# Pell Collocation Pseudo Spectral Scheme for One-Dimensional Time-Fractional Convection Equation with Error Analysis

Authors: A. S. Mohamed
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In the current manuscript, we implement the collocation method to obtain an approximate solution of one-dimensional time-fractional convection equation. The operational matrices of Pell polynomials are applied to solve the fractional partial differential equations. In the Caputo sense, we describe the time-fractional derivatives. So, this algorithm allows us to transform the fractional differential equation with initial (boundary) conditions into a system of algebraic equations in a good form. The convergence and error analysis of Pell polynomials are carefully investigated, and some examples are explored to show the comparison and efficiency of our method with others.


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## Body

## 1. Introduction

The investigation of differential and integral equations of fractional order is a very important subject. These kinds of equations describe many phenomena in different fields such as mathematics, mechanics, engineering, physics, biology, fluid, and chemistry. It is difficult to evaluate the exact solution of these equations. It had to evaluate the approximate solution instead. For example, the variational iteration method is used to solve fractional wave and Burgers equations, time-fractional partial differential equations, and fractional Klein-Gordon and Black Scholes equations (see [1–3]). The linear space-time-fractional convection-diffusion equations with fractional derivatives for both space and time, the fractional kinetic equations of diffusion or dispersion with time, and space-fractional differential equations with variable diffusivity coefficient are studied by the finite difference method (see [4–6]). The finite element is another method for solving time-fractional partial differential equations, space-fractional partial advection-diffusion equation with non-homogeneous initial-boundary conditions, and linear Riesz space-fractional partial differential equations with a second-order time derivative (see [7–9]).Various types of ordinary and partial fractional differential (integral) equations can be solved by the well-known methods which are called spectral methods. These methods decrease the number of unknowns and supply small errors. The way of solution in these methods is expressed as expansion of polynomials. One of the spectral methods is shifted Jacobi polynomials which are used to approximate the solution of Volterra-Fredholm integral equation, the space-fractional advection-dispersion equation, and linear multi-term fractional differential equations (see [10–12]). Another spectral method is shifted Chebyshev which is used for solving one-dimensional linear hyperbolic first-order partial differential equations (see [13]). Solving multi-term fractional differential equations and a system of high-order linear differential equations with variable coefficients are using Lucas method (see [14, 15]). Evaluating the solution of Volterra-Fredholm integral equation, multi-term fractional differential equations, multi-term initial value problems, and fractional pantograph differential equations are studied using generalized Lucas polynomials (see [16–19]). Another way of solving such problems is connecting between generalized Lucas polynomials together with third and fourth kinds of Chebyshev polynomials (see [20]). The coupled systems of a fractional differential equation is solved by generalized Fibonacci method (see [21]). However, for multidimensional Burgers-type equations, one- and two-dimensional nonlinear heat-type equations, two-dimensional linear and nonlinear Sobolev equations, and nonlinear reactionâ€“diffusion Brusselator system with applications in chemical processes, all of them are using mixed Lucas and Fibonacci polynomials (see [22–25]). Also, Fermat polynomials make use for solving the Bagly-Torvik equation (see [26]). In our paper, we shall use Pell collocation (PLC) method to solve partial fractional differential equations and to the best of our knowledge this is the first to be done, also compare our results with the non-uniform L1-discontinuous Galerkin (DG) method (see [27]), and show that our method is more accurate although it takes a longer time.We considered the fractional derivative in the Caputo sense because it is mathematically rigorous than other derivatives and this was discussed in details (see [17]).Now our aim is to solve the one-dimensional time-fractional convection equation using Pell polynomials (see [27])
(1)cD0,tλυy,t+βvyy,t=hy,ty,t∈Λ×0,τυy,0=υ0y,y∈Λυd,t=0,t∈0,τ,where υy,t is an unknown function, Λ=d,a is a bounded interval, and β is a non zero given const. hy,t∈L∞0,τ;L2Λ and υ0y∈L2Λ are known functions. cD0,tλ is the λ-th fractional order Caputo derivative which is given by
(2)cD0,tλυy,t=1Γ1−λ∫0tt−r−λ∂υ∂rdr,0⋖λ⋖1,We organize this paper as follows: In Section2, we demonstrate the basic properties and preliminaries of Pell polynomials which will be frequent. Section 3 displays the derivatives of integer and fractional orders for these polynomials. The spectral collocation algorithm for solving one-dimensional time-fractional convection equation is analyzed and described in Section 4. The basic results which show the investigation of convergence, error analysis, and the evaluation of the global error for Pell expansion are included in Section 5. Two numerical examples are demonstrated and we compare the results with another method which will be in Section 6. The last section deals with the conclusions.


## 2. Preliminaries and Properties of Pell Polynomials

It is known that Pell polynomials are special cases of Fibonacci polynomials and the recurrence form of Pell polynomials is(3)pLj+1y=2ypLj+1y+pLjy.Another form of this relation is(4)Wj+1y=ρyWj+1y+θyWjy,j≥0,with initial conditions
(5)W0y=0,W1y=1.The Binet formula of Pell polynomials is(6)Wjy=γjy−ηjyγy−ηy,where
(7)γy=ρy+ρ2y+4θy2,ηy=ρy−ρ2y+4θy2.We propose to chooseρy=2y and θy=1. Then, Pell polynomials will take the form
(8)pLjy=y+y2+1j−y−y2+1j2jy2+1,j≥0.The analytical form of these polynomials can be deduced from equation (3) [21], pLj+1y,j≥0(9)pLj+1y=∑r=0j22j−2rj−rryj−2r,where j is the largest integer less than or equal j From Theorem (4) [21], we have
(10)yj=12j∑r=0j2−1rjrj−2r+1j−r+1pLj−2r+1y,j≥0.


## 3. Integer and Fractional Derivatives of Pell Polynomials

In this section, we get the integer and fractional derivatives of Pell polynomials. This section will be divided into two subsections; the first explores the integer derivatives of Pell polynomials and the second concerns the fractional derivatives of these polynomials.Letυy be expressed in terms of Pell polynomials
(11)υy=∑j=1∞cjpLjy.The approximate solution of this function is(12)υy≈υMy=∑j=1M+1cjpLjy=CTPLy,where
(13)CT=c1,c2,⋯,cM+1,PLy=pL1y,pL2y,⋯,pLM+1yT.


### 3.1. The Integer Derivatives of Pell Polynomial

The first derivative ofPLy can be written as (see [21])
(14)dPLydy=F1PLy,where F1=fnm1 is the M+1×M+1 matrix of derivatives.
(15)fnm1=2−1n−m−1/2m+1,n>m,n+modd0otherwise.IfM=5, then F1 has the form
(16)F1=000000200000040000−2060000−4080020−60100.Also, the integer derivatives ofPLy can be easily written in the form (see [21])
(17)dιPLydyι=FιPLy=F1ιPLy.


### 3.2. The Fractional Derivatives of Pell Polynomial

Ifλ is not an integer, then the fractional derivative of PLy takes the form (see [21])
(18)DλPLy=y−λFλPLy,where Fλ=fnmλ is the M+1×M+1 Pell operational matrix of fractional derivatives of order λ, which has the form
(19)Fλ=000⋯0⋮⋮⋮⋮ζλλ,1ζλλ,20⋯0⋮⋮⋮⋮ζλi,1⋯ζλi,i⋯0⋮⋮⋮⋮ζλM+1,1ζλM+1,2ζλM+1,3⋯ζλM+1,M+1.fnmλ can be written in the form
(20)fnmλ=ζλn,m,n≥λ,n≥m0otherwise,where
(21)ζλn,m=mξm∑l=λn−11+l−m/2l!n+l−1/2!n−l−1/2!1+l−m/2!1+m+l/2!Γ1+l−λ,ξk=12,k=0,1,otherwise.Sinceab=Γa+b/Γa,
(22)n+l−1/2!n−l−1/2!=n−l+12l.Clearly,ζλn,m can be rewritten in the form
(23)ζλn,m=m∑l=λn−11+l−m/2l1+l−m/2n−l+1/2l1+m+l/2Γ1+l−λ.So the fractional derivative has the relation(24)DλpLny=y−λ∑m=1nζλn,mpLmy.


## 3.1. The Integer Derivatives of Pell Polynomial

The first derivative ofPLy can be written as (see [21])
(14)dPLydy=F1PLy,where F1=fnm1 is the M+1×M+1 matrix of derivatives.
(15)fnm1=2−1n−m−1/2m+1,n>m,n+modd0otherwise.IfM=5, then F1 has the form
(16)F1=000000200000040000−2060000−4080020−60100.Also, the integer derivatives ofPLy can be easily written in the form (see [21])
(17)dιPLydyι=FιPLy=F1ιPLy.


## 3.2. The Fractional Derivatives of Pell Polynomial

Ifλ is not an integer, then the fractional derivative of PLy takes the form (see [21])
(18)DλPLy=y−λFλPLy,where Fλ=fnmλ is the M+1×M+1 Pell operational matrix of fractional derivatives of order λ, which has the form
(19)Fλ=000⋯0⋮⋮⋮⋮ζλλ,1ζλλ,20⋯0⋮⋮⋮⋮ζλi,1⋯ζλi,i⋯0⋮⋮⋮⋮ζλM+1,1ζλM+1,2ζλM+1,3⋯ζλM+1,M+1.fnmλ can be written in the form
(20)fnmλ=ζλn,m,n≥λ,n≥m0otherwise,where
(21)ζλn,m=mξm∑l=λn−11+l−m/2l!n+l−1/2!n−l−1/2!1+l−m/2!1+m+l/2!Γ1+l−λ,ξk=12,k=0,1,otherwise.Sinceab=Γa+b/Γa,
(22)n+l−1/2!n−l−1/2!=n−l+12l.Clearly,ζλn,m can be rewritten in the form
(23)ζλn,m=m∑l=λn−11+l−m/2l1+l−m/2n−l+1/2l1+m+l/2Γ1+l−λ.So the fractional derivative has the relation(24)DλpLny=y−λ∑m=1nζλn,mpLmy.


## 4. The Algorithm of the Method

In this section, we study the method for solving one-dimensional time-fractional convection equation with initial (boundary) conditions by using Pell polynomials.The functionυy,t of two variables can be expanded in terms of double Pell polynomials
(25)υy,t=∑k=1∞∑ℓ=1∞ckℓpLkypLℓt,and the approximate solution of this function is
(26)υy,t≈∑k=1M+1∑ℓ=1M+1ckℓpLkypLℓt=CTΨy,t,where ckℓ, K,ℓ=1,2,⋯,M+1 are the unknown coefficients
(27)CT=ckℓT=c11c12c13⋯c1M+1c21c22c23⋯c2M+1⋮⋮⋮⋯⋮cM1cM2cM3⋯cM+1M+1,where M is any arbitary positive integer. pLky and pLℓt are Pell polynomials defined in equation (9). Also, Ψy,t is an M+1×M+1 matrix introduced by
(28)Ψy,t=ψ11y,t,ψ21y,t,⋯,ψM+1y,t;ψ12y,t,ψ22y,t,⋯,ψM+12y,t;ψ1M+1y,t,ψ2M+1y,t,⋯,ψM+1M+1y,t,where
(29)ψkℓy,t=pLkypLℓt.Substitute equation (26) into equation (1), we obtain
(30)cD0,tλ∑k=1M+1∑ℓ=1M+1ckℓpLkypLℓt+β∂∂y∑k=1M+1∑ℓ=1M+1ckℓpLkypLℓt=hy,t.Let(31)ϕkℓy,t=pLkycD0,tλpLℓ′t+βpLℓt∂∂ypLk′y.Consequently, equation (30) can be rewritten as
(32)∑k=1M+1∑ℓ=1M+1ckℓϕkℓy,t=hy,t.Collocating this equation atM+1×M+1 roots of Pell polynomials, we have a system of equations
(33)∑k=1M+1∑ℓ=1M+1ckℓϕkℓyi,tj=hyi,tj,i,j=1,2,⋯,M+1.The matrix form of this equation is(34)ΦTC=H,where Φ=ϕkℓij=ϕkℓyi,tj,k,ℓ,i,j=1,2,⋯,M+1, and
(35)H=hy1,t1,hy2,t1,⋯,hyM+1,t1;hy1,t2,hy2,t2,⋯,hyM+1,t2;hy1,tM+1,hy2,tM+1,⋯,hyM+1,tM+1T.Finally, we can determine the constants by(36)C=ΦT−1H,with initial (boundary) conditions in the forms
(37)CTPLyPL0=υ0y,y∈Λ,CTPLdPLt=0,t∈0,τ.Now we summarize the steps of our scheme in Algorithm1 below.Algorithm 1: Coding algorithm for the proposed scheme.
Inputβ∈R\0; the functions υy,t and hy,t.Step 1. Define Pell polynomials by (9).Step 2. Compute the basis function of Pell polynomials by (29).Step 3. Define the basis function vector Ψy,t by (28).Step 4. Substituting Eqs. (2) and (26) into Eq. (1).Step 5. Collocating Eq. (32) in M+1×M+1 roots of the polynomial pLkypLℓt.Step 6. Compute the matrix H using (35).Step 7. Define the M+1×M+1 unknown vectors CT.Step 8. Use NSolve command to solve the system ΦTC=H.Output The approximate solution: υy,t=CTΨy,t.


## 5. The Convergence and Error Analysis

In this section, we explore our main results concerning the convergence and error analysis of Pell polynomials for fractional differential equations. Let us start with the following three Lemmas.Lemma 1.
Ifυy is an infinitely differential function, then it can be expressed in the form of the Pell polynomials.
(38)υy=∑k=1∞∑ρ=0∞−1ρk2ρ+k−1!ù2ρ+k−122ρ+k−1ρ!ρ+k!pLky,
whereùj=υj0/j!.Proof.
From equation (43) (see [21]).
(39)υy=∑k=1∞ckGka,by,where Gka,by are the generalized Fibonacci polynomials. Taking a=2, b=1, and equations (4.3,5.3) (see [28]), the result holds.Lemma 2.
The modified Bessel function satisfies the following inequality.(40)Iky≤ykcoshy2kΓk+1.Proof.
See [29].Lemma 3.
The following relations are satisfied.(41)pLky≤pLk,y∈Λ,where
(42)pLk=pLk1≤3k,pLk=pLk1≤3k+1.Proof.
By simplifying the relation in equation (8) and y=1, we obtain the required inequalities.The following results are important in itself and will be helpful in discussing the examples given in the coming sections.Theorem 4.
Ifυy,t=∑k=1∞∑ℓ=1∞ckℓpLkypLℓt and the exact solution of equation (1) is defined on Λ×0,τ, then the following hold:
(43)1ckℓ≤μk+ℓ−22k+ℓ−2k−1!ℓ−1!cosh2μ,where υj0,0≤μj,j≥0 where μ is a positive constant.
(2)
The series converges absolutelyProof.
From Lemma1.
(44)ckl≤∑i=0∞∑j=0∞k2i+k−1!ù2i+k−122i+k−1i!i+k!ℓ2j+ℓ−1!ù2j+ℓ−122j+ℓ−1j!j+ℓ!.
Using the given assumptions, we have(45)ckl≤kℓ∑i=0∞∑j=0∞μ2i+2j+k+ℓ−222i+2j+k+ℓ−2i!j!i+k!j+ℓ!=4kℓμ2IkμIℓμ.
By applying Lemma2, we obtain the desired results of the first part. For the second part, from Lemma 3, we obtain
(46)ckℓpLkypLℓt≤μk+ℓ−23k+ℓ2k+ℓ−2k−1!ℓ−1!cosh2μ.
So(47)∑k=1∞∑ℓ=1∞ckℓpLkypLℓt≤cosh2μ∑k=1∞∑ℓ=1∞μk+ℓ−23k+ℓ2k+ℓ−2k−1!ℓ−1!=9e3μcosh2μ.
Thus, the proof completes.Theorem 5.
If Theorem4 is satisfied and the truncation error isεMy,t=υy,t−υM+1y,t, where υM+1y,t=∑k=1M+1∑ℓ=1M+1ckℓpLkypLℓt is the approximate solution of equation (1), then the following estimation is true:
(48)εMy,t⋖18e3μM+1!3μ2M+1cosh2μ.Proof.
From the definition ofεMy,t, we have
(49)εMy,t=∑k=1∞∑ℓ=1∞ckℓpLkypLℓt−∑k=1M+1∑ℓ=1M+1ckℓpLkypLℓt=∑k=1M+1∑ℓ=M+2∞ckℓpLkypLℓt+∑k=M+2∞∑ℓ=1∞ckℓpLkypLℓt.
Using Theorem4 and Lemma 3 and collecting the summation, we get
(50)εMy,t≤9e3μ1−ΓM+1,3μ/22ΓM+12cosh2μ,where Γ. and Γ.,. are gamma and incomplete gamma functions, respectively. So
(51)ΓM+1−ΓM+1,3μ2⋖∫03μ/2yMdy=3μ/2M+1M+1.
Then, the right hand side of the above inequality can be simplified as(52)εMy,t⋖9e3μM+1!3μ2M+12−3μ/2M+1M+1!cosh2μ,⋖18e3μM+1!3μ2M+1cosh2μ.
Thus, the inequality holds.Theorem 6.
Ifυy,t satisfies the hypotheses of Theorem 5, and by assuming
(53)RM+1y,t=cD0,tλυM+1y,t+βυyM+1y,t−hy,t,RM+1=maxy∈Λ,t∈0,τRM+1y,t,and 1+β≤σ (a positive constant), then we have the following global error estimation:
(54)RM+1≤54σe3μM+1!3μ2M+1cosh2μ.Proof.
From equation (1), we have
(55)hy,t=cD0,tλυy,t+βυyy,t.
So, we obtain(56)RM+1y,t≤cD0,tλυM+1y,t−υy,t+βυM+1y,t−υy,t.
Using Theorem (2) and Lemma (3), this estimation is transformed to
(57)RM+1y,t≤1+βcosh2μ∑k=1M+1∑ℓ=M+2∞+∑k=M+2∞∑ℓ=1∞μk+ℓ−23k+ℓ+12k+ℓ−2k−1!ℓ−1!.
By evaluating the summation, we obtain(58)RM+1≤27σe3μ1−ΓM+1,3μ/22ΓM+12cosh2μ.
The proof of the required result is similar to the proof in the given last theorem.


## 6. Numerical Examples

In this section, we demonstrate two examples using Pell polynomials for solving equation (1). Then, the comparison of the used method and the method in [27] shows the efficiency of our method.Example 1.
Consider the one-dimensional time-fractional convection equation (1) with conditions
(59)υy,0=0,y∈0,1,υ0,t=0,t∈0,1.
where(60)hy,t=Γλ+1+2t2−λΓ3−λsinπy+πt2+tλcosπy.
The exact solution of this equation isυy,t=t2+tλsinπy.In Table1, we observe that the absolute errors obtained by Pell collocation PLC method at M=2, 3, and 4 are better than obtained by the L1-discontinuous Galerkin L1/DG method [27]. Figures 1–3 display the numerical solutions at λ =0.4, 0.6, and 0.8. The absolute errors at the same values are plotted in Figure 4. It is clear that the last figure of the absolute errors decreases drastically with decreasing the number of steps.Example 2.
Consider the one-dimensional time-fractional convection equation (1) with conditions
(61)υy,0=y21−y2,y∈0,12−y21−y2,y∈12,1,υ0,t=0,t∈0,1.
where(62)hy,t=Γλ+1+2t2−λΓ3−λy21−y2+t2+tλ+12y1−y1−2y,y∈0,12−Γλ+1+2t2−λΓ3−λy21−y2−t2+tλ+12y1−y1−2y,y∈12,1.
The exact solution of this equation isυy,t=t2+tλ+1y21−y2,y∈0,1/2−t2+tλ+1y21−y2,y∈1/2,1.Table 1
Maximum absolute errorE with PLC method.
(a)λMEMEME0.401.1×10−141.2×10−130.62031.8×10−1546.2×10−140.801.6×10−158.9×10−15(b)
L2 errors with L1/DG method
Mλ=0.4λ=0.6λ=0.8647.3×10−41×10−32.1×10−31282.7×10−44.2×10−49.3×10−42569.5×10−51.7×10−44.1×10−45123.3×10−56.4×10−51.8×10−410241.1×10−52.5×10−57.9×10−5(c)CPU timeCN98.4361.6×10−15696.9987.3×10−152333.649.2×10−15Figure 1
Graph of the numerical solutions atλ=0.4.Figure 2
Graph of the numerical solutions atλ=0.6.Figure 3
Graph of the numerical solutions atλ=0.8.Figure 4
Graph of the absolute errors atM =2, 3, 4.Table2 lists the numerical solutions of our method at M=2 and different values of λ. These solutions and the absolute errors are plotted in Figures 5–8. From Figure 8, we observe that the convergence is exponential, and the errors are better with decreasing the number of steps. We notice that the obtained errors are the best and the least.Table 2
Maximum absolute errorE with PLC method
(a)λME0.400.622.6×10−130.85.7×10−14(b)
L2 errors with L1/DG methods
Mλ=0.4λ=0.6λ=0.842.8×10−32.7×10−32.7×10−383.5×10−43.5×10−43.5×10−4164.4×10−54.4×10−54.2×10−5325.4×10−65.1×10−65.2×10−6(c)CPU timeCN374.5632.6×10−13405.8282.03×10−13350.3299.2×10−13Figure 5
Graph of the numerical solutions atλ=0.4.Figure 6
Graph of the numerical solutions atλ=0.6.Figure 7
Graph of the numerical solutions atλ=0.8.Figure 8
Graph of the absolute errors atM =2 and different values of λ.


## 7. Conclusion

Due to this method, we demonstrate two examples which show the high accuracy of our suggested method and also it is easy to apply. The Mathematica software was agreeable surprised with the results of this method. We also show that any system of linear equations can borrow the given method. The comparison of the method and the non-uniform L1-discontinuous Galerkin (DG) method shows that the spectral solution in ours is more efficient. The truncation and global errors are discussed. Finally, we conclude that the proposed method can solve different kinds of fractional differential equations and integral equations. Finally, Pell polynomial is not the best choice to approximate the solutions of differential equations. I think—as far as I know—the orthogonal Chebyshev polynomials are better.



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