A TUTORIAL INTRODUCTION TO THE TWO-SCALE FRACTAL CALCULUS AND ITS APPLICATION TO THE FRACTAL ZHIBER–SHABAT OSCILLATOR

Ji-Huan He and Yusry O. El-Dib
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Ji-Huan He: School of Science, Xi’an University of Architecture and Technology, Xi’an, P. R. China†School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, P. R. China‡National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, P. R. China
Yusry O. El-Dib: �Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

FRACTALS (fractals), 2021, vol. 29, issue 08, 1-9

Abstract: In this paper, a tutorial introduction to the two-scale fractal calculus is given. The two-scale fractal derivative is conformable with the traditional differential derivatives. When the fractal dimensions tend to an integer value, its basic properties are discussed, and the fractal Zhiber–Shabat oscillator is used as an example to reveal the basic properties of a fractal differential equation. The two-scale transform is used to convert the nonlinear Zhiber–Shabat oscillator with the fractal derivatives to the traditional model. The homotopy perturbation method has been demonstrated under a suitable transformation of the system containing several exponential nonlinear terms to the famous Helmholtz–Duffing oscillator. Stability behavior is discussed. Several numerical illustrations are also provided to exhibit the integrity of the introduced formulation. It is demonstrated that the proposed formulation is accurate enough for highly nonlinear differential equations containing large nonlinear terms.

Keywords: Zhiber–Shabat Wave Equation; Fractal Nonlinear Oscillator; Homotopy Perturbation Method; He’s Frequency Formula; Stability Behavior (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21502686

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