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

Radwan Abu-Gdairi,Mohammad Al-Smadi,Shaher Momani,Shrideh Al-Omari,Shatha Hasan

doi:10.32604/cmes.2022.017010

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.

Highlights

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]
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
Several examples of linear and non-linear stiff systems have been given to show the efficiency of the proposed method

Summary

Introduction

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]. Increasing the number of nodes leads to the need for large computer memory along with long operating times, and cumulative errors may affect the accuracy of the solutions To overcome such drawbacks, an advanced numerical algorithm will be formulated based on dividing any temporal interval into small subintervals and applying the standard RKHS method on each subinterval. 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. 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.

Reproducing Kernel Hilbert Space Method

Multi-Step Reproducing Kernel Hilbert Space Method

Numerical Examples

Conclusion