Abstract
We extend for the first time the applicability of the optimal homotopy asymptotic method (OHAM) to find the algorithm of approximate analytic solution of delay differential equations (DDEs). The analytical solutions for various examples of linear and nonlinear and system of initial value problems of DDEs are obtained successfully by this method. However, this approach does not depend on small or large parameters in comparison to other perturbation methods. This method provides us with a convenient way to control the convergence of approximation series. The results which are obtained revealed that the proposed method is explicit, effective, and easy to use.
1. Introduction
Delay differential equation (DDE) is a form of differential equations in which derivative of the unknown function in a given time is specified in terms of the values at an earlier point in time.
DDEs have the general form where is the delay function.
Many problems of physics, biological models, control system, and medical and biochemical fields are modelled by DDEs. Recent studies in such diverse fields have shown that DDEs play an important role in explaining many different phenomena. Patel et al. [1] introduced an iterative scheme for the optimal control systems described by DDEs with a quadratic cost functional. In physiology, Glass and Mackey [2] applied time delays to many physiological models. Busenberg and Tang [3] created a model for cell cycle by delay equations. In recent years, DDEs are used to design models as HIV-1 therapy for fighting a virus with another virus [4].
In the last years, a great deal of attention has been devoted to study DDEs. Hence, they are solved by numerical method and approximation approach, such as Adomian decomposition method [5, 6], homotopy perturbation method (HPM) [7, 8], multiquadric approximation scheme [9, 10], variational iteration method (VIM) [8, 11, 12], spline methods [13], homotopy analysis method (HAM) [14], Chebyshev polynomials [15], Galerkin method [16], Legendre wavelet method [17], differential transform method [18], and Runge-Kutta method [19]. Recently, a new approach of homotopy which is called optimal homotopy asymptotic method (OHAM) was proposed and developed by Marinca et al. [20–24] for the approximate solution nonlinear problems of thin film flow of a fourth-grade fluid and for the study of the behavior of nonlinear mechanical vibration of electrical machines. In OHAM, the control and adjustment of the convergence region are provided in a convenient way. Furthermore, the OHAM has been built in convergence criteria similar to those of HAM but with greater degree of flexibility. Islam et al. [25] have applied this method successfully to nonlinear problems and have also shown its effectiveness and accuracy. Idrees et al. [26] used OHAM to study the squeezing flow between two infinite planar plates slowly approaching each other.
The aim of this paper is to apply OHAM to get an approximate analytic solution of DDEs. The capability of this approach is tested upon several examples which offer an approximate solution in a series form that converges to exact solution in few terms. The rest of this paper is organized as follows. In Section 2, we describe the basic idea of OHAM. In Section 3, we provide the convergent theorem for this type of equations. Section 4 presents several examples to demonstrate the efficiency of the framework. The conclusion of this study is presented in Section 5.
2. Description of the Method
In this section, framework of the proposed method is given and represented in the following differential equation: where are the linear operators and are the nonlinear operators contain delay function, is an unknown function, denotes an independent variable, is a known function, and are the delay functions.
According to OHAM, we construct a homotopy which satisfies where , is an embedding parameter, is a nonzero auxiliary function for , and is an unknown function. Obviously, when and it holds that and , respectively. Thus, as varies from to , the solution approach from to , where is the initial guess that satisfies the linear operator and the initial conditions Next, we choose the auxiliary function in the form where are convergence control parameters which can be determined later. can be expressed in another form as reported by Herişanu and Marinca [24].
To get an approximate solution, we expand in Taylor’s series about in the following manner: By substituting (6) into (3) and equating the coefficient of like powers of , we obtain the following linear equations. Define the vectors where and . The zeroth-order problem is given by (4), and the first- and second-order problems are given as The general governing equations for are where and is the coefficient of in the expansion of about the embedding parameter : It has been observed that the convergence of the series (6) depends upon the auxiliary constants . If it is convergent at , one has The result of the th-order approximation is given as Substituting (12) into (2) yields the following residual: If , then will be the exact solution. Generally such a case will not arise for nonlinear problems, but we can minimize the functional where and are the endpoints of the given problem. The unknown convergence control parameters can be calculated from the system of equations It should be noted that our process included the auxiliary function which provides us an easy way to set and optimally control the convergent area and the rate of the solution series.
3. Convergence Theorem
In this section, we introduce the convergence of the solution for DDEs.
Theorem 1. If the series (12) converges to , where is produced by (8) and the -order deformation (10), then is the exact solution of (2).
Proof. Since the series converges, it can be written as and it holds that The left hand-side of (10) satisfies According to (18) we have Using the linear operator , which satisfies Also the right hand side can be written as Now, if the , is properly chosen, then (24) leads to which is the exact solution.
4. Applications
In this section, we will present a few examples with a known analytic solution in order to demonstrate the effectiveness and high precision of this algorithm.
Example 1. Consider the following linear delay differential equation [5]:
with the exact solution
Applying the procedure which is described in Section 2, the linear and nonlinear operators are
where is the expansion Taylor series of with respect to , which can be written as
Now, apply (4) to to give the zeroth-order problem as
The solution of the zeroth-order deformation is
The first-order deformation which is obtained from (8) is given as
and has the solution
The second-order deformation is given by (9):
with initial condition
The solution of (34) is given by
According to (10), the third-order deformation is defined as
with initial condition
and has the solution
By using (31), (33), (36), and (39), the third-order approximate solution by OHAM for is
By using the proposed method of Section 2 on , we use the residual error:
The Less Square error can be formed as
Thus, the following optimal values of ’s are obtained:
In this case, our approximate solution is
Equations (44) and (41) are plotted in Figures 1(a) and 1(b), respectively. Figure 1(a) shows a comparison between the approximate solution which is obtained by using OHAM and exact solution (27). The residual error is plotted in Figure 1(b). We noted that the absolute maximum error for solving this example via HAM is while the absolute maximum error via OHAM is , which leads to conclude that OHAM is more accurate than HAM.
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(b)
Example 2. Consider the linear delay differential equation of third order [5] with exact solution According to the method which was described in the above section, we start with By applying OHAM, we have the following zero-, first-, second-, and the third-order approximate solutions: By adding (48)–(50) and (51), we obtain Following the procedure described in Section 2 regarding the domain between and , we use the residual The following optimal values of ’s are obtained: By substituting values in (52), we have The comparison between the approximate solution and the exact solution is shown in Figures 2(a) and 2(b). We observe that the results agree very well with the exact solution.
(a)
(b)
Example 3. Consider the first order of nonlinear delay differential equation [14]
which has the exact solution
By applying the same method as in Examples 1 and 2, we have the following:
According to OHAM, we have the following zero-, first-, second- and the third-order approximate solutions:
From (59), the third-order approximate solution by OHAM is given as
By using (60) in (14) and applying the method as discussed in (15) and (16), we obtain the following values of ’s:
The approximate solution now becomes
From Figures 3(a) and 3(b), we observe that the results agree very well with the exact solution; as we increase the order of the problem the accuracy increases and the residual error will decrease as shown in Figure 3(b). We observed that the absolute maximum error for solving this example via HAM is while the absolute maximum error by using OHAM is , which revealed that the proposed method is more accurate than HAM.
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(b)
Example 4. Consider the third-order nonlinear delay differential equation [14] The exact solution of the above problem is given as By applying the present method, the linear and nonlinear operators are defined as According to OHAM, we have the following zero-, first-, second- and third-order approximate solutions: From (66), the third-order approximate solution by OHAM is given as By using (67) in (14) and applying the method as discussed in (15) and (16), we obtain the following values of ’s: The approximate solution now becomes Numerical results of the solution are displayed in Figures 4(a) and 4(b).
(a)
(b)
Example 5. Consider the system of delay differential equation [14] with initial conditions Following the same procedure, we have According to OHAM formulation, we have the following: zeroth-order solution: first-order solution: second-order solution: Making use of (73)–(75) and extending the solutions up to a fifth order, the approximate solutions by OHAM for are By using the proposed procedure which is described in Section 2 on , we use the residual error The following values of ’s, ’s and ’s are obtained: By using the above values, the approximate solutions are From Figure 5, we can observe the accuracy of the solution obtained by the five-term approximate solution using OHAM which is quite good.
5. Conclusions
In this work, OHAM is employed for the first time to propose a new analytic approximate solution of delay differential equations (DDEs). This method has been tested in various examples of linear and nonlinear and system of initial value problems of DDEs and was seen to yield satisfactory results. The OHAM provides us with a simple way to optimally control and adjust the convergence solution series and it gives a good approximation in few terms which is converged to the exact solution and proved the efficiency and reliability of the method. This fact is obvious from the use of the auxiliary function . In OHAM, it is important to solve a set of nonlinear algebraic equations with unknown convergence control parameters, , and this makes it time consuming, especially for large .