New Implementation of Reproducing Kernel Method for Solving Functional-Differential Equations

In this paper, we apply the new algorithm of reproducing kernel method to give the approximate solution to some functional-differential equations. The numerical results demonstrate the accuracy of the proposed algorithm.


Introduction
The subject of differential equations is a wide field in pure and applied mathematics, engineering, physics, chemistry, biology, psychology and other fields.All of these disciplines are associated with the properties of differential equations of various types.Pure mathematics discusses existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions.Differential equations have a prominent role in modelling virtually each physical, technical and/or biological processes, from celestial motion, to bridge design, to interactions between neurons.
This paper investigates the approximate solution of the following linear functional-differential equation using new implementation of the reproducing kernel method (RKM): , , , , 0 1 0, 0 1 0, 0 1. y x g x y y y Ay By Cy Dy x ′′ ′ ′′ ′ where g is a function of its variables, A, B, C, D are real constants and unknown function ( ) y x is continuous on the interval [0, 1].These problems arise in many areas of applied mathematics, physics and engineering, such as fluid mechanics, gas dynamics, reaction diffusion process, nuclear physics, chemical reactor theory, geophysics, studies of atomic structures and etc.Several numerical techniques such as finite difference approximation [1], cubic splines [2] [3], B-splines [4], Adomian decomposition method [5], differential transformation method [6] and others [7] [8] have been proposed to obtain approximate solution of these problems by some authors.The application of RKM in linear and nonlinear problems has been developed by many researchers [9]- [12].The RKM has been treated singular linear two-point boundary value problem, singular nonlinear twopoint periodic boundary value problem, nonlinear system of boundary value problem, singular integral equations, nonlinear partial differential equations and etc. in recent years in [13]- [17].
As we know, Gram-Schmidt orthogonalization process is numerically unstable, in addition it may take a lot of time to produce numerical approximation.Here, instead of using orthogonal process, we successfully make use of the basic functions which are obtained by RKM.
This paper is organized as follows.In the next section, two reproducing kernel Hilbert space (RKHS) are introduced.Section 3 is devoted to solve Equation ( 1) by new implementation of RKM.Some numerical examples are presented in Section 4.Last section is a brief conclusion.

Reproducing Kernel Spaces
In this section, we follow the recent work of [18] [19] and present some useful materials.
where δ is the Dirac delta function, therefore the following theorem holds.
x W y R y x = Proof.Applying integration by parts three times, since [ ] x R t is the solution of the following constant linear homogeneous differential equ- ation with 6 order, ( ) with the boundary condition: We know that Equation ( 3) has characteristic equation 6 0 λ = , and the eigenvalue 0 λ = is a root whose multiplicity is 6.Hence, the general solution of Equation ( 2) is ( ) , .
Now, we are ready to calculate the coefficients ( ) , 0, , Then, using Equations ( 4) and ( 6), the unknown coefficients of Equation ( 5) are uniquely obtained.Therefore, [ ] The inner product and the norm in the function space [ ] 1 2 0,1 W are defined as follows: ( ) ( ) ( ) ( ) x t t x r t x t x

The New Implementation of the Method
In this section, we shall give the exact or approximate solution of Equation ( 1   is the adjoint operator of  .
, where the subscript t in the operator  indicates that the operator  applies to the function of t.
Proof.For each fixed [ ]  which means that ( ) .
Usually, a normalized orthogonal system is constructed from by using the Gram-Schmidt algorithm, and then the approximate solution will be obtained by calculating a truncated series based on these functions.However, Gram-Schmidt algorithm has some drawbacks such as numerical instability and high volume of computations.Here, to fix these flaws, we state the following Theorem in which the following notations are used.
a F  It is necessary to mention that here we solve the system .Ψ = a F which obtained without using the Gram- Schmidt algorithm.

Numerical Examples
To illustrate the effectiveness of the proposed method, some numerical examples are considered in this section.The numerical results in Table 1 and Table 2 show that the approximate solution and its derivatives up to second order, converge to the exact solution and its derivatives respectively.The examples are computed by using Maple 18.
Example 1.Consider the following second-order ordinary differential equation with singular coefficients with exact solution ( )

Conclusion
In this paper, it is shown that the RKM without Gram-Schmidt algorithm is quiet efficient and well suited for finding the approximate solutions to some two-order initial value problems.We obtained the numerical solutions, with high accuracy and moderate CPU time.The numerical results obtained here, indicate the high performance of this method for approximating solution of functional-differential equations.

Figure 1 .
Figure 1.The absolute error between exact and approximate solutions for Example 1 with N = 20.

Figure 2 .
Figure 2. The absolute error between exact and approximate solutions for Example 2 with N = 20.
) in the reproducing kernel space

Table 1 .
Absolute errors for Example 1 with CPU time 46.582 in seconds.

Table 1 ,
Table 2 and Figure 1, Figure 2. The results shown in

Table 1 ,
Table2, indicate that the approximate solution and its derivatives up to second order, converge to the exact solution and its derivatives respectively.

Table 2 .
Absolute errors for Example 2 with CPU time 14.321 in seconds.