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

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Received 10 April 2016; accepted 19 June 2016; published 22 June 2016

1. 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. Differen- tial 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):

(1)

where g is a function of its variables, A, B, C, D are real constants and unknown function 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, geo- physics, studies of atomic structures and etc. Several numerical techniques such as finite difference approxi- mation [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 two- point 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.

2. Reproducing Kernel Spaces

In this section, we follow the recent work of [18] [19] and present some useful materials.

Definition 1. For a nonempty set, let be a Hilbert space of real-valued functions on some set. A function is said to be the reproducing kernel of if and only if

1.,

2., , (reproducing property).

Also, a Hilbert space of functions that possesses a reproducing kernel k is a RKHS and we de-

note it by. In the following we often denote by k_{x} the function.

Definition 2. is an absolute continuous real-valued function on the interval

. The inner product and the norm in the function

space are defined as follows:

Suppose that function satisfies the following generalized differential equations

(2)

where is the Dirac delta function, therefore the following theorem holds.

Theorem 1 Under the assumptions of Equation (2), Hilbert space is a RKHS with the reproducing kernel function namely for any and each fixed

Proof. Applying integration by parts three times, since, we have

Therefore, Equation (2) implies that

While, function is the solution of the following constant linear homogeneous differential equ- ation with 6 order,

(3)

with the boundary condition:

(4)

We know that Equation (3) has characteristic equation, and the eigenvalue is a root whose multiplicity is 6. Hence, the general solution of Equation (2) is

(5)

Now, we are ready to calculate the coefficients and. Since

we have

(6)

Then, using Equations (4) and (6), the unknown coefficients of Equation (5) are uniquely obtained. Therefore,

Definition 3. { is an absolute continuous real-valued function on the interval, }. The inner product and the norm in the function space are defined as follows:

Theorem 2 Hilbert space is a reproducing kernel space with the reproducing kernel function

Theorem 3 (see [20] ) Let be a dense subset of interval then is a basis of.

3. The New Implementation of the Method

In this section, we shall give the exact or approximate solution of Equation (1) in the reproducing kernel space. We suppose that Equation (1) has a unique solution. We consider Equation (1) as

(7)

where and such that f is an analytical function. It

is clear that is the bounded linear operator of into. We shall give the representation of

analytical solution of Equation (7) in the space. Put and

where is the reproducing kernel of and is the adjoint operator of.

Theorem 4 Let be a dense subset of interval, then is a complete system of

and, where the subscript t in the operator indicates that the operator applies to the function of t.

Proof. For each fixed, let which means that

Note that is dense on, hence,. Due to the existence of, then. There-

fore, is the complete system of. Now, note 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.

where And

where

Theorem 5 Suppose that is a linearly independent set in and be a norma-

lized orthogonal system in, such that. If

then

Proof. Suppose that then Now, by truncating N-term of the two series, because of and since so

Due to the linear independence of, therefore

(8)

Equation (7), imply. For we have

Equation (8), imply, hence

It is necessary to mention that here we solve the system which obtained without using the Gram- Schmidt algorithm.

4. 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

(9)

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

where. The exact solution is.

Example 2. We consider the following second-order delay differential equation

(10)

with exact solution.

Remark 1. The RKM is tested on these problems with gird points for. The

numerical results for Example 1 and 2 are listed in Table 1, Table 2 and Figure 1, Figure 2. The results shown in Table 1, Table 2, 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.

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

Figure 2. The absolute error between exact and approximate solutions for Example 2 with N = 20.

5. 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.

Acknowledgements

We thank the Editor and the referees for their comments.

NOTES

^{*}Corresponding author.

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