M2uθM1M2+uθ−uθM1M2uM1M2.Now, using bounds obtained in the previous section, an error bound of (42) can be calculated as follows:
(45)RM1M2L2πΘ1M1!M2−μ!2M1k1−12M2k2−123+6πΘ2,0M1!M2!2M1k1−12M2k2−123×πΘ2,1M1−1!M2!2M1k1−12M2k2−123+uθM1M2L2+πΘ2,1uM1M2L2M1−1!M2!2M1k1−12M2k2−123+πΘ2,3uM1M2L2M1−3!M2!2M1k1−12M2k2−123.Obviously, the right-hand side of (44) tends to zero, when M1,M2 are sufficiently large.


## 5. Numerical Examples

The two models given in Section3 are considered to illustrate the accuracy and applicability of the proposed scheme. Maximum absolute errors are computed when the derivative order is an integer (classical case). The results are compared to the exact ones. Computations and simulations are handled by Maple 16.Example 1.
As a first example, the following linear inhomogeneous time-fractional KdV equation:(46)D0Cϑμuθ,ϑ+∂uθ,ϑ∂θ+∂3uθ,ϑ∂θ3=2ϑ2−μΓ3−μcosθ,θ,ϑ∈J,μ∈0,1,subject to the initial and boundary conditions:
(47)uθ,0=0,u0,ϑ=0,∂u0,ϑ∂θ=0,∂2u0,ϑ∂θ2=0.The exact solution isuθ,ϑ=ϑ2cosθ, if μ=1. Maximum absolute errors (MAE) are listed in Table 1 for μ=1,k1=k2=1,M1=M2=2,3,4,5. As seen, the errors decrease when M1,M2 increase. Values of absolute errors of the exact and numerical solutions, at equally spaced points θi=ϑj=0.2i,i=0,1,⋯,5, are seen in Table 2 for k1=k2=1,M1=M2=4,μ=0.7,0.8,0.9,1. The results have more accuracy as μ⟶1. Plots of numerical solutions are depicted in Figure 1 for k1=k2=1,M1=M2=4,μ=0.7,0.8,0.9,1, and ϑ=3. It can be found that the approximate solutions approach the exact one when μ⟶1.Example 2.
Consider the time-fractional nonlinear KdV equation as follows:(48)D0Cϑμuθ,ϑ+6uθ,ϑ∂uθ,ϑ∂θ+∂3uθ,ϑ∂θ3=0,θ,ϑ∈J,μ∈0,1,with
(49)uθ,0=12sech2θ2,u0,ϑ=12sech2ϑ2,∂u0,ϑ∂θ=12sech2ϑ2tanhϑ2,∂2u0,ϑ∂θ2=12sech2ϑ2tanh2ϑ2+12sech2ϑ2−12+12tanh2ϑ2.Table 1
Maximum absolute errors forμ=1 and different values of M1,M2 for Example 1.
M1=M22345MAE5.2885×10−37.0306×10−42.3882×10−52.0614×10−6Table 2
Absolute errors fork1=k2=1,M1=M2=4 at equally spaced points for Example 1.
θi=ϑiμ=0.7μ=0.8μ=0.9μ=104.8935×10−75.4663×10−74.4250×10−71.4522×10−110.26.1889×10−76.0321×10−74.6254×10−71.2627×10−70.48.5473×10−78.0229×10−77.3242×10−76.8164×10−70.65.8306×10−65.6182×10−63.6174×10−61.1872×10−60.81.7757×10−51.6597×10−51.0738×10−51.7657×10−619.7134×10−57.8075×10−53.3765×10−52.3881×10−5Figure 1
Exact and approximate solutions fork1=k2=1,M1=M2=4, and μ=0.7,0.8,0.9,1 at time ϑ=3 for Example 1.The exact solution isuθ,ϑ=1/2sech2θ/2−ϑ/2, if μ=1. Plots of exact and approximate solutions and the absolute error function are depicted in Figure 2 for k1=k2=1,M1=M2=5, and μ=1. A graphical comparison between exact and approximate solutions is observed in Figure 3 for k1=k2=1;M1=M2=4; μ=0.3,0.5,0.7,0.9,1; and ϑ=0.45. It can be found that the approximate solutions approach the exact one when μ⟶1. Values of absolute errors are listed in Table 3 for M1=M2=6;μ=0.3,0.5,0.7,0.9,1; and θi=ϑi=0.2i,i=0,1,⋯,5. It can be seen from Figure 3 and Table 3 that the approximate solutions approach the exact one when μ⟶1.Figure 2
(a) Exact solution, (b) approximate solution, and (c) absolute error function fork1=k2=1,M1=M2=5, and μ=1 for Example 2.
(a)(b)(c)Figure 3
Exact and approximate solutions fork1=k2=1,M1=M2=6, and μ=0.3,0.5,0.7,0.9,1 at time ϑ=0.45 for Example 2.Table 3
Absolute errors fork1=k2=1,M1=M2=6 at equally spaced points for Example 2.
θi=ϑiμ=0.3μ=0.5μ=0.7μ=0.9μ=106.6378×10−86.8466×10−85.5505×10−82.4525×10−81.3170×10−90.28.6724×10−56.7852×10−54.4262×10−51.5860×10−53.1335×10−90.46.5214×10−45.1748×10−43.4307×10−41.2522×10−43.7353×10−100.62.0195×10−31.6274×10−31.0975×10−34.0827×10−41.2169×10−80.84.2767×10−33.5051×10−32.4073×10−39.1284×10−43.0611×10−817.2448×10−36.0506×10−34.2372×10−31.6379×10−34.9306×10−8


## 6. Conclusion

In this paper, the second-kind Chebyshev wavelets were employed to solve time-fractional inhomogeneous KdV and time-fractional nonlinear KdV equations. Using the presented scheme, the main problem was converted into a system of algebraic equations wherein obtaining its solution is easier than finding the solution of the problem under study. In comparison with the Adomian decomposition, homotopy analysis, and homotopy perturbation methods, the SKCW method possesses fewer computational costs. The few numbers of the basis functions lead to an approximate solution with appropriate accuracy. As seen from Table1, by increasing values of Mi,i=1,2, the maximum absolute errors decrease. In Tables 2 and 3, values of absolute errors at equally spaced points decrease as μ⟶1; then, approximate solutions are getting close to exact ones. This can be seen in Figures 1 and 3. It was seen from illustrative examples that the method is an efficient numerical scheme to find an approximate solution for linear or nonlinear PDEs. The authors intend to test the proposed approach on other nonlinear fractional partial differential equations such as Newell–Whitehead–Segel and Phi-four.



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