A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions.
In this paper, we consider the following singularly perturbed multipantograph delay equation:
Singularly perturbed problems arise frequently in applications including geophysical fluid dynamics, oceanic and atmospheric circulation, chemical reactions, and optimal control. These problems are characterized by the presence of a small parameter that multiplies the highest order derivative, and they are stiff and there exist boundary layers where the solutions change rapidly.
Functional differential equations with proportional delays are usually referred to as pantograph equations. These equations arise in a variety of applications, such as number theory, electrodynamics, astrophysics, nonlinear dynamical systems, probability theory on algebraic structure, quantum mechanics, and cell growth.
Recently, singularly perturbed delayed differential equations have attracted significant attention. The numerical treatment of such problems presents some major computational difficulties, and therefore discussion on numerical solutions of singularly perturbed delayed differential equation is rare. Amiraliyev et al. [
Reproducing kernel theory has important applications in numerical analysis, differential equations, probability, and statistics, amongst other fields [
In this paper, based on the RKM presented in [
The rest of the paper is organized as follows. In the next section, the numerical technique for (
We divide the domain
We seek the regular region solution as an asymptotic expansion of the form
Substituting
By the RKM, the solutions of the above equations
Therefore, the solution of regular region
In the following, we will show how to solve (
Consider the following operator equation:
The reproducing kernel space
In (
For (
Note here that
For each fixed
If
Applying Theorem
Note that
The approximate solution
Consider
Consider the following singular perturbation problem with pantograph delay
Absolute errors
Absolute errors
Absolute errors
For comparison, we consider the following singular perturbation problem without delay [
Comparison of maximum absolute error with other methods for

[ 
[ 
Present method 









Comparison of maximum absolute error with other methods for

[ 
[ 
Present method 









Comparison of computed solution with [

Exact solution  Present method  [ 
[ 
[ 

0.01  −0.987875  −0.987921  −0.989854  −1.398575  −2.607798 
0.03  −0.967160  −0.967161  −0.969099  −1.603239  −1.940047 
0.05  −0.945600  −0.945602  −0.947500  −1.579587  −1.698810 
0.07  −0.923240  −0.923242  −0.925100  −1.550175  −1.598463 
0.09  −0.900080  −0.900082  −0.901900  −1.519601  −1.544179 
0.10  −0.888200  −0.888202  −0.890000  −1.503912  −1.509999 
0.30  −0.608600  −0.608603  −0.610000  −1.134145  −1.144100 
0.50  −0.249000  −0.249005  −0.250000  −0.657717  −0.664829 
0.70  0.190600  0.190594  0.190000  −0.074626  −0.078895 
0.90  0.710200  0.710193  0.710000  0.615126  0.613702 
1.00  1.000000  1.000000  1.000000  1.000000  1.000000 
In this paper, a new method is proposed for solving singularly perturbed multipantograph delay equations. The present method is based on the RKM, the asymptotic expansion technique, and the method of scaling. The major advantage of the method is that it can produce good globally continuous approximate solutions. The results from the numerical example show that the present method is an accurate and reliable analytical technique for treating singularly perturbed multipantograph delay equations.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments. The work was supported by the NSFC (Grant nos. 11201041 and 11026200), the Special Funds of the National Natural Science Foundation of China (Grant no. 11141003), and Qing Lan Project of Jiangsu Province.