Abstract
This paper deals with the singularly perturbed delay differential equations under boundary conditions. A numerical approximation based on the exponential functions is proposed to solve the singularly perturbed delay differential equations. By aid of the collocation points and the matrix operations, the suggested scheme converts singularly perturbed problem into a matrix equation, and this matrix equation corresponds to a system of linear algebraic equations. Also, an error analysis technique based on the residual function is introduced for the method. Four examples are considered to demonstrate the performance of the proposed scheme, and the results are discussed.
1. Introduction
The mathematical models of many practical phenomena in many areas of sciences often result in boundary-value problems of singularly perturbed delay differential equations, for example, the study of bistable devices [1], description of the human pupil-light reflex [2], a variety of models for physiological processes or diseases [3, 4], evolutionary biology [4], variational problems in control theory [5, 6], and so forth. These problems mainly depend on a small positive parameter and a delay parameter in such a way that the solution varies rapidly in some parts of the domain and varies slowly in some other parts of the domain. Also, this class of problems possesses boundary layers, that is, regions of rapid change in the solution near one of the boundary points.
In the recent years, many researchers have a great interest in singularly perturbed delay differential equation problems. For example, for these problems, Patidar and Sharma [7] have studied nonstandard finite difference methods, Kadalbajoo and Ramesh [8] have applied the hybrid method, Kadalbajoo and Sharma [9] have presented a numerical study, Rai and Sharma [10] have worked on a numerical scheme based fitted operator methods, Kadalbajoo and Ramesh [11] have studied the finite difference scheme, the hybrid method, and the fitted mesh methods, Amiraliyev and Erdogan [12] have presented a numerical study based on finite difference scheme and piecewise-uniform mesh, Kadalbajoo and Kumar [13] have applied the fitted mesh B-spline collocation method, and Kadalbajoo and Sharma [14] have presented a numerical study involved which finite difference scheme. In addition, Rai and Sharma [15] have solved the singularly perturbed differential difference equation arising in the modeling of neuronal variability by using fitted operator scheme, Lange and Miura [16] have given the singular perturbation analysis of boundary-value problems for differential difference equations and boundary-value problems, and Kadalbajoo and Sharma [17, 18] have studied the numerical solutions of singularly perturbed delay differential equations by various methods.
On the other hand, exponential polynomials or exponential functions have interesting applications in many optical and quantum electronics [19], some nonlinear phenomena modeled by partial differential equations [20], many statistical discussions (especially in data analysis) [21], the safety analysis of control synthesis [22], the problem of expressing mean-periodic functions [23], and the study of spectral synthesis [24, 25]. These polynomials are based on the exponential base set .
Recently, Yüzbaşı and Sezer have studied the exponential polynomial solutions of the systems of linear differential equations in [26].
In this study, we consider the singularly perturbed delay differential equation with the boundary conditions where is a small positive parameter (), is a small shifting parameter , and are given constants, is an unknown function, and and are the known functions defined on interval .
The aim of this paper is to give an approximate solution of the problems (1)-(2) in the form where the exponential basis set is defined by and , are unknown coefficients.
To find a solution in the form (3) of (1) under the conditions (2), we will use the equally spaced collocation points
2. Matrix Relations for Exponential Functions
In this section, we construct the matrix relations related to the exponential solution (3). Note that these relations will be used in Section 3.
Firstly, the approximate solution defined by (3) of (1) can be written in the matrix form where Also, the relation between and its first derivative is given by and the relation between and its second derivative is in the form where By placing the relation (7) into first derivative of (5), we have Similarly, from the relations (5) and (8), we obtain the matrix form By writing in (10), we get the relation The relation between and is as follows: where By substituting (13) into (12), we have the matrix form
3. Exponential Collocation Method
In this section, to compute the unknown coefficients in the approximate solution (3), we use the following procedure by using the matrix relations in Section 2.
Firstly, let us substitute the matrix relations (5), (11), (12), and (15) into (1) as follows: The collocation points defined by (4) are placed into (16), and we have the system of the matrix equations as The system can be written in the matrix form where We note that the matrix equation (18) corresponds to a system of algebraic equations with the unknown coefficients .
Briefly, (18) can be expressed in the form where From the relation (5), the matrix forms of the conditions (2) are written as Briefly, we write the above matrix forms of the conditions as follows: where To obtain the solution of (1) under conditions (2), we replace the row matrices (23) with any two rows of the matrix (20), and thus we have the augmented matrix For simplicity, if the last two rows of the matrix (20) are replaced, the augmented matrix (25) becomes However, we do not have to replace the last rows. For example, if the matrix is singular, then the rows that have the same factor or all zeros are replaced.
If , then the coefficients are uniquely determined by where By substituting the determined coefficients into (3), we obtain the approximate solution On the other hand, when , if , then we may find a particular solution. Otherwise if , then there is not a solution.
4. Error Estimation Based on Residual Function and Improvement of Solution
In this section, we apply the error estimation technique in [27–29] and the residual correction method in [30, 31] for our method and the problems (1)-(2). For our purpose, let us define the residual function for the present method as where denotes the approximate solution (29) of the problems (1)-(2). Therefore, satisfies and the conditions If is the exact solution of the problems (1)-(2), then becomes the error function. By substituting (33) into the problem (1)-(2) and by using (30), we obtain the error differential equation By using (33), the inhomogeneous conditions (2) and (32) are reduced to the homogeneous conditions From (34) and (35), we can clearly write the error problem By solving the problem (36) in the same way as Section 3, the approximation is obtained for .
Consequently, by summing the exponential polynomial solution and the estimated error function , we obtain the corrected exponential solution We note that the errors , () can be estimated by the error function when the exact solution of (1) is unknown.
5. Numerical Examples
In this section, we apply the presented method to some examples. In examples, the terms , , , and , respectively, represent the exact solution, the approximate solution, the corrected approximate solution, and the estimated absolute error function. Also, denotes the estimated maximum error for the values , , , and .
Example 1 (see [13]). Firstly, let us consider the singularly-perturbed delay differential equation with the boundary conditions Firstly, we obtain the approximate solutions for various values of by the presented method in Section 3. Secondly, the approximate solutions are corrected by the residual correction technique for various values of . Hence, the corrected approximate solutions are obtained. In Table 1, we give the estimated maximum absolute errors for various values of , , and . Figures 1(a), 1(b), and 1(c) display the corrected approximate solutions for some values of , , and .
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Example 2 (see [7]). Now, we consider the singularly perturbed delay differential equation with the boundary conditions In Table 2, the estimated maximum absolute errors for various values of , , and are presented. For various values of , , and , Figures 2(a), 2(b), 2(c), and 2(d) show the graphs of the corrected approximate solutions .
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Example 3 (see [9]). Let us consider the boundary-value problem The exact solution of this problem is given by where For some values of , the corrected approximate solutions are compared with the exact solution in Figures 3(a) and 3(b).
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Example 4 (see [14]). Finally, we consider the problem Figure 4(a) displays the corrected approximate solutions for , , and various values of and . The estimated error functions for these approximate solutions are shown in Figure 4(b). It is seen from Figure 4(b) that the absolute errors decrease while value of increases. In Figure 4(c), we show the corrected approximate solutions for , , and different values of . Figure 4(d) shows the estimated error functions for the approximate solutions in Figure 4(c).
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6. Conclusions
In this paper, a numerical scheme based on the exponential functions and the collocation points is presented for the singularly perturbed delay differential equations. Numerical examples are given to demonstrate the applicability and the efficiency of the method. Since the exact solutions of the problems in Examples 1, 2, and 4 are not available, we have computed the estimated maximum absolute errors. For Examples 1 and 2, the maximum absolute errors for some values of , , , and are tabulated in Tables 1 and 2. Also, the approximate solutions for different values of , , , and are compared with the exact solution. It is seen from Figure 3(a) that the accuracies of the approximate solutions increase while and increase. However, when values of and are increased, it is observed from Tables 1 and 2 and Figure 4(c) that the errors usually increase. In addition, the estimated absolute error functions are shown in Figure 4(b) for Example 3. It is seen from Figure 4(b) that the errors decrease as and increase. Moreover, the approximate solutions can be very easily obtained using the software programs.
Acknowledgments
The authors would like to thank the referees for their constructive comments and suggestions to improve the paper. The first author is supported by the Scientific Research Project Administration of Akdeniz University.