Abstract
We applied a new analytic approximate technique, optimal homotopy asymptotic method (OHAM), for treatment of differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Volterra equation in different form. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier, and explicit.
1. Introduction
In physical science the DDEs play a vital role in modeling of the complex physical phenomena. The DDEs models are used in vibration of particles in lattices, the flow of current in a network, and the pulses in biological chains. The models containing DDEs have been investigated by numerical techniques such as discretizations in solid state physics and quantum mechanics. In the last decades most of the research work has been done on DDEs. Levi and Yamilov [1, 2] attribute their work to classification of DDEs and connection of integrable partial differential equation (PDEs) and DDEs [3, 4]. In [5–8] the exact solutions of the DDEs have been studied. For the solution of the DDEs, Zou et al. extended the homotopy analysis method (HAM) [9, 10]. Wang et al. extended Adomian decomposition method (ADM) for solving nonlinear difference-differential equations (NDDEs) and got a good accuracy with the analytic solution [11]. Recently, Marinca et al. introduced optimal homotopy asymptotic method (OHAM) [12–16] for the solution of nonlinear problems. The validity of OHAM is independent of whether or not the nonlinear problems contain small parameters.
The motivation of this paper is to extend OHAM for the solution of NDDEs. In [17–20] OHAM has been proved to be useful for obtaining an approximate solution of nonlinear boundary value problems of differential equations. In this work, we have proved that OHAM is also useful and reliable for the solution of NDDEs, hence showing its validity and great potential for the solution of NDDEs phenomenon in science and engineering.
In the succeeding section, the basic idea of OHAM [12–16] is formulated for the solution of boundary value problems. In Section 3, the effectiveness of the modified formulation of OHAM for NDDEs has been studied.
2. Basic Mathematical Theory of OHAM for NDDE
Let us take OHAM to the following differential-difference equation: with boundary conditions: where is a differential operator, is an unknown function, and denote spatial and temporal independent variables, respectively, is the boundary of , and is a known analytic function. can be divided into two parts and such that where is simpler part of the partial differential equation which is easier to solve and contains the remaining part of . So (1) can be written as According to OHAM, one can construct an optimal homotopy satisfying where is an embedding parameter, is an unknown function, and is a nonzero auxiliary function. The auxiliary function is nonzero for and . Equation (5) is the structure of OHAM homotopy. Clearly we have Obviously, when and , we obtain respectively. Thus, as varies from to , the solution varies from to , where is the zeroth order solution and can be obtained from (10).
Expanding , , by Taylors series and choosing as given where , are constants and to be determined and .
Now substituting (9) into (5) and equating the coefficient of like powers of , we obtain We obtained the zeroth order solution and first, second, third, and the general order solutions by solving (10)–(14). In the same way the remaining solutions can be determined. It has been observed that the convergence of the series (9) depends on the auxiliary constants . If it is convergent at , one has Substituting (15) into (1), it results in the following expression for residual: If , then is the exact solution of the problem. Generally it does not happen, especially in nonlinear problems.
For the determinations of auxiliary constants, , , there are different methods like Galerkin’s method, Ritz method, least squares method, and collocation method. One can apply the method of least squares as follows: where and are two values, depending on the nature of the given problem.
The auxiliary constants can be optimally calculated as The order approximate solution can be obtained by these obtained constants.
The convergence of OHAM is directly proportional to the number of optimal constants .
3. Application of OHAM to Difference-Differential Equations
To show the validity and effectiveness of OHAM formulation to difference-differential equation, we apply it to Voltera equations in different form.
Model 1. We consider Voltera equation of the form
with initial conditions
The exact solution of (3.1) is given by
and we take
According to (1), we have
Using (19), (20), and (22) into (5) and using the method discussed in Section 2 lead to the following.
Zeroth Order Problem. Consider
with
from which we obtain
First Order Problem. One has
with
Its solution is
Second Order Problem. Consider
with
Third Order Problem. One has
with
Adding (26), (29), (32), and (35), we obtain
For the computation of the constants , , and applying the method of least square mentioned in (16)–(18), we get
Putting these values in (37), we obtained the approximate solution of the form
The exact solution of the problem (3.1) is [11]
The Adomian solution with Padé approximation of (3.1) is given by [11]
Model 2. Now consider another form of Volterra equation
with initial conditions
The exact solution of (42) is given by
We take
According to (1), we have
Using (42), (43), and (45) into (5) and using the method discussed in Section 2 lead to the following.
Zeroth Order Problem. Consider
with
from which we obtain
First Order Problem. One has the following:
with
Its solution is
Second Order Problem. Consider
with
Third Order Problem. One has
with
Adding (49), (52), (55), and (58), we obtain
For the computation of the constants , , and applying the method of least square mentioned in (16)–(18), we get
The exact solution of problem (42) is [9]
The HAM solution of (42) is given by [9]
4. Results and Discussions
The formulation presented in Section 2 provides highly accurate solutions for the problems demonstrated in Section 3. We have used Mathematica 7 for most of our computational work. Table 1 and Figure 1 give the comparisons of OHAM with ADM and exact solutions for Model 1. Also the absolute errors at different values of at are given (Figure 4). Figure 3 shows the comparisons of OHAM results with HAM, exact at different values of at for Model 2. The convergence of approximate orders to exact solutions is given in Table 2 for Model 1 and Figure 5 for Model 2. The residuals have been plotted for Model 1 in Figure 2 and for Model 2 in Figure 5. We have concluded that the results obtained by OHAM are identical to the results obtained by ADM, HAM, and exact. OHAM converges rapidly with increasing of the order of approximation.
5. Conclusion
In this work, we have seen the effectiveness of OHAM [12–16] to differential-difference equations. By applying the basic idea of OHAM to differential-difference equations, we found it simpler in applicability, more convenient to control convergence, and involved less computational overhead. Therefore, OHAM shows its validity and great potential for the differential-difference equations arising in science and engineering.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are thankful to the anonymous reviewers for their constructive and valuable suggestions.