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 time-delay 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 real-life 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: 𝐹𝑥^(𝑛)ℎ_𝑛(𝑡),𝑥^(𝑛−1)ℎ_𝑛−1(𝑡),𝑥^(𝑛−2)ℎ_𝑛−2(𝑡),…,𝑥⁽¹⁾ℎ₁(𝑡),𝑥ℎ₀(𝑡),𝑡=0, (1) together with the initial conditions ^𝑛−1_𝑖=0𝑟_𝑖𝑗𝑥^(𝑖)(𝛼)=𝑠_𝑗,𝑗∈1,2,…,𝑛. (2) Here 𝐹 is a function which satisfies such conditions as necessary to ensure that the problem (1)-(2) admits a unique solution, 𝑟_𝑖𝑗 and 𝑠_𝑗 are real constants, and the functions ℎ_𝑘(𝑡), 𝑘∈0,1,…,𝑛, are polynomial functions in the 𝑡 variable, 𝑡∈[𝛼,𝛽].

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 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 𝜃-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 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.

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 𝐷 be the operator associated with the differential equation (1): 𝐷(𝑥)=𝐹𝑥^(𝑛)ℎ_𝑛(𝑡),𝑥^(𝑛−1)ℎ_𝑛−1(𝑡),𝑥^(𝑛−2)ℎ_𝑛−2(𝑡),…,𝑥⁽¹⁾ℎ₁(𝑡),𝑥ℎ₀(𝑡),𝑡. (3)

The error obtained by replacing the exact solution 𝑥 with an approximate solution 𝑥_𝛼 is given by the so-called remainder: 𝑅𝑡,𝑥_𝛼=𝐷𝑥_𝛼(𝑡),𝑡∈𝛼,𝛽. (4)

As a consequence, we will search for approximate polynomial solutions 𝑥_𝛼 of (1)-(2) on the [𝛼,𝛽] interval, solutions which satisfy the following conditions: 𝑅𝑡,𝑥_𝛼<𝜖,𝜖∈ℝ₊,_{𝑛−1𝑖=0}𝑟_𝑖𝑗𝑥^(𝑖)(𝛼)=𝑠_𝑗,𝑗∈1,2,…,𝑛. (5)(6)

Definition 1. (a) We call an 𝜖-approximate polynomial solution of the problem (1)-(2) an approximate polynomial solution 𝑥_𝛼 satisfying the relations (5) and (6).
(b) We call a weak 𝛿-approximate polynomial solution of the problem (1)-(2) an approximate polynomial solution 𝑥_𝛼 satisfying the relation ^𝛽_𝛼𝑅²𝑡,𝑥_𝛼𝑑𝑡≤𝛿,𝛿∈ℝ₊, (7) 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, 𝑃_𝑚(𝑡)=𝛽₀+𝛽₁𝑡+⋯+𝛽_𝑚𝑡^𝑚, 𝛽_𝑖∈ℝ, 𝑖=0,1,…,𝑚, satisfying the conditions ^𝑛−1_𝑖=0𝑟_𝑖𝑗𝑃_𝑚^(𝑖)(𝛼)=𝑠_𝑗,𝑗∈1,2,…,𝑛, (8) such that the sequence of polynomials 𝑃_𝑚(𝑡) is convergent to the solution of the problem (1)-(2); that is, lim_𝑚→∞𝐷(𝑃_𝑚(𝑡))=0.

Remark 3. Any 𝜖-approximate polynomial solution of the problem (1)-(2) is also a weak 𝜖²⋅(𝛽−𝛼)-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 𝜖-polynomial solution of the type 𝑥(𝑡)=^𝑚_𝑘=0𝑐_𝑘⋅𝑡^𝑘,𝑚>𝑛, (9) where the constants 𝑐₀,𝑐₁,…,𝑐_𝑚 are calculated using the following steps.(1)In the first step we substitute the approximate solution (9) in (1) and obtain the remainder: ℜ𝑡,𝑐₀,𝑐₁,…,𝑐_𝑚=𝑅(𝑡,𝑥)=𝐹𝑥^(𝑛)ℎ_𝑛(𝑡),𝑥^(𝑛−1)ℎ_𝑛−1(𝑡),𝑥^(𝑛−2)×ℎ_𝑛−2(𝑡),…,𝑥⁽¹⁾ℎ₁(𝑡),𝑥ℎ₀(𝑡),𝑡. (10)(2)Next we compute 𝑐₀,𝑐₁,…,𝑐_𝑛−1 as functions of 𝑐_𝑛,…,𝑐_𝑚 by using the initial conditions: ^𝑛−1_𝑖=0𝑟_𝑖𝑗𝑥^(𝑖)(𝛼)=𝑠_𝑗,𝑗∈1,2,…,𝑛. (11)(3)We attach to the problem (1)-(2) the following real functional: 𝐽𝑐_𝑛,…,𝑐_𝑚=^𝛽_𝛼ℜ²𝑡,𝑐₀,𝑐₁,…,𝑐_𝑚𝑑𝑡. (12)(4)Next we compute the values of 𝑐⁰_𝑛,…,𝑐⁰_𝑚 as the values which give the minimum of the functional (12) and the values of 𝑐⁰₀,𝑐⁰₁,…,𝑐⁰_𝑛−1 again as functions of 𝑐⁰_𝑛,…,𝑐⁰_𝑚 by using the initial conditions.(5)By using the constants 𝑐⁰₀,𝑐⁰₁,…,𝑐⁰_𝑚 thus determined, we consider the polynomial: 𝑇_𝑚(𝑡)=^𝑚_𝑘=0𝑐⁰_𝑘⋅𝑡^𝑘. (13)
Based on the way the coefficients of polynomial 𝑇_𝑚(𝑡) are computed and taking into account the relations (10)–(13), the following inequality holds: 0≤^𝛽_𝛼𝑅²𝑡,𝑇_𝑚(𝑡)𝑑𝑡≤^𝛽_𝛼𝑅²𝑡,𝑃_𝑚(𝑡)𝑑𝑡,∀𝑚∈ℕ. (14) It follows that 0≤lim_𝑚→∞^𝛽_𝛼𝑅²𝑡,𝑇_𝑚(𝑡)𝑑𝑡≤lim_𝑚→∞^𝛽_𝛼𝑅²𝑡,𝑃_𝑚(𝑡)𝑑𝑡=0. (15) We obtain lim_𝑚→∞^𝛽_𝛼𝑅²𝑡,𝑇_𝑚(𝑡)𝑑𝑡=0. (16) From this limit we obtain that ∀𝜖>0, ∃𝑚₀∈ℕ such that ∀𝑚∈ℕ, 𝑚>𝑚₀. It follows that 𝑇_𝑚(𝑡) is a weak 𝜖-approximate polynomial solution of the problem (1)-(2).

As a consequence of the second remark, in order to find 𝜖-approximate polynomial solutions of the problem (1)-(2) by PLSM, we will first determine weak approximate polynomial solutions, 𝑥_𝛼 following the steps 1 to 5 previously described. If |𝑅(𝑡,𝑥_𝛼)|<𝜖, then 𝑥_𝛼 is also an 𝜖-approximate polynomial solution of the problem.

3. Applications

3.1. Application 1: Nonlinear Riccati Equation

Our first test problem is the following Cauchy problem: 𝑥^′(𝑡−2)−𝑡²⋅𝑥(2⋅𝑡−3)−𝑥²(𝑡−1)+5𝑡⁴−20𝑡³+19𝑡²𝑥(0)−2𝑡−3=0,=−2. (17)

The exact solution of this problem is 𝑥_𝑒(𝑡)=𝑡²−𝑡−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): 𝑥_BES=0.1⋅10⁻¹⁹𝑡³+𝑡²−1.0000000000000000001𝑡−2. (18) 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 𝑥_PLSM=𝑐₀+𝑐₁⋅𝑡+𝑐₂⋅𝑡². (19)(ii)Taking into account the fact that, by using the initial condition, 𝑐₀ must be equal to −2, the functional (12) corresponding to the problem (17) is 𝐽𝑐₁,𝑐₂=𝑐⁴₁5−2𝑐³₁𝑐₂3+6𝑐³₁5+6𝑐²₁𝑐²₂7−134𝑐²₁𝑐₂+21701𝑐²₁−𝑐105¹𝑐³₂2+107𝑐₁𝑐²₂−141049𝑐₁𝑐₂+701963𝑐₁+𝑐210₄₂9−25𝑐³₂9+1206𝑐²₂−3518341𝑐₂+3158543.315 (20)(iii)To find the minimum of this functional we compute the stationary points as the solutions of the system 𝜕𝐽𝜕𝑐₁=0,𝜕𝐽𝜕𝑐₂=0. (21) Since the only stationary point is 𝑐₁=−1, 𝑐₂=1 and it is easy to show that this point is indeed a minimum, we obtain the following polynomial approximate solution of (17): 𝑥_PLSM=−2−𝑡+𝑡², (22) 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: Second-Order Neutral Functional-Differential Equation with Proportional Delays

Our second test problem is the following Cauchy problem: 𝑥^′′(𝑡)−34𝑥(𝑡)𝑡−𝑥2−𝑥^′𝑡2−12𝑥^′′𝑡2+𝑡²𝑥(0)+𝑡−1=0,0≤𝑡≤1,=0,𝑥^′(0)=0. (23)

The exact solution of this problem is 𝑥_𝑒(𝑡)=𝑡².

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 𝜃-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 𝑥_PLSM=𝑐₀+𝑐₁⋅𝑡+𝑐₂⋅𝑡². (24)(ii)Taking into account the initial conditions we obtain the following values of the constants 𝑐₀=0, 𝑐₁=0. In this case the functional (12) corresponding to the problem (23) is 𝐽𝑐₂=11𝑐²₂−3011𝑐₂+1511.30 (25)(iii)To find the minimum of this functional we compute the stationary points as the solutions of the equation 𝐽^′(𝑐₂)=0. The only stationary point is 𝑐₂=1 and it is easy to show that this point is indeed a minimum.(iv)We obtain the following polynomial approximate solution of (23): 𝑥_PLSM=𝑡². (26) Again we obtained the exact solution of the problem.

3.3. Application 3: Pantograph-Type Nonlinear Equation

Our third test problem is the following Cauchy problem: 𝑥^′′(𝑡)−𝑒^−𝑡𝑥^′1𝑡−5−()𝑥(𝑡)𝑡𝑡−1+2𝑥3−𝑥²(𝑡)−−14sin²𝑡3+49𝑡sin3𝑡+sin9()×1−exp−𝑡6𝑡cos3−1−1156𝑡sin2−1−1109cos²𝑡2+14𝑡cos2+23𝑡cos6−13𝑡sin3𝑡cos2−1+𝑡2𝑡sin3−13𝑡cos2(0)=0,3𝑥+6𝑥^′(0)(0)=2,−2𝑥+𝑥^′(0)1=−2. (27)

The exact solution of this problem is 𝑥_𝑒(𝑡)=(1/2)sin(𝑡/3)+(1/3)cos(𝑡/2).

In [11], Shakeri and Dehghan used the homotopy perturbation method (HPM) to compute approximate solutions 𝑥_HPM of (27).

In [12], Yildirim et al. used the variational iteration method (VIM) to compute approximate solutions 𝑥_VIM of (27).

Using our method we obtained the following polynomial approximate solution of (27): 𝑥_PLSM=−3.8337691⋅10⁻⁶⋅𝑡⁵+0.00088928⋅𝑡⁴−0.00309479⋅𝑡³−0.0416659⋅𝑡²+161⋅𝑡+3. (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 𝑥_HPM from [11], to the approximate solution 𝑥_VIM from [12], and to our approximate solution 𝑥_PLSM. The absolute errors corresponding to the approximate solution 𝑥_VIM are not explicitly given in [12], but we extracted some approximate values from a figure included in [12] (namely, Figure 2(b)).

Table 1 

Comparison of VIM, HPM, and PLSM for (27).

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 Pantograph-Type Equation

Our third test problem is the following Cauchy problem: 𝑥^′′′(𝑡)+𝑥(𝑡)+𝑥(𝑡−0.3)−𝑒^−𝑡+0.3,𝑥(0)=1,𝑥^′(0)=−1,𝑥^′′(0)=1,0≤𝑡≤1. (29)

The exact solution of this problem is 𝑥_𝑒(𝑡)=𝑒^−𝑡.

In [13], Doha et al. used the Jacobi rational-Gauss collocation method (JRC) to compute approximate solutions 𝑥_JRC of (29). In [14], Sezer and Akyuz-Dascioglu used the Taylor series method (TM) to compute approximate solutions 𝑥_TM of (29). In [15], Ozturk and Gulsu used a Chebyshev method (CM) to compute approximate solutions 𝑥_CM of (29). In [16], Yalçinbaş et al. used a Hermite collocation method (HCM) to compute approximate solutions 𝑥_HCM of the same equation.

Using our method we obtained the following polynomial approximate solution of (29): 𝑥_PLSM=−1.6853299⋅10⁻⁶⋅𝑡⁹+0.0000227752⋅𝑡⁸−0.000196271⋅𝑡⁷+0.00138757⋅𝑡⁶−0.00833288⋅𝑡⁵+0.0416666⋅𝑡⁴−0.166667⋅𝑡³+𝑡²2−𝑡+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 𝑥_TM from [14], to the approximate solution 𝑥_CM from [15], to the approximate solution 𝑥_HCM from [16], and to the (best) approximate solution 𝑥_JRC from [13], as given in [13], together with the errors corresponding to our approximate solution 𝑥_PLSM.

Table 2 

Comparison of TM, CM, HCM, JRC, and PLSM for (29).

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 Time-Delay Model in Biology

Our next test problem is 𝑥^′(𝑡)(𝑡)𝑥()−2⋅𝑥1−𝑡−0.1𝑥(0)0.5=0,=1. (31)

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 𝑥_VIM and 𝑥_ADM of (31).

Using our method we obtained the following polynomial approximate solution of (31): 𝑥_PLSM=−2.57841⋅𝑡⁷+15.8186⋅𝑡⁶−39.5946⋅𝑡⁵+52.059⋅𝑡⁴−38.6665⋅𝑡³+16.3752⋅𝑡²−3.89839⋅𝑡+1. (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 𝑥_ADM and 𝑥_VIM from [17] and to our approximate solution 𝑥_PLSM.

Table 3 

Comparison of VIM, HPM, and PLSM for (31).

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 𝑥^′(𝑡)+0.2𝜋𝑥(𝑡−𝜋)sin²(𝑡)+0.2𝜋𝑥(𝑡−2𝜋)cos²(𝑡)𝑥(0)=0,=1. (33)

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 𝑥^′(𝑡)+^𝑚_𝑘=0𝛼_𝑘(𝑡)𝑥ℎ_𝑘(𝑡)=0. (34) 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): 𝑥_PLSM=−7.0502⋅10⁻⁶⋅𝑡⁵−0.0000361911⋅𝑡⁴+0.000175863⋅𝑡³+0.000388792⋅𝑡²−0.00895158⋅𝑡+0.1. (35)

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.

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 (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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.