Analytical Approximate Solutions for a General Class of Nonlinear Delay Differential Equations

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.


Introduction
Delay differential equations are frequently used to model reallife phenomena in various fields such as mechanics, biology, computer science, and chemistry. Some of the recent studies involving delay differential equations include topics as varied as epidemic models that describe the fraction of a population infected by a virus [1], complex oscillator network [2], and neural networks [3].
It is known that the computation of exact solutions for delay differential equations is only possible in particular cases. It follows that, in most cases, in order to obtain information about the phenomena modeled, the computation of approximate solutions becomes a necessity.
(i) In [4] a numerical approximation based on the Bessel functions of the first kind was applied to (1) of the Riccati type.
(ii) In [5] a two-stage order-one Runge-Kutta method was applied to (1) of the neutral-function differential type. The same equation was studied in [6,7] by using the one-leg -method, in [8] by using the variational iteration method, in [9] by using the homotopy perturbation method, and in [10] by using a method based on Chebyshev polynomials.
(iii) In [11] the homotopy perturbation method was applied to (1) of the pantograph type. The same equation was studied in [12] by using the variational iteration method.
(iv) In [13] the Jacobi rational-Gauss collocation method was applied to (1) of the generalized pantograph type. The same equation was studied in [14] by using the 2 The Scientific World Journal Taylor series method, in [15] by using a Chebyshev method, and in [16] by using a Hermite collocation method. (v) In [17] the variational iteration method and the Adomian decomposition method were applied for the case of a nonlinear time-delay model in biology which is also a particular case of (1). (vi) In [18,19] local fractional methods (local fractional variation iteration method and local fractional Adomian decomposition method, resp.) were applied.

The Polynomial Least Squares Method
Let be the operator associated with the differential equation (1): The error obtained by replacing the exact solution with an approximate solution is given by the so-called remainder: As a consequence, we will search for approximate polynomial solutions of (1)-(2) on the [ , ] interval, solutions which satisfy the following conditions: together with the initial condition (6).
Remark 3. Any -approximate polynomial solution of the problem (1)-(2) is also a weak 2 ⋅ ( − )-approximate polynomial solution, but the opposite is not always true. It follows that the set of weak approximate solutions of the problem (1)-(2) also contains the approximate solutions of the problem. Proof. Taking into account the definition, we will find a weak -polynomial solution of the typẽ where the constants 0 , 1 , . . . , are calculated using the following steps.
(4) Next we compute the values of 0 , . . . , 0 as the values which give the minimum of the functional (12) and the values of 0 0 , 0 1 , . . . , 0 −1 again as functions of 0 , . . . , 0 by using the initial conditions. (5) By using the constants 0 0 , 0 1 , . . . , 0 thus determined, we consider the polynomial: Based on the way the coefficients of polynomial ( ) are computed and taking into account the relations (10)-(13), the following inequality holds: The Scientific World Journal 3 It follows that We obtain From this limit we obtain that ∀ > 0, ∃ 0 ∈ N such that ∀ ∈ N, > 0 . It follows that ( ) is a weakapproximate polynomial solution of the problem (1)-(2).
As a consequence of the second remark, in order to find -approximate polynomial solutions of the problem (1)-(2) by PLSM, we will first determine weak approximate polynomial solutions,̃following the steps 1 to 5 previously described. If | ( ,̃)| < , theñis also an -approximate polynomial solution of the problem.

Application 1: Nonlinear Riccati Equation.
Our first test problem is the following Cauchy problem: The exact solution of this problem is ( ) = 2 − − 2.
In [4], by using a numerical approximation based on the Bessel functions of the first kind, Yüzbaşi computed the following approximate solution of (17): The maximum absolute error of this approximation is reported as 1.2171 ⋅ 10 −19 .
Using the steps described in the previous section we performed the following computations.
(i) We computed a polynomial solution of the form (ii) Taking into account the fact that, by using the initial condition, 0 must be equal to −2, the functional (12) corresponding to the problem (17) is (iii) To find the minimum of this functional we compute the stationary points as the solutions of the system Since the only stationary point is 1 = −1, 2 = 1 and it is easy to show that this point is indeed a minimum, we obtain the following polynomial approximate solution of (17): which is actually the exact solution of the problem.
We remark that while in this simple case we were able to compute the exact minimum of the functional (12) in most of the applications the direct computation of the minimum is not possible and some numerical techniques are employed.

Application 2: Second-Order Neutral Functional-Differential Equation with
Proportional Delays. Our second test problem is the following Cauchy problem: The exact solution of this problem is ( ) = 2 . In [5], Bellen and Zennaro used a two-stage order-one Runge-Kutta method to compute a numerical solution of the problem (23) and the absolute error of their approximation is of the order 10 −3 . In [6,7], Wang et al. used the one-leg -method to compute approximate solutions of (23) and the absolute error of their best approximation is of the order 10 −3 . In [8], Chen and Wang used the variational iteration method to compute approximate solutions of (23) and the absolute error of their best approximation is of the order 10 −6 . In [9], Biazar and Ghanbari used the homotopy perturbation method (HPM) to compute approximate solutions of (23) and the absolute error of their best approximation is of the order 10 −6 . In [10], Sedaghat et al. used a method based on Chebyshev polynomials to compute a numerical solution of the problem (23) and the absolute error of their approximation is of the order 10 −17 .
Using our method we performed the following computations.
(i) We compute a polynomial solution of the form (ii) Taking into account the initial conditions we obtain the following values of the constants 0 = 0, 1 = 0.

4
The Scientific World Journal In this case the functional (12) corresponding to the problem (23) is (iii) To find the minimum of this functional we compute the stationary points as the solutions of the equation ( 2 ) = 0. The only stationary point is 2 = 1 and it is easy to show that this point is indeed a minimum.
(iv) We obtain the following polynomial approximate solution of (23): Again we obtained the exact solution of the problem.
(28) Table 1 presents the comparison between the absolute errors (as the difference in absolute value between the approximate solution and the exact solution) corresponding to the approximate solution HPM from [11], to the approximate solution VIM from [12], and to our approximate solution PLSM . The absolute errors corresponding to the approximate solution VIM are not explicitly given in [12], but we extracted some approximate values from a figure included in [12] (namely, Figure 2(b)).
It is easy to see that the approximate solution given by PLSM is much closer to the exact solution than the previous ones from [11,12]. We mention the fact that our solution not only is more precise but also, at the same time, has a much simpler form.

Application 4: Generalized Pantograph-Type Equation.
Our third test problem is the following Cauchy problem: The exact solution of this problem is ( ) = − . In [13], Doha et al. used the Jacobi rational-Gauss collocation method (JRC) to compute approximate solutions JRC of (29). In [14], Sezer and Akyuz-Dascioglu used the Taylor series method (TM) to compute approximate solutions TM of (29). In [15], Ozturk and Gulsu used a Chebyshev method (CM) to compute approximate solutions CM of (29). In [16], Yalçinbaş et al. used a Hermite collocation method (HCM) to compute approximate solutions HCM of the same equation.
Using our method we obtained the following polynomial approximate solution of (29): − + 1.
(30) Table 2 presents the comparison between the absolute errors (as the difference in absolute value between the approximate solution and the exact solution) corresponding to the approximate solution TM from [14], to the approximate solution CM from [15], to the approximate solution HCM from [16], and to the (best) approximate solution JRC from [13], as given in [13], together with the errors corresponding to our approximate solution PLSM .
The Scientific World Journal 5  Again it is easy to see that the approximate solution given by PLSM is much closer to the exact solution than the previous ones.

Application 5: Nonlinear Time-Delay Model in Biology.
Our next test problem is The exact solution of this problem is not known.
In [17], Dehghan and Salehi used the variational iteration method (VIM) and the Adomian decomposition method (ADM) to compute approximate solutions VIM and ADM of (31).
Using our method we obtained the following polynomial approximate solution of ( (32) Table 3 presents the comparison between the absolute errors (as the difference in absolute value between the approximate solution and the numerical solution presented in [17]) corresponding to the approximate solutions ADM and VIM from [17] and to our approximate solution PLSM .
The approximate solution given by PLSM is closer to the numerical solution than the previous ones from [17].
The exact solution of this problem is not known. In [20], Berezansky and Braverman studied the existence of positive solutions for equations of the type In the case of (33) it was shown that there exists indeed such a positive solution, but the solution was not effectively computed. Using our method on the interval [0, 1] we obtained the following polynomial approximate solution of (33): PLSM = −7.0502 ⋅ 10 −6 ⋅ 5 − 0.0000361911 ⋅ 4 + 0.000175863 ⋅ 3 + 0.000388792 ⋅ 2 − 0.00895158 ⋅ + 0.1.
The error obtained by replacing the approximate solution back in the equation and computing the remainder is of the order 10 −7 .
It is easy to see that the solution is positive on the interval, where the computation was performed.

Conclusions
The polynomial least squares method (PLSM) was presented as a straightforward and efficient method to compute approximate polynomial solutions for nonlinear delay differential equations.
The applications presented clearly illustrate the accuracy of the method. Indeed, for the equations of the type (1) considered, namely, (17)-(33), the solutions obtained by using PLSM are more precise than the ones previously computed by using other methods. Moreover, for some problems PLSM was able to compute the exact solution while the other methods only produced approximate ones.