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

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. (e fractional derivative is defined in Caputo manner. (e 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. (e 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.


Introduction
Fractional calculus is used in various branches of mathematics due to its numerous applications in modeling different physical phenomena in engineering and science. e concept of fractional calculus has been derived from the fact D α (f(x)), 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 realworld physical problems, FDEs have been widespread due to their numerous applications in engineering and real-life sciences problems [6][7][8][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], fluiddynamic traffic model [15], colored noise [16], solid mechanics [11], anomalous transport [17], and bioengineering [18][19][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][22][23][24][25].
In the present work, CPM is extended for the solutions of FDDEs. e 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 with the following boundary conditions: where h is the delay function which is to be assumed continues in the interval [c, d] and satisfies the inequality c 0 ≤ h(u) ≤ u for some fix real constant c 0 , for u ∈ [c, d] and ζ ∈ C[c 0 , c] e following is a summary of the paper's structure. In Section 2, we introduce some fundamental fractional calculus definitions and mathematical techniques that will be useful in our later study. e approximation of the fractional derivative D c u f(u) 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.

Basic Definitions of Fractional Derivatives
Definition 1. A real function, g(u), u > 0, is said to be in the space C μ , μ ∈ R, if there exists a real number p > μ, such that g(u) � u p g 1 (u), where g 1 (u) ∈ [0, ∞), and it is said to be in the space C m μ if and only if g (m) ∈ C μ , m ∈ N.
Definition 2. In Caputo manner, the derivative having fractional-order D c g(u) is given as below: e order of the derivative is c > 0, and the lowest integer greater than c is j ∈ N and g ∈ C n − 1 . We have the Caputo derivative [39]: where the lowest integer larger than or equal to c is denoted by the ceiling function ⌈ c ⌉ and N 0 � 1, 2, . . .. Remember that the Caputo differential operator is the same as the normal differential operator of the integer order for c ∈ N. Fractional differentiation is a linear operation, just like integer-order differentiation: where ϕ and μ are constants.

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] where T 0 (u) � 1 and T 1 (u)u. e Chebyshev polynomial analytical form for degree j is defined as [41] T j (u) � j 2 If we apply the Chebyshev polynomials over the [0, 1] interval, we explain the Chebyshev shifted polynomials T j (u).
ese are described in the sense of Chebyshev polynomials T j (u) as [41] T j (u) � T j (2u − 1).
And recurrence formula is as follows: where T 0 (u) � 1 and T 1 (u) � 2u − 1. e orthogonality condition is [42] Now, we can use the well-known relation, 2 Complexity and equation (8) to get shifted Chebyshev polynomials analytical form having order j as A function f(u) ∈ L 2 [0, 1] may be described in terms of Chebyshev shifted polynomials as where the coefficients c j , j � 1, 2,. . ., are given by Only Chebyshev shifted polynomials first (m + 1)-terms are considered in practice. us, 3.1. Chebyshev Truncation eorem [43]. e sum of the absolute values of all the disregarded coefficients limits the inaccuracy in approximating f(u) by the sum of its first m terms. at is, assuming then, for all f(u), all m, and all u ∈ [− 1, 1], we obtain Proof. For any u ∈ [− 1, 1] and all k, the Chebyshev polynomials are bounded by 1, |T k (u)| ≤ 1. As a result, the kth term is restricted by |c k |. 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. e following theorem contains the main approximate formula for the fractional derivative of f(u). [44]. Assume α > 0 and that f(u) is estimated by the shifted Chebyshev polynomials as in (16). en, where b α j,r is given by Proof. Since Caputo fractional differentiation is a linear operation, we have Now, to evaluate D α (T j (u)), applying to equations (4) and (5)- (13), Since T j (u) is a polynomial having degree j, we obtain e following is the result of combining (21)- (23): which is the desired result. Test example: consider formula (19) with and which yields the same result as evaluating D 1/2 (u 2 ) by relation (5).

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 solution f(u) is defined in terms of a finite number m of shifted Chebyshev polynomials, i.e., We can use theorem (4.1) and equation (22) to solve equation (1): Using (22) in the boundary conditions (2), we may construct the following ⌈ α ⌉ equations: We get (m + 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 c j , j � 0, 1, 2, . . . m, to find out an estimated solution μ m (ψ).

Numerical Representation
In this section, we solve some delay problems. e results we obtained are compared with other methods. All the numerical results are obtained using MAPLE.

Problem 1. Consider the FDDE:
subject to the initial conditions f(0) � 1, having accurate solution f(u) � exp u at α � 1. e exact solution and CPM solution are given in Table 1. 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.

Problem 2. Consider the nonlinear DDE,
subjects to the initial condition f(0) � 1, f ′ (0) � 0. e accurate solution of this equation for α � 2 is f(u) � cos(u). e 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. e 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. e results of the presented method are better than those of the MWBA method for example 2.
Problem 3. Consider the fractional DDE of the form with initial conditions 4 Complexity e exact solution of this equation for α � 2 is f(u) � u 2 . e 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. e results of the presented method are better than those of the MWBA method for example 3.
e accurate solution of this equation for α � 2 is f(u) � 1 + u − u 3 . e 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. e 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.
e accurate solution of this problem for α � 3 is f(u) � exp − u . e 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. e results of the presented method are better than those of the CWM method for this problem.  8 Complexity Table 5: Exact, CPM solution, and CPM A.E of problem 3 at m � 3.

Conclusion
In this study, we applied the Chebyshev pseudospectral method for solving fractional delay differential equations. e 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. e 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.

Data Availability
e numerical data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.