Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.017010

ARTICLE

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis

1Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, 13132, Jordan

2Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun, 26816, Jordan

3Department of Physics and Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, 11134, Jordan

4Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, 20550, United Arab Emirates

5Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan

*Corresponding Author: Shrideh Al-Omari. Email: shridehalomari@bau.edu.jo

Received: 19 April 2021; Accepted: 27 July 2021

Abstract: In this paper, an efficient multi-step scheme is presented based on reproducing kernel Hilbert space (RKHS) theory for solving ordinary stiff differential systems. The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform form for a rapidly convergent series in the posed Sobolev space. Using the Gram-Schmidt orthogonality process, complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction. Consequently, by applying the standard RKHS method to each subinterval, approximate solutions that converge uniformly to the exact solutions are obtained. For this purpose, several numerical examples are tested to show proposed algorithm’s superiority, simplicity, and efficiency. The gained results indicate that the multi-step RKHS method is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

Keywords: Multi-step approach; reproducing kernel Hilbert space method; stiffness system; error analysis; numerical solution

During studying and modeling many basic physical phenomena, such as chemical kinematics, aerodynamics, electrical circuits, ballistics, control models, and missile guidance, a type of differential equations appears that is difficult to solve through traditional numerical procedures, called differential stiffness system, which was first highlighted in the work of Curtiss and Hirschfelder [1–6]. Mathematical stiffness models reflect the different growth rates and various dynamic processes of the considered physical systems. It arises when some of the solution components decay much more rapidly than other components because they contain the term

subject to the initial conditions

By classical RKHS method to solve this system with 100 nodes, the numerical results will be achieved under the rapid increase of the error as shown in Fig. 1. Simultaneously, increasing the number of nodes leads to the need for large computer memory along with long operating times, and cumulative errors may also affect the accuracy of the solutions. However, to overcome such drawbacks, an advanced numerical algorithm will be formulated based on dividing any temporal interval into small subintervals and then applying the standard RKHS method on each subinterval. This technique is called a multi-step reproducing Hilbert space (MS-RKHS) method. It is worth noting here that there are advanced methods in the literature to address many different engineering and physical problems, for more detail we refer to [25–30] and references therein.

By and large, there are no conventional analytical or semi-approximate methods that produce precise approximate or closed-form solutions for stiff differential systems. Therefore, there has become an urgent need for effective numerical algorithms to find accurate solutions for such models especially for large periods of time. This gives us the incentive, in this work, to explore effective accurate solutions [31–39]. Motivated by the previous discussion, our study aims to design a novel iterative algorithm to generate an analytical solution to stiff models of ordinary differential equations over a large duration through the use of a multi-step technique. To begin with, two reproducing kernel functions are established to generate a complete orthonormal basis in the Hilbert space. Based on reproducing kernel property, linear, bounded, and invertible differential operator is defined to create an analytical solution of the proposed model over a dense partition of the time period. In this direction, the approximate solution converges uniformly to the analytical solution. Error analysis is discussed as well. Lastly, some numerical examples are presented to illustrate the reliability and efficiency of the suggested multi-step approach. This paper is organized in five sections including the introduction. In Section 2, a brief description of the RKHS method is given. In Section 3, the MS-RKHS method is presented. In Section 4, several examples are given. Finally, a short conclusion is presented in Section 5.

2 Reproducing Kernel Hilbert Space Method

A reproducing kernel space is a Hilbert space

1.

2.

where K is the reproducing kernel function of

The reproducing kernel function possesses many nice properties, including it being unique, positive definite, conjugate symmetric. For the theory and applications of RKHS, we refer to [40–45].

Definition 2.1 [46]: The function space

The inner product for

Theorem 1.1 [46]: The space

Definition 2.2 [47]: The function space

with inner product for

Theorem 2.2 [47]: The space

Hereinafter, we consider the following stiff system:

along with the following initial conditions (ICs)

To solve system (3)–(4) using the RKHS method, we first homogenize ICs (4) in light of the following transformation:

which leads to the following system:

Subsequently, we define the differential operator

Herein, we have to construct an orthogonal function system of the space

Next, we will use the Gram-Schmidt orthogonalization process on

Let

The following theorems give the form of the solution of system (6).

Theorem 2.3: If

The N-term approximate solution

Eventually, the solution of system (3) is obtained as

3 Multi-Step Reproducing Kernel Hilbert Space Method

To clarify the MS-RKHS method that used in this work to find approximate solutions of system (3), we divide the interval

to get the approximate solution:

Second, we apply the standard RKHS method again to the problem:

to get the approximate solution:

Continuing this process at each subinterval and applying the standard RKHS method to the problems:

to get the approximate solutions:

So, the solution of system (3) using the MS-RKHS method is given by

In this section, we will apply the MS-RKHS method described in Section 3 to solve some linear and nonlinear examples of stiff systems. In each example, we compare the exact solution with the approximate one when N = 100. The results are given in tables and graphs. Computations will be performed via Mathematica 10.0 software package.

Example 4.1: Consider the following homogeneous linear stiff system:

subject to the initial conditions

The exact solution of system (12) is

Example 4.2: Consider the following nonhomogeneous linear stiff system:

subject to the initial conditions

The exact solution of system (13) is

Example 4.3: Consider the following nonlinear stiff system:

subject to the initial conditions

The exact solution of system (14) is

Example 4.4: Consider the following stiff system:

subject to the initial conditions

The exact solution of system (15) is given as

In the following, some numerical and graphical simulation of Example 4.4 are performed in Table 6 and Fig. 6 for

In this work, a modified multi-step algorithm, the MS-RKHS method, has been lucratively implemented based on the standard RKHS method to obtain approximate solutions of stiff systems of ordinary differential equations. Several examples of linear and non-linear stiff systems have been given to show the efficiency of the proposed method. The achieved results have been presented numerically and graphically as well. By comparing our results with the exact solutions and classical RKHS method, we observe that the MS-RKHS method yields accurate approximations. Moreover, using the MS-RKHS method, the intervals of convergence for the series solution will increase without needing large computer memory, which takes less time to give accurate numerical results. For the near future work, the presented multi-step approach will be applied for solving stiff systems of fractional and partial differential equations along with nonclassical conditions.

Acknowledgement: Authors would like to thank reviewers for their insightful comments and suggestions. The first author is grateful for the support provided by Zarqa University, Jordan.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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