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

被引:114
作者
He, Ji-Huan [1 ,2 ,3 ]
El-Dib, Yusry O. [4 ]
机构
[1] Xian Univ Architecture & Technol, Sch Sci, Xian, Peoples R China
[2] Henan Polytech Univ, Sch Math & Informat Sci, Jiaozuo, Henan, Peoples R China
[3] Soochow Univ, Coll Text & Clothing Engn, Natl Engn Lab Modern Silk, 199 Ren Ai Rd, Suzhou, Peoples R China
[4] Ain Shams Univ, Fac Educ, Dept Math, Cairo, Egypt
关键词
Zhiber-Shabat Wave Equation; Fractal Nonlinear Oscillator; Homotopy Perturbation Method; He's Frequency Formula; Stability Behavior; TRAVELING-WAVE SOLUTIONS; MODEL;
D O I
10.1142/S0218348X21502686
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
引用
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页数:9
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