Abstract
We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay. The Bezier curves method can solve boundary value problems for singularly perturbed differential-difference equations. The approximation process is done in two steps. First we divide the time interval, into subintervals; second we approximate the trajectory and control functions in each subinterval by Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degree and determined Bezier curves on any subinterval by control points. The proposed method is simple and computationally advantageous. Several numerical examples are solved using the presented method; we compared the computed result with exact solution and plotted the graphs of the solution of the problems.
1. Introduction
In recent years, there has been a growing interest in the singularly perturbed delay differential equation (see [1–4]). A singularly perturbed delay differential equation is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and involving at least one delay term. Such types of differential equations arise frequently in applications, for example, the first exit time problem in modeling of the activation of neuronal variability [5], in a variety of models for physiological processes or diseases [6], to describe the human pupil-light reflex [7], and variational problems in control theory and depolarization in Stein’s model [8]. Investigation of boundary value problems for singularly perturbed linear second-order differential-difference equations was initiated by Lange and Miura [5, 9, 10]; they proposed an asymptotic approach in study of linear second-order differential-difference equations in which the highest order derivative is multiplied by small parameters. Kadalbajoo and Sharma [11–14] discussed the numerical methods for solving such type of boundary value problems. Amiraliyev and Erdogan [15] and Amiraliyeva and Amiraliyev [16] developed robust numerical schemes for dealing with singularly perturbed delay differential equation.
In the present work we suggest a technique similar to the one which was used in [17, 18] for solving singularly perturbed differential-difference equation with delay in the following form (see [13]): where is small parameter, , and is also a small shifting parameter, , , , , and are assumed to be smooth, and is a constant. For , the problem is a boundary value problem for a singularly perturbed differential equation and then as the singular perturbation parameter tends to zero, the order of the corresponding reduced problem is decreased by one, so there will be one layer. It may be a boundary layer or an interior layer depending on the nature of the coefficient of the convection term.
The current paper is organized as follows. In Section 2, function approximation will be introduced. Numerical examples will be stated in Section 3. Finally, Section 4 will give a conclusion briefly.
2. Function Approximation
Consider the problem (1). Divide the interval into a set of grid points such that where , , and is a positive integer. Let for . Then, for , the problem (1) can be decomposed to the following suboptimal control problems: where .
We mention that is defined where . Also where denotes the integer part of.
Let where is the characteristic function of for . It is trivial that .
Our strategy is using Bezier curves to approximate the solutions by where is given below. Individual Bezier curves that are defined over the subintervals are joined together to form the Bezier spline curves. For , define the Bezier polynomials of degree that approximate the action of over the interval as follows: where is the Bernstein polynomial of degree over the interval and is the control points (see [17]). By substituting (5) in (3), one may define for as Let where is the characteristic function of for . Beside the boundary conditions on , at each node, we need to impose continuity condition on each successive pair of to guarantee the smoothness. Since the differential equation is of first order, the continuity of (or ) and its first derivative give where is the th derivative with respect to at .
Thus, the vector of control points () must satisfy (see [17])
Ghomanjani et al. [17] proved the convergence of this method where .
Now, the residual function can be defined in as follow: where is the Euclidean norm and is a sufficiently large penalty parameter. Our aim is solving the following problem over : The mathematical programming problem (11) can be solved by many subroutine algorithms. Here, we use Mapleto solve this optimization problem.
3. Numerical Results and Discussion
Consider the following examples which can be solved by using the presented method.
Example 1. First we consider the problem (see [11]) under the boundary conditions A boundary layer exists on left side of the interval. For this problem, the exact solution is where Also, we have plotted the graphs of the exact and computed solution of the problem in Figure 1. The maximum errors are shown in Table 1.
Example 2. Next we consider the problem (see [11]) under the boundary conditions A boundary layer exists on right side of the interval. For this problem, the exact solution is where Also, we have plotted the graphs of the exact and computed solution of the problem in Figure 2. The maximum errors are shown in Table 2.
4. Conclusions
We have described a numerical algorithm for solving BVPs for singularly perturbed differential-difference equation with small shifts. Here, we have discussed both the cases by using Bezier curves, when boundary layer is on the left side and when boundary layer is on the right side of the underlying interval. Numerical examples show that the proposed method is efficient and very easy to use.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the anonymous reviewers for their careful reading, constructive comments, and nice suggestions which have improved the paper very much.