Solving Singularly Perturbed Multipantograph Delay Equations Based on the Reproducing Kernel Method

and Applied Analysis 3 0.2 0.4 0.6 0.8 1 x 2 ×10 −6


Introduction
In this paper, we consider the following singularly perturbed multipantograph delay equation: where 0 <  ≪ 1, 0 <   ≤ 1,  is a positive integer and (),   () and () are assumed to be sufficiently smooth, such that (1) has a unique solution with a boundary layer at  = 0.
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. [1,2] proposed a uniform numerical method for dealing with singularly perturbed delay initial value problems.Kadalbajoo et al. [3][4][5][6] presented some effective methods for solving singularly perturbed delay boundary value problems.In [7], Amiraliyev and Cimen also introduced a numerical method for singularly perturbed delay boundary value problems.In [8], Rai and Sharma described a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts.
In this paper, based on the RKM presented in [9,11], an effective numerical method will be presented for solving singularly perturbed delayed boundary value problem (1).
The rest of the paper is organized as follows.In the next section, the numerical technique for (1) is introduced.Error analysis is introduced in Section 2. The numerical example is given in Section 3. Section 4 ends this paper with a brief conclusion.

Numerical Method
We divide the domain [0, 1] into two subdomains, namely, [0, ] and [, 1], where  is a positive real number.The asymptotic approximation technique and the RKM are combined to solve (1) in the regular domain [𝐾𝜀, 1].And then the value of asymptotic approximation in the regular domain is used as the boundary condition at the so-called transition point  = .In the boundary layer domain [0, ], (1) is solved by combining the method of scaling and the RKM.After solving both the regular and boundary layer domain problems their solutions are combined to obtain an approximate solution to the original problem over the entire domain [0, 1].

Solution of the Regular Domain.
We seek the regular region solution as an asymptotic expansion of the form where   () are unknown functions to be determined.Substituting   () into (1) and equating the coefficients of like powers of , we obtain By the RKM, the solutions of the above equations  0 ,  1 ,  2 , . . .,   () can be approximated by where Therefore, the solution of regular region   () can be approximated by In the following, we will show how to solve (3) using the RKM in detail.
Consider the following operator equation: Under the assumption that (3) has a unique solution, we will give the approximate solution of (3) in the reproducing kernel space The inner product and norm in  2 2 [0, 1] are given, respectively, by Its reproducing kernel is Abstract and Applied Analysis For detailed method of obtaining reproducing kernel   (), please refer to [9,15].
=1 is dense on [0, 1] and the solution of (6) is unique, then the solution of (6) is Proof.Applying Theorem 1, it is easy to see that {  ()} ∞ =1 is the complete orthonormal basis of   Note that ((),   ()) = (  ) for each () ∈  1 2 [0, 1]; hence we have and the proof of the theorem is complete.
The approximate solution V  () can be obtained by taking finitely many terms in the series representation of V() and

Numerical Examples
Example 1.Consider the following singular perturbation problem with pantograph delay where () is given such that its exact solution is ()

Conclusion
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.

Table 1 :
Comparison of maximum absolute error with other methods for  = 64.

Table 2 :
Comparison of maximum absolute error with other methods for  = 256.