Research Article  Open Access
Analytical Approximate Solutions for a General Class of Nonlinear Delay Differential Equations
Abstract
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 timedelay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.
1. 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.
In the present paper we used the polynomial least squares method (PLSM) to compute approximate solutions for the following class of nonlinear delay differential equations: together with the initial conditions Here is a function which satisfies such conditions as necessary to ensure that the problem (1)(2) admits a unique solution, and are real constants, and the functions , , are polynomial functions in the variable, .
Among the methods recently used to compute approximate solutions for various delay differential equations of (1) type we mention the following.(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 twostage orderone RungeKutta method was applied to (1) of the neutralfunction differential type. The same equation was studied in [6, 7] by using the oneleg 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 rationalGauss collocation method was applied to (1) of the generalized pantograph type. The same equation was studied in [14] by using the 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 timedelay 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.
In the next section we introduce the polynomial least squares method (PLSM), which allows us to find analytical approximate polynomial solutions for the problem (1)(2), and in the third section we compare our approximate solutions with approximate solutions presented in [4–17].
2. 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 socalled remainder:
As a consequence, we will search for approximate polynomial solutions of (1)(2) on the interval, solutions which satisfy the following conditions:
Definition 1. (a) We call an approximate polynomial solution of the problem (1)(2) an approximate polynomial solution satisfying the relations (5) and (6).
(b) We call a weak approximate polynomial solution of the problem (1)(2) an approximate polynomial solution satisfying the relation
together with the initial condition (6).
Remark 2. Taking into account the way the problem (1)(2) is defined, from the Weierstrass approximation theorem it follows that there exists the sequence of polynomials, , , , satisfying the conditions such that the sequence of polynomials is convergent to the solution of the problem (1)(2); that is, .
Remark 3. Any approximate polynomial solution of the problem (1)(2) is also a weak 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.
Theorem 4. The problem (1)(2) admits a sequence of weak approximate polynomial solutions.
Proof. Taking into account the definition, we will find a weak polynomial solution of the type
where the constants are calculated using the following steps.(1)In the first step we substitute the approximate solution (9) in (1) and obtain the remainder:
(2)Next we compute as functions of by using the initial conditions:
(3)We attach to the problem (1)(2) the following real functional:
(4)Next we compute the values of as the values which give the minimum of the functional (12) and the values of again as functions of by using the initial conditions.(5)By using the constants 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:
It follows that
We obtain
From this limit we obtain that , such that , . It follows that is a weak approximate 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 , then is also an approximate polynomial solution of the problem.
3. Applications
3.1. Application 1: Nonlinear Riccati Equation
Our first test problem is the following Cauchy problem:
The exact solution of this problem is .
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 .
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, must be equal to , 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 , 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.
3.2. Application 2: SecondOrder Neutral FunctionalDifferential Equation with Proportional Delays
Our second test problem is the following Cauchy problem:
The exact solution of this problem is .
In [5], Bellen and Zennaro used a twostage orderone RungeKutta method to compute a numerical solution of the problem (23) and the absolute error of their approximation is of the order . In [6, 7], Wang et al. used the oneleg method to compute approximate solutions of (23) and the absolute error of their best approximation is of the order . 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 . 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 . 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 .
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 , . 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 . The only stationary point is 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.
3.3. Application 3: PantographType Nonlinear Equation
Our third test problem is the following Cauchy problem:
The exact solution of this problem is .
In [11], Shakeri and Dehghan used the homotopy perturbation method (HPM) to compute approximate solutions of (27).
In [12], Yildirim et al. used the variational iteration method (VIM) to compute approximate solutions of (27).
Using our method we obtained the following polynomial approximate solution of (27):
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 from [11], to the approximate solution from [12], and to our approximate solution . The absolute errors corresponding to the approximate solution are not explicitly given in [12], but we extracted some approximate values from a figure included in [12] (namely, Figure ).

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.
3.4. Application 4: Generalized PantographType 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 rationalGauss collocation method (JRC) to compute approximate solutions of (29). In [14], Sezer and AkyuzDascioglu used the Taylor series method (TM) to compute approximate solutions of (29). In [15], Ozturk and Gulsu used a Chebyshev method (CM) to compute approximate solutions of (29). In [16], Yalçinbaş et al. used a Hermite collocation method (HCM) to compute approximate solutions of the same equation.
Using our method we obtained the following polynomial approximate solution of (29):
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 from [14], to the approximate solution from [15], to the approximate solution from [16], and to the (best) approximate solution from [13], as given in [13], together with the errors corresponding to our approximate solution .

Again it is easy to see that the approximate solution given by PLSM is much closer to the exact solution than the previous ones.
3.5. Application 5: Nonlinear TimeDelay 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 and of (31).
Using our method we obtained the following polynomial approximate solution of (31):
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 and from [17] and to our approximate solution .

The approximate solution given by PLSM is closer to the numerical solution than the previous ones from [17].
3.6. Application 6: Scalar Differential Equation with Several Delays
Our last test problem is
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 we obtained the following polynomial approximate solution of (33):
The error obtained by replacing the approximate solution back in the equation and computing the remainder is of the order .
It is easy to see that the solution is positive on the interval, where the computation was performed.
4. 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 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
References
 C. Zhang and H. Chen, “Block boundary value methods for delay differential equations,” Applied Numerical Mathematics. An IMACS Journal, vol. 60, no. 9, pp. 915–923, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Li and W. Jiang, “Nonlinear waves in complex oscillator network with delay,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 11, pp. 3226–3237, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 F. Zhang and Y. Zhang, “State estimation of neural networks with both timevarying delays and normbounded parameter uncertainties via a delay decomposition approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 12, pp. 3517–3529, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 Ş. Yüzbaşı, “A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differentialdifference equations,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1691–1705, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations. Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, NY, USA, 2003.
 W. Wang and S. Li, “On the oneleg $\theta $methods for solving nonlinear neutral functional differential equations,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 285–301, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 W. Wang, T. Qin, and S. Li, “Stability of oneleg $\theta $methods for nonlinear neutral differential equations with proportional delay,” Applied Mathematics and Computation, vol. 213, no. 1, pp. 177–183, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 X. Chen and L. Wang, “The variational iteration method for solving a neutral functionaldifferential equation with proportional delays,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2696–2702, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 J. Biazar and B. Ghanbari, “The homotopy perturbation method for solving neutral functionaldifferential equations with proportional delays,” Journal of King Saud University—Science, vol. 24, no. 1, pp. 33–37, 2012. View at: Publisher Site  Google Scholar
 S. Sedaghat, Y. Ordokhani, and M. Dehghan, “Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4815–4830, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 F. Shakeri and M. Dehghan, “Solution of delay differential equations via a homotopy perturbation method,” Mathematical and Computer Modelling, vol. 48, no. 34, pp. 486–498, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 A. Yildirim, H. Koçak, and S. Tutkun, “Reliable analysis for delay differential equations arising in mathematical biology,” Journal of King Saud University—Science, vol. 24, no. 4, pp. 359–365, 2012. View at: Publisher Site  Google Scholar
 E. H. Doha, A. H. Bhrawy, D. Baleanu, and R. M. Hafez, “A new Jacobi rationalGauss collocation method for numerical solution of generalized pantograph equations,” Applied Numerical Mathematics, vol. 77, pp. 43–54, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 M. Sezer and A. AkyuzDascioglu, “A Taylor method for numerical solution of generalized pantograph equations with linear functional argument,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 217–225, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Ozturk and M. Gulsu, “Approximate solution of linear generalized pantograph equations with variable coefficients on ChebyshevGauss grid,” Journal of Advanced Research in Scientific Computing, vol. 4, no. 1, pp. 36–51, 2012. View at: Google Scholar  MathSciNet
 S. Yalçinbaş, M. Aynigül, and M. Sezer, “A collocation method using Hermite polynomials for approximate solution of pantograph equations,” Journal of The Franklin Institute, vol. 348, no. 6, pp. 1128–1139, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 M. Dehghan and R. Salehi, “Solution of a nonlinear timedelay model in biology via semianalytical approaches,” Computer Physics Communications, vol. 181, no. 7, pp. 1255–1265, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 X.J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 2, pp. 625–628, 2013. View at: Publisher Site  Google Scholar
 X.J. Yang, D. Baleanu, and W.P. Zhong, “Approximate solutions for diffusion equations on Cantor spacetime,” Proceedings of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013. View at: Google Scholar  MathSciNet
 L. Berezansky and E. Braverman, “Positive solutions for a scalar differential equation with several delays,” Applied Mathematics Letters, vol. 21, no. 6, pp. 636–640, 2008. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2014 Bogdan Căruntu and Constantin Bota. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.