# Analysis of the Fractional-Order Delay Differential Equations by the Numerical Method

Authors: Saadia Masood; Muhammad Naeem; Roman Ullah; Saima Mustafa; Abdul Bariq
Journal: Complexity
 (2022)

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## Abstract

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations.


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## Body

## 1. Introduction

Fractional calculus is used in various branches of mathematics due to its numerous applications in modeling different physical phenomena in engineering and science. The concept of fractional calculus has been derived from the factDαfx, where alpha is noninteger. Later on, different scientists such as Riemann–Liouville, Euler, Leibniz, L’Hospital, Bernoulli, and Wallis have devoted their work to this research area. Fractional calculus has numerous applications in different field of sciences. For example dynamic of viscoelastic materials [1], electromagnetism [2], fluid mechanics [3], propagation of spherical flames [4], and viscoelastic materials [5].In our real life, DEs are used to develop a different number of physical problems. Some are more complex and cannot be modeled with the help of simple differential equations. For these complex problems, a new technique has been used by the researchers known as fractional differential equations (FDEs). In the mathematical modeling of real-world physical problems, FDEs have been widespread due to their numerous applications in engineering and real-life sciences problems [6–9], such as economics [10], solid mechanics [11], continuum and statistical mechanics [12], oscillation of earthquakes [13], dynamics of interfaces between soft-nanoparticles and rough substrates [14], fluid-dynamic traffic model [15], colored noise [16], solid mechanics [11], anomalous transport [17], and bioengineering [18–20].Delay differential equations (DDEs) have a wide range of applications in engineering and science. Delay differential equation simplifies the ordinary differential equation, depends on the past data, and is suitable for physical systems. Nowadays, researchers pay more attention to FDDEs as compared to DEs because a slight delay has a large effect. In this regard, numerous papers have been dedicated to the study of the numerical solution of FDDEs. FDDs have been widespread in mathematical modelings, such as population dynamics, epidemiology, immunology, physiology, and neural networks [21–25].In literature, there is no precise technique for finding an exact or analytical solution for every FDDEs; the researcher’s effort is to find the numerical solution of FDDEs. Various methods have been implemented for solving these problems numerically. The well-known among these methods are new predictor corrector method (NPCM) [26], adomian decomposition method (ADM) [27], Legendre pseudospectral method (LSM) [28], kernel method (KM) [29], LMS method (LMSM) [30], Adams–Bashforth–Moulton algorithm (ABMA) [31], extend predictor corrector method (EPCM) [32], simplified reproducing kernel method (SRKM) [29], variation iteration method (VIM) [33], homotopy perturbation method (HPM) [34], Galerkin method (GM) [35], Runge–Kutta-type methods (RKM) [36], Bernoulli wavelet method (BWM) [37], and modified Laguerre wavelet method [38] have been used for the analytical and numerical solution of FDDEs.In the present work, CPM is extended for the solutions of FDDEs. The results we obtained are compared with other methods, which show that CPM has good convergence rate than other methods. We focus on FDDE of the form(1)Duγfu=gu,fu,fhu,c≤u≤d,m<γ≤m+1,m=1,2,3,…,with the following boundary conditions:(2)fc=α0,fd=α1,fu=ζu,u∈c0,c,where h is the delay function which is to be assumed continues in the interval [c, d] and satisfies the inequality c0≤hu≤u for some fix real constant c0, for u∈c,d and ζ∈Cc0,cThe following is a summary of the paper’s structure. In Section2, we introduce some fundamental fractional calculus definitions and mathematical techniques that will be useful in our later study. The approximation of the fractional derivative Duγfu is obtained in Section 3. Section 4 describes the Chebyshev collocation method’s application to the solution of eq. (1). As a result, a set of algebraic equations is created, and the solution to the problem in question is presented. Section 5 provides some numerical results to help clarify the method.


## 2. Basic Definitions of Fractional Derivatives

Definition 1.
A real function,gu,u>0, is said to be in the space Cμ,μ∈ℝ, if there exists a real number p>μ, such that gu=upg1u, where g1u∈0,∞, and it is said to be in the space Cμm if and only if gm∈Cμ, m∈ℕ.Definition 2.
In Caputo manner, the derivative having fractional-orderDγgu is given as below:(3)Dγgu=1Γj−γ∫0uu−tj−γ−1gntdt,u>0,j−1<γ<j.
The order of the derivative isγ>0, and the lowest integer greater than γ is j∈ℕ and g∈C−1n.
We have the Caputo derivative [39]:(4)DγC=0,Cis a constant,(5)Dγuα=0forα∈ℕ0andα<γΓα+1Γα+1−γuα−γforα∈ℕ0andα≥γ,where the lowest integer larger than or equal to γ is denoted by the ceiling function γ and ℕ0=1,2,…. Remember that the Caputo differential operator is the same as the normal differential operator of the integer order for γ∈ℕ. Fractional differentiation is a linear operation, just like integer-order differentiation:(6)Dγϕgu+μhu=ϕDγgu+μDγhu,where ϕ and μ are constants.


## 3. Chebyshev Series Expansion Is Used to Approximate a Caputo Derivative

On the interval−1,1, Chebyshev polynomials are defined and, with the help of recurrence formulae, explained as [40, 41](7)Tj+1u=2uTju−Tj−1u,j=1,2,…,where T0u=1 and T1uu. The Chebyshev polynomial analytical form for degree j is defined as [41](8)Tju=j2∑r=0j/2−1rj−r−1!r!j−2r!2uj−2r.If we apply the Chebyshev polynomials over the0,1 interval, we explain the Chebyshev shifted polynomials T^ju. These are described in the sense of Chebyshev polynomials Tju as [41](9)T^ju=Tj2u−1.And recurrence formula is as follows:(10)T^j+1u=22u−1T^ju−T^j−1u,j=1,2,…,where T^0u=1 and T^1u=2u−1. The orthogonality condition is [42](11)∫01T^juT^muu−u2du=0m≠j,π2m=j≠0,πm=j=0.Now, we can use the well-known relation,(12)T^ju=T2ju,and equation (8) to get shifted Chebyshev polynomials analytical form having order j as(13)Tj^u=∑r=0j−1r22j−2rj2j−r−1!r!2j−2r!xj−2r.A functionfu∈L20,1 may be described in terms of Chebyshev shifted polynomials as(14)fu=∑j=1∞cjT^ju,where the coefficients cj, j = 1, 2,…, are given by(15)c0=1π∫01guT^0uu−u2duandcn=2π∫01guT^juu−u2du.Only Chebyshev shifted polynomials firstm+1-terms are considered in practice. Thus,(16)fmu=∑j=0mcjT^ju.


### 3.1. Chebyshev Truncation Theorem [43]

The sum of the absolute values of all the disregarded coefficients limits the inaccuracy in approximatingfu by the sum of its first m terms. That is, assuming(17)fmu=∑k=0mckTku,then, for all fu, all m, and all u∈−1,1, we obtain(18)ETm=fu−fmu≤∑k=m+1∞ck.Proof.
For anyu∈−1,1 and all k, the Chebyshev polynomials are bounded by 1, Tku≤1. As a result, the kth term is restricted by ck. By subtracting the reduced series from the infinite series, bounding each term in the difference, and then summing the bounds, the theorem can be derived.
The following theorem contains the main approximate formula for the fractional derivative offu.


### 3.2. Theorem [44]

Assumeα>0 and that fu is estimated by the shifted Chebyshev polynomials as in (16). Then,(19)Dαfmu=∑j=αm∑r=0n−αcjbj,rαuj−r−α,where bj,rα is given by(20)bj,rα=−1r22j−2rj2j−r−1!j−r!r!2j−2r!Γj−r+1−α.Proof.
Since Caputo fractional differentiation is a linear operation, we have(21)Dαfmu=∑j=0mcnDαTj^u.
Now, to evaluateDαTj^u, applying to equations (4) and (5)–(13),(22)DαTj^u=∑r=0j−1r22j−2rj2j−r−1!r!2j−2r!Dαuj−r,j=α,α+1,…m.
SinceTj^u is a polynomial having degree j, we obtain(23)DαTj^u=0forallj=0,1,2,…,α−1,α>0.
The following is the result of combining (21)–(23):(24)Dαfmu=∑j=αm∑r=0n−αcj−1r22j−2rj2j−r−1!j−r!r!2j−2r!Γj−r+1−αuj−r−α=∑j=αm∑r=0n−αcjbj,rαuj−2r−α,which is the desired result.
Test example: consider formula (19) with fu=u2,m=2. The shifted series of u2 is(25)u2=c0T0^u+c1T1^u+c2T2^u=38T0^u+12T1^u+18T2^uand(26)D12u2=∑j=12∑r=0n−1cnbn,r12uj−r−12=c1b1,012u12+c2b2,012u32+c2b2,112u12=2πu12+83πu32−2πu12=83πu32,which yields the same result as evaluating D1/2u2 by relation (5).


## 3.1. Chebyshev Truncation Theorem [43]

The sum of the absolute values of all the disregarded coefficients limits the inaccuracy in approximatingfu by the sum of its first m terms. That is, assuming(17)fmu=∑k=0mckTku,then, for all fu, all m, and all u∈−1,1, we obtain(18)ETm=fu−fmu≤∑k=m+1∞ck.Proof.
For anyu∈−1,1 and all k, the Chebyshev polynomials are bounded by 1, Tku≤1. As a result, the kth term is restricted by ck. By subtracting the reduced series from the infinite series, bounding each term in the difference, and then summing the bounds, the theorem can be derived.
The following theorem contains the main approximate formula for the fractional derivative offu.


## 3.2. Theorem [44]

Assumeα>0 and that fu is estimated by the shifted Chebyshev polynomials as in (16). Then,(19)Dαfmu=∑j=αm∑r=0n−αcjbj,rαuj−r−α,where bj,rα is given by(20)bj,rα=−1r22j−2rj2j−r−1!j−r!r!2j−2r!Γj−r+1−α.Proof.
Since Caputo fractional differentiation is a linear operation, we have(21)Dαfmu=∑j=0mcnDαTj^u.
Now, to evaluateDαTj^u, applying to equations (4) and (5)–(13),(22)DαTj^u=∑r=0j−1r22j−2rj2j−r−1!r!2j−2r!Dαuj−r,j=α,α+1,…m.
SinceTj^u is a polynomial having degree j, we obtain(23)DαTj^u=0forallj=0,1,2,…,α−1,α>0.
The following is the result of combining (21)–(23):(24)Dαfmu=∑j=αm∑r=0n−αcj−1r22j−2rj2j−r−1!j−r!r!2j−2r!Γj−r+1−αuj−r−α=∑j=αm∑r=0n−αcjbj,rαuj−2r−α,which is the desired result.
Test example: consider formula (19) with fu=u2,m=2. The shifted series of u2 is(25)u2=c0T0^u+c1T1^u+c2T2^u=38T0^u+12T1^u+18T2^uand(26)D12u2=∑j=12∑r=0n−1cnbn,r12uj−r−12=c1b1,012u12+c2b2,012u32+c2b2,112u12=2πu12+83πu32−2πu12=83πu32,which yields the same result as evaluating D1/2u2 by relation (5).


## 4. Chebyshev Collocation Method

We solve the FDDE (1) with the boundary conditions (2) stated in the next part using Chebyshev’s collocation method in this section. Assume that the approximate solutionfu is defined in terms of a finite number m of shifted Chebyshev polynomials, i.e.,(27)fmu=∑j=0mcjT^ju.We can use theorem (4.1) and equation (22) to solve equation (1):(28)∑j=αm∑r=0n−αcjbj,rαuj−2r−α=gu,∑j=0mcjT^ju,∑j=0mcjT^jhu,0<u<1,m+1<α<m.Now, we collocate (23) at points up, p=0,1,2,…,m−α:(29)∑j=αm∑r=0n−αcjbj,rαupj−2r−α=gup,∑j=0mcjT^jup,∑j=0mcjT^jhup,up,p=0,1,…m−α,m+1<α<m.Using (22) in the boundary conditions (2), we may construct the following α equations:(30)∑i=0m−1ici=α0,∑i=0mci=α1.We getm+1−α algebraic equations from (24) and α algebraic equations from (26). As a result, we have total (m + 1) linear or nonlinear algebraic equations that can be easily solved using matrices for unknowns cj,j=0,1,2,…m, to find out an estimated solution μmψ.


## 5. Numerical Representation

In this section, we solve some delay problems. The results we obtained are compared with other methods. All the numerical results are obtained using MAPLE.Problem 1.
Consider the FDDE:(31)dαfudu=12expu2fu2+12fu,0<α≤1,subject to the initial conditions f0=1, having accurate solution fu=expu at α=1.
The exact solution and CPM solution are given in Table1. Table 2 shows CPM and CWM error comparison at m=4 which confirm that CPM converges quickly as compared to CWM. We illustrate the accurate and estimated solutions for m=4 in Figure 1, while Figure 2 shows the error comparison of both methods. Also, Figure 3 provides the graphical layout of the solution of example 1 at various fractional orders. It can be seen that the solutions of CPM are in good agreement to the actual solution than that of CWM.Table 1
Exact, CPM solution, and CPM A.E of problem1 for m = 4.
uExactCPMCPM error01.000 000 000 000 0001.000 000 000 000 0000.0 000 000 000E +000.011.010 050 167 084 1701.010 050 167 197 0001.1 283 000 000E −100.021.020 201 340 026 7601.020 201 341 818 6701.7 919 100 000E −090.031.030 454 533 953 5201.030 454 542 957 0209.0 035 000 000E −090.041.040 810 774 192 3901.040 810 802 432 0602.8 239 670 000E −080.051.051 271 096 376 0201.051 271 164 791 8206.8 415 800 000E −080.061.061 836 546 545 3601.061 836 687 312 3001.4 076 694 000E −070.071.072 508 181 254 2201.072 508 439 997 5602.5 874 334 000E −070.081.083 287 067 674 9601.083 287 505 579 6304.3 790 467 000E −070.091.094 174 283 705 2101.094 174 979 518 5406.9 581 333 000E −070.101.105 170 918 075 6501.105 171 970 002 3501.0 519 267 000E −06Table 2
Absolute error (A.E) comparison of CPM and other different methods of problem 1 atm = 4.
uCPM A.ECWM A.E00.0 000 000 000E +000.0 000 000 000E +000.011.1 283 000 000E −101.82 567E −040.021.7 919 100 000E −093.68 446E −040.039.0 035 000 000E −095.57 408E −040.042.8 239 670 000E −087.49 213E −040.056.8 415 800 000E −089.43 609E −040.061.4 076 694 000E −071.14 034E −030.072.5 874 334 000E −071.33 912E −030.084.3 790 467 000E −071.53 969E −030.096.9 581 333 000E −071.74 174E −030.101.0 519 267 000E −061.94 497E −03Figure 1
The exact and CPM solution graph for problem1.Figure 2
Error graph of CWM and CPM for problem1.Figure 3
The graph of the absolute error at various fractional orders for problem1.Problem 2.
Consider the nonlinear DDE,(32)dαfudu=1−2f2u2,0≤u≤1,1<α≤2,subjects to the initial condition f0=1,f′0=0.
The accurate solution of this equation forα=2 is fu=cosu. The exact solution and CPM solution are shown in Table 3. Table 4 shows the error comparison of CPM at m=3 and MWBA at m=20 which confirm that CPM converges quickly as compare to MLWM. The estimated and accurate solutions are illustrated in Figure 4, whereas Figure 5 shows the error comparison of both methods. In addition, the convergence phenomena of the solutions at different fractional orders can be seen in Figure 6. The results of the presented method are better than those of the MWBA method for example 2.Table 3
Exact, CPM solution, and CPM A.E of problem2 at m = 10.
uExactCPMCPM error01.000 000 000 000 0001.000 000 000 000 0000.0 000 000 000E +000.100.995 004 165 278 0260.995 004 165 278 0262.9 000 000 000E −190.200.980 066 577 841 2420.980 066 577 841 2422.7 250 000 000E −160.300.955 336 489 125 6060.955 336 489 125 6211.5 097 140 000E −140.400.921 060 994 002 8850.921 060 994 003 1382.5 278 786 000E −130.500.877 582 561 890 3730.877 582 561 892 5442.1 711 958 200E −120.600.825 335 614 909 6780.825 335 614 921 7381.2 059 545 750E −110.700.764 842 187 284 4880.764 842 187 333 1954.8 707 056 530E −110.800.696 706 709 347 1650.696 706 709 498 8941.5 172 838 626E −100.900.621 609 968 270 6640.621 609 968 640 6093.6 994 453 098E −101.00.540 302 305 868 1400.540 302 306 536 3946.6 825 465 360E −10Table 4
Absolute error (A.E) comparison of CPM and other different methods for problem2.
uCPM A.E at (m = 3)MLWM A.E at (m = 20)002.10 000E −080.102.9 000 000 000E −192.11 000E −080.202.7 250 000 000E −162.09 000E −080.301.5 097 140 000E −142.09 000E −080.402.5 278 786 000E −132.08 000E −080.502.1 711 958 200E −122.06 000E −080.601.2 059 545 750E −112.04 000E −080.704.8 707 056 530E −112.03 000E −080.801.5 172 838 626E −102.00 000E −080.903.6 994 453 098E −101.99 000E −081.06.6 825 465 360E −101.97 000E −08Figure 4
The graph of absolute error at various fractional orders for problem2.Figure 5
Exact and CPM solution graph for problem2.Figure 6
Error graph of MLWM and CPM for problem2.Problem 3.
Consider the fractional DDE of the form(33)dαfudu=fu2+34fu−u2+2,0≤u≤1,1<α≤2,with initial conditions f0=f′0=0.
The exact solution of this equation forα=2 is fu=u2. The exact solution and CPM solution are shown in Table 5. Table 6 shows the error comparison of CPM at m=3 and MWLM at m=5 which confirm that CPM converges quickly as compare to MLWM. We illustrate the accurate and estimated solutions for m=3 in Figure 7, while Figure 8 shows the error comparison of both methods. The results of the presented method are better than those of the MWBA method for example 3.Table 5
Exact, CPM solution, and CPM A.E of problem 3 atm = 3.
uExactCPMCPM error00.000 000 000 000 000.000 000 000 000 004.0 000 000 000E +000.100.010 000 000 000 000.010 000 000 000 001.2 100 000 000E −290.200.040 000 000 000 000.040 000 000 000 002.4 800 000 000E −290.300.090 000 000 000 000.090 000 000 000 003.7 400 000 000E −290.400.160 000 000 000 000.160 000 000 000 005.0 000 000 000E −290.500.250 000 000 000 000.250 000 000 000 006.4 000 000 000E −290.600.360 000 000 000 000.360 000 000 000 007.8 000 000 000E −290.700.490 000 000 000 000.490 000 000 000 009.4 000 000 000E −290.800.640 000 000 000 000.640 000 000 000 001.1 100 000 000E −280.900.810 000 000 000 000.810 000 000 000 001.2 800 000 000E −281.01.000 000 000 000 001.000 000 000 000 001.4 000 000 000E −28Table 6
Absolute error (A.E) comparison of CPM and other different methods of problem3.
uCPM A.E at (m = 3)MLWM A.E at (m = 5)04.0 000 000 000E +001.41 421E −090.101.2 100 000 000E −294.75 800E −080.202.4 800 000 000E −299.69 300E −080.303.7 400 000 000E −291.47 010E −070.405.0 000 000 000E −291.98 200E −070.506.4 000 000 000E −292.50 900E −070.607.8 000 000 000E −293.05 500E −070.709.4 000 000 000E −293.62 400E −070.801.1 100 000 000E −284.22 000E −070.901.2 800 000 000E −284.84 800E −071.01.4 000 000 000E −285.51 000E −07Figure 7
The exact and CPM solution graph for problem3.Figure 8
Error graph of MLWM and CPM for problem3.Problem 4.
Consider the following nonlinear delay differential equation with boundary conditionsf0=1andf1=1:(34)dαduαfu=83ddufu2fu+8u2fu2−43−223u−7u2−53u3,1<α≤2.
The accurate solution of this equation forα=2 is fu=1+u−u3. The exact and CPM solution are shown in Table 7. Table 8 shows the error comparison of CPM at m=4 and MWBA at m=8 which confirm that CPM converges quickly as compare to MWBA. The estimated and accurate solutions are illustrated in Figure 9, while Figure 10 shows the error comparison of both methods. It can be seen that our method is more accurate.Table 7
Exact, CPM solution, and CPM A.E atm = 4 of problem 4.
uExactCPMCPM error01.000 000 000 000 0001.000 000 000 000 0002.0 000 000 000E −400.101.099 000 000 000 0001.099 000 000 000 0003.0 000 000 000E −390.201.192 000 000 000 0001.192 000 000 000 0006.0 000 000 000E −390.301.273 000 000 000 0001.273 000 000 000 0007.0 000 000 000E −390.401.336 000 000 000 0001.336 000 000 000 0001.0 000 000 000E −380.501.375 000 000 000 0001.375 000 000 000 0001.3 000 000 000E −380.601.384 000 000 000 0001.384 000 000 000 0001.6 000 000 000E −380.701.357 000 000 000 0001.357 000 000 000 0001.9 000 000 000E −380.801.288 000 000 000 0001.288 000 000 000 0002.3 000 000 000E −380.901.171 000 000 000 0001.171 000 000 000 0002.5 000 000 000E −381.01.000 000 000 000 0001.000 000 000 000 0002.9 000 000 000E −38Table 8
Absolute error (A.E) comparison of CPM and other different methods for problem4.
uCPM error at (m = 4)MWBA error at (m = 8)02.0 000 000 000E −401.20 000E –290.103.0 000 000 000E −391.00 000E −290.206.0 000 000 000E −391.00 000E −290.307.0 000 000 000E −391.00 000E −290.401.0 000 000 000E −381.00 000E −290.501.3 000 000 000E −381.00 000E −290.601.6 000 000 000E −381.00 000E −290.701.9 000 000 000E −381.00 000E −290.802.3 000 000 000E −382.00 000E −290.902.5 000 000 000E −382.00 000E −291.02.9 000 000 000E −381.50 000E −29Figure 9
The exact and CPM solution graph for problem4.Figure 10
Error graph of MWBA and CPM for problem4.Problem 5.
Consider the FDDE(35)dαfudu+fu+fu−0.3=exp−u+0.3,1<α<2,0<α<1,having initial conditions f0=1,f′0=−1,andf′′0=1.
The accurate solution of this problem forα=3 is fu=exp−u. The exact and CPM solutions are shown in Table 9. Table 10 shows the error comparison of CPM and CWM at m=6 which confirm that CPM converges quickly as compared to CWM. We illustrate the accurate and estimated solutions for m=6 in Figure 11, while Figure 12 shows the error comparison of both methods. In Figure 13, the solution for example 4.5 at different fractional orders is calculated. It is confirmed that the solution at various fractional order approaches towards the integer-order solution. The results of the presented method are better than those of the CWM method for this problem.Table 9
Exact, CPM solution, and CPM error of problem5 for m = 6.
uExactCPMCPM (A.E)01.000 000 000 000 0001.000 000 000 000 0000.0 000 000 000E +000.010.990 049 833 749 1680.990 049 833 749 1682.6 506 231 572E −160.020.980 198 673 306 7550.980 198 673 306 7381.6 837 479 038E −140.030.970 445 533 548 5080.970 445 533 548 3181.9 035 198 323E −130.040.960 789 439 152 3230.960 789 439 151 2621.0 614 659 655E −120.050.951 229 424 500 7140.951 229 424 496 6954.0 185 154 936E −120.060.941 764 533 584 2490.941 764 533 572 3411.1 907 912 782E −110.070.932 393 819 905 9480.932 393 819 876 1512.9 797 628 776E −110.080.923 116 346 386 6360.923 116 346 320 7526.5 884 200 594E −110.090.913 931 185 271 2280.913 931 185 138 6941.3 253 379 871E −100.100.904 837 418 035 9600.904 837 417 788 5122.4 744 798 291E −10Table 10
Absolute error (A.E) comparison of CPM and other different methods of problem5 at m = 6.
uCPM A.ECWM A.E00.0 000 000 000E +000.0 000 000 000E +000.012.6 506 231 572E −168.20 000E −090.021.6 837 479 038E −146.68 000E −080.031.9 035 198 323E −132.28 800E −070.041.0 614 659 655E −125.50 500E −070.054.0 185 154 936E −121.09 130E −060.061.1 907 912 782E −111.91 420E −060.072.9 797 628 776E −113.08 520E −060.086.5 884 200 594E −114.67 410E −060.091.3 253 379 871E −106.75 420E −060.102.4 744 798 291E −109.40 260E −06Figure 11
The exact and CPM solution graph for problem5.Figure 12
Error graph of CWM and CPM for problem5.Figure 13
The graph of absolute error at various fractional orders for problem5.


## 6. Conclusion

In this study, we applied the Chebyshev pseudospectral method for solving fractional delay differential equations. The technique is easy to implement and show good convergence rate than other methods. Some examples are solved which shows the effectiveness of the present method. The results we obtained are compared with other methods such as modified wavelet-based algorithm (MWBA), modified Laguerre wavelet method (MLWM), Chebyshev wavelet method (CWM). It is clear from comparison that CPM has higher accuracy than all these methods. Although, CPM can easily be extended to other fractional delay or nondelay models of physics and real-life sciences.



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*Source: 3218213-2022-03-14.xml*