# A Specific Method for Solving Fractional Delay Differential Equation via Fraction Taylor’s Series

Authors: Ming-Jing Du
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

It is well known that the appearance of the delay in the fractional delay differential equation (FDDE) makes the convergence analysis very difficult. Dealing with the problem with the traditional reproducing kernel method (RKM) is very tricky. The feature of this paper is to gain a more credible approximate solution via fractional Taylor’s series (FTS). We use the FTS to deal with the delay for improving the accuracy of the approximate solutions. Compared with other methods, the five numerical examples demonstrate the accuracy and efficiency of the proposed method in this paper.


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## Body

## 1. Introduction

For the past few years, FDDE have been applied in various fields of neoteric science and engineering such as economics, physics, dynamics, hydrology, finance, signal processing, neural network, and control theory [1]. There has been a growing interest in researching numerical methods for this equation. Many experts and scholars use various methods to study this kind of model, including Legendre seudospectral method [2], a new Legendre operational technique [3], the homotopy perturbation method [4], a linearized compact finite difference scheme [5], the techniques of Tau method and collocation method [6], the adaptation of a fractional backward difference method [7], extended Chebyshev wavelet method [8], and the variational iteration and Adomian decomposition method [9]. It is worth mentioning that in [10], Esra investigates a couple of different financial and economic models by market equilibrium and option pricing with three different fractional derivatives and proves the precision of the Sumudu transform and decomposition series method constructed by the Laplace transform. In [11], the Rabotnov exponential kernel is used to define the fundamental calculus of the fractional derivative and solve the nonlinear dispersive wave model. In [12], Asif uses three methods (the expansion method, the finite difference method, and the Laplace perturbation method) to solve the time-fractional nonlinear Burger–Fisher equation. In [13], Laiq uses the optimal auxiliary function method for solving the general partial differential equations. In [14], Abdulnasir adopts a new simple and effective new algorithm based on one of the Appell polynomials, namely, Genocchi polynomials, to solve the generalized Pantograph equations, the FDDE with neutral terms, and delay differential system with constant and variable coefficients. In [15], Phang uses an operational matrix method for solving the FDDEs. In [16], Salem gives the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations (FLE) with multipoint and nonlocal integral boundary conditions (NIBCs). In [17], Salem studies the existence and uniqueness of the solution for the same class of the equation in a slightly different form with antiperiodic and NIBCs. In [18], Selam and Dosari conduct research on the existence results of solutions for boundary value problems of inclusion type. In [19], Salem and Alghamdi give the existence and uniqueness of solutions for the Langevin equation (LE) that has Caputo fractional derivatives of two different orders. In [20], Salem et al. give the existence results for an infinite system of LEs involving generalized derivatives of two distinct fractional orders with three-point boundary conditions.In this study, we focus on a more general case in math, considering the following FDDE arising from economic models.(1)Dνut=atut−τ+btu′t−σ+ctut+ft,t∈0,T,ut=φt,t∈−τ,0∪−σ,0.τ and σ are the constant delays. at, bt, ct, φt, and ft are the continuous functions. Dν is the Riemann–Liouville fractional derivative of order ν>0.This paper is organized as follows: in Section2, some important definitions about the fractional Taylor’s series, reproducing kernels, and their spaces are reviewed. The process of solving problem (1) is presented in Section 3. The five numerical examples are given in Section 4. Then, we give the conclusion in Section 5.


## 2. Fractional Taylor’s Series

Definition 1.
TheR−L fractional derivative of order α is defined as [1](2)Dαux=1Γn−αdndxn∫0xutx−tn−αdt,where Γ⋅ is the gamma function and n=α+1, a denotes the greatest integer smaller than α.Lemma 1.
Let us suppose thatm<α≤m+1, m∈N−0 and that ut has derivatives of order k (integer), 1≤k≤m. Let us suppose further that umt has a FTS of order α−m=:β provided by the expression (the symbol = means that the left side is defined by the right one) [21]:(3)umt+h=∑k=0∞hkα−mΓ1+kα−mDkα−mumt,m<α≤m+1.
Next, on integrating this series with respect toh yields(4)ut+h=∑k=0mhkk!ukt+∑k=1∞hkβ+mΓkβ+m+1ukβ+mt,β≔α−m.Definition 2.
W2n+10,T=ut|ut,…,unt is one-variable absolutely continuous function ACF,un+1t∈L20,T. The inner product is given by(5)ut,wtW2n+1=∑i=0nui0wi0+∫0Tun+1twn+1tdt.Lemma 2.
The spaceW2n+10,T is a reproducing kernel space (RKS) and its reproducing kernel (RK) Rtn+1y is denoted by [12](6)Rtm+1y=∑i=0ntiyi+∫0tt−xny−xndx,t<y,∑i=0ntiyi+∫0yt−xny−xndt,y<t.


## 3. Solution of the Equation

By Lemma1, ft can be expressed via fractional Taylor’s series.(7)ft=Dνut−at⋅∑k=01−τkk!ukt+−τβ+1Γβ+2uβ+1t−bt⋅∑k=01−σkα−1Γ1+kα−1Dka−1u′t−ctut.Using TRKM, we need to define a bounded linear operatorL [22]: W230,T⟶W210,T, let(8)Lut=ft,so the solutions of (1) is the solution of (8). Rx3y [14] is the RK of the RKS W230,T; Rx1y is the RK of the RKS W210,T. According to [15], the exact solution of (7) is expanded by ut=∑j=1∞ftjζjt. The approximate solution of (7) is expanded by uNt=∑j=1Nftjζjt. For the convergence theorem, please refer to [23]. Error estimate is as follows.Theorem 1.
∀t∈0,T,∃H=ut−uNt⋅∂Rη,t/∂t|t=θ; there is alwaystjj=1∞satisfyingt−tj=1/N, sout−uNt≤H/N.Proof.
Due toLutj=LuNtj, then(9)Lut−LuNt=Lut−Lutj−LuNt−LuNtj.
Due tout=uη,Rη,t,Lut=uη,LRη,t, then(10)Lut−LTNt=Lut−Lutj−LuNt−LuNtj=uη,LRη,t−LRη,tj−uNη,LRη,t−LRη,tj=uη−uNη,LRη,t−LRη,tj.
So,(11)ut−uNt=L−1Lut−LuNt≤uη−uNη,L−1LRη,t−L−1LRη,tj≤u−uN⋅Rη,t−Rη,tj.
Because we take the norm ofRη,t−Rη,tj, for ∀η, the RK function Rη,t is derived on t in the interval of 0,T; therefore, Rη,t−Rη,tj=∂Rη,t/∂t|t=θ⋅t−tj, let H=ut−uNt⋅∂Rη,t/∂t|t=θ, hence(12)ut−uNt≤ut−uNt.∂Rη,t∂t|t=θ⋅t−tj=HN.


## 4. Numerical Examples

Example 1.
Consider the following FDDE:(13)Dνut=−u′t−1−ut+ft,t>0,ut=t2sinπt,t∈−1,0,with ν=3/2, ft=t−1⋅−πt−1cosπt−2sinπt+t2sinπt+D3/2t2sinπt which is compatible with the exact solution ut=t2sinπt. We estimated the parameter m=1, k=1, α=1.5, by Mathematical 7.0, N=14; Figure 1 shows the absolute errors by two methods, the left one is given by the traditional RKM, and the right one is given by the proposed method in this paper. It is obvious that the proposed method in this paper is more accurate than the traditional RKM.Figure 1
Absolute errors of Example1 by the traditional RKM (left) and the proposed method in this paper (right, m=1, k=1, α=1.5).Example 2.
Consider the following delay integro-differential equation [24]:(14)−tuνt+tu′t+ut+tu′t−1+3ut+1=ft+∫−11tu2x−1dx,u0=u′0=0,with ν=2, ft=23/48−13cos4/16+t−1tcost−1+t−2tcost−3sin4/8−tsin1−t+2+ttsint+31+tsin1+t; the exact solution of (14) is ut=tsint. The delay term in the equation is expanded by fractional Taylor’s series. We estimated the parameters m=1, k=1, α=1.1, and β=0.1. By Mathematical 7.0, N=10, Figure 2 presents the absolute errors by two methods, the left one is given by the traditional RKM, and the right one is given by the proposed method in this paper. It is obvious that the proposed method in this paper is more precise than the traditional RKM.Figure 2
Absolute errors of Example2 by the traditional RKM (left) and the proposed method in this paper (right, m=1, k=1, α=1.1, β=0.1).Example 3.
Consider the following FDDE:(15)uνt−1−tut−1=183−5t+3t3−t4,u0=0,with ν=1, the exact solution ut=t/23. The delay term in the equation is expanded by fractional Taylor’s series. We estimated the parameters m=1, k=1, α=1.5, and β=0.5. By Mathematical 7.0, N=12, Figure 3 shows the absolute errors by two methods, the left one is given by the traditional RKM, and the right one is given by the proposed method in this paper. It is obvious that the proposed method in this paper is more precise than the traditional RKM.Figure 3
Absolute errors of Example3 by the traditional RKM (left) and the proposed method in this paper (right, m=1, k=1, α=1.5, β=0.5).Example 4.
Consider the following FDDE [25]:(16)Dνut+ut−ut−τ=2Γ3−νt2−ν−1Γ2−νt1−ν+2τt−τ2−τ,u0=0.
We estimated the parametersν=0.2, τ=0.1, m=1, k=1, α=1.5, and β=0.5. The exact solution of (16) is ut=t2−t. By Mathematical 7.0, N=16, Figure 4 presents the absolute errors by two methods. The left one is given by the traditional RKM and the right one is given by the proposed method in this paper. It is obvious that the proposed method in this paper is more precise than the traditional RKM.Figure 4
Absolute errors of Example4 by traditional method (blue line) and the proposed method in this paper (red line, m=1, k=1, α=1.5, β=0.5).Example 5.
Consider the following FDDE [25–27]:(17)Dνut=−ut−ut−τ+e−t+τ,u0=1,u′0=−1,u″0=1.
The exact solution isut=e−t when ν=3. We estimated the parameters τ=0.3, m=1, k=1, α=1.5, and β=0.5. By Mathematical 7.0, N=12, Table 1 shows the numerical comparison by the five methods.Table 1
Numerical solution comparison of Example5 by the five methods.
xiExact solutionADM [26]Method [25]Hermite wavelet [27]Traditional RKMThe proposed method0.01.000 01.000 01.000 01.000 01.000 01.000 00.20.818 70.818 70.818 70.818 70.818 90.818 70.40.670 30.670 30.670 30.670 30.671 80.670 30.60.548 80.548 80.548 80.548 80.552 90.548 80.80.449 30.449 30.449 40.449 30.457 50.449 3


## 5. Conclusion

In this study, the FTS and the RKs are successfully used to solve a class of PDDEs. The comparison between the proposed method and other methods is given in Section4, which shows that the proposed method in this paper is a very satisfactory numerical method. In addition, we can also use this new method to solve the option pricing model.



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