Guest Editors
Prof. JI-Huan He, Soochow University, China
Dr. Muhammad Nadeem, Yibin University, China
Summary
Fractal geometry, two-scale fractal, fractional calculus, discontinuity, fractal MEMS system, fractal vibration system, fractal solitary theory, fractal variational theory.
An engineer is familiar with differential models for complex
problems, but they become inaccurate or even invalid for discontinuous
problems, e.g., porous medium, stochastic diffusion and permeability. To overcome the shortcoming of
the traditional calculus, the fractal theory and fractional calculus have to be adopted.
This special issue focuses on the last development of the
fractal-fractional theory for discontinuous problems for applications in engineering & sciences,
and submissions on the following specific topics are welcome:
1) Fractal theory for
natural phenomena and meta-materials and other artificial materials;
2) Fractal theory for
mechanical and architectural designs;
3) Fractal patterns in
natural phenomena and chaotic structures;
4) Fractal-fractional
differential models for practical problems
5) Physical laws in fractal space or fractal
spacetime;
6) Two-scale economics
and two-scale mathematics;
7) Analytical methods
and Numerical methods for fractal–fractional differential equations
Keywords
Fractal geometry, two-scale fractal, fractional calculus, discontinuity, fractal MEMS system, fractal vibration system, fractal solitary theory, fractal variational theory.
Published Papers
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Open Access
ARTICLE
Modifications of the Optimal Auxiliary Function Method to Fractional Order Fornberg-Whitham Equations
Hakeem Ullah, Mehreen Fiza, Ilyas Khan, Abd Allah A. Mosa, Saeed Islam, Abdullah Mohammed
Computer Modeling in Engineering & Sciences, Vol.136, No.1, pp. 277-291, 2023, DOI:10.32604/cmes.2023.022289
（This article belongs to this Special Issue:
Fractal-Fractional Models for Engineering & Sciences)
Abstract In this paper, we present a new modification of the newly developed semi-analytical method named the Optimal Auxilary Function Method (OAFM) for fractional-order equations using the Caputo operator, which is named FOAFM. The mathematical theory of FOAFM is presented and the effectiveness of this method is proven by using it with well-known Fornberg-Whitham Equations (FWE). The FOAFM results are compared with other method results along with their exact solutions with the help of tables and plots to prove the validity of FOAFM. A rapidly convergent series solution is obtained from FOAFM and is validated by comparison with other results. The…
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Open Access
ARTICLE
Bifurcation Analysis and Bounded Optical Soliton Solutions of the Biswas-Arshed Model
Fahad Sameer Alshammari, Md Fazlul Hoque, Harun-Or-Roshid, Muhammad Nadeem
Computer Modeling in Engineering & Sciences, Vol.135, No.3, pp. 2197-2217, 2023, DOI:10.32604/cmes.2023.022301
（This article belongs to this Special Issue:
Fractal-Fractional Models for Engineering & Sciences)
Abstract We investigate the bounded travelling wave solutions of the Biswas-Arshed model (BAM) including the low group velocity dispersion and excluding the self-phase modulation. We integrate the nonlinear structure of the model to obtain bounded optical solitons which pass through the optical fibers in the non-Kerr media. The bifurcation technique of the dynamical system is used to achieve the parameter bifurcation sets and split the parameter space into various areas which correspond to different phase portraits. All bounded optical solitons and bounded periodic wave solutions are identified and derived conforming to each region of these phase portraits. We also apply the…
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Open Access
ARTICLE
Novel Analysis of Two Kinds Hybrid Models in Ferro Martial Inserting Variable Lorentz Force Past a Heated Disk: An Implementation of Finite Element Method
Enran Hou, Umar Nazir, Samaira Naz, Muhammad Sohail, Muhammad Nadeem, Jung Rye Lee, Choonkil Park, Ahmed M. Galal
Computer Modeling in Engineering & Sciences, Vol.135, No.2, pp. 1393-1411, 2023, DOI:10.32604/cmes.2022.022500
（This article belongs to this Special Issue:
Fractal-Fractional Models for Engineering & Sciences)
Abstract In this article, the rheology of Ferro-fluid over an axisymmetric heated disc with a variable magnetic field by considering the dispersion of hybrid nanoparticles is considered. The flow is assumed to be produced by the stretching of a rotating heated disc. The contribution of variable thermophysical properties is taken to explore the momentum, mass and thermal transportation. The concept of boundary layer mechanism is engaged to reduce the complex problem into a simpler one in the form of coupled partial differential equations system. The complex coupled PDEs are converted into highly nonlinear coupled ordinary differential equations system (ODEs) and the…
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Open Access
ARTICLE
Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with Caputo Fractional Derivative
Dandan Dai, Xiaoyu Li, Zhiyuan Li, Wei Zhang, Yulan Wang
Computer Modeling in Engineering & Sciences, Vol.135, No.2, pp. 1371-1392, 2023, DOI:10.32604/cmes.2022.022323
（This article belongs to this Special Issue:
Fractal-Fractional Models for Engineering & Sciences)
Abstract Although some numerical methods of the fractional-order chaotic systems have been announced, high-precision numerical methods have always been the direction that researchers strive to pursue. Based on this problem, this paper introduces a high-precision numerical approach. Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method. We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies. We investigate the influence of , , on the numerical solution of fractional-order Lorenz chaotic systems. The simulation results of integer order are in…
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Open Access
ARTICLE
Cushioning Performance of Hilbert Fractal Sandwich Packaging Structures under Quasi-Static Compressions
Xingye Xu, Haiyan Song, Lijun Wang
Computer Modeling in Engineering & Sciences, Vol.135, No.1, pp. 275-292, 2023, DOI:10.32604/cmes.2022.022637
（This article belongs to this Special Issue:
Fractal-Fractional Models for Engineering & Sciences)
Abstract The sandwich structure of cushioning packaging has an important influence on the cushioning performance. Mathematical fractal theory is an important graphic expression. Based on Hilbert fractal theory, a new sandwich structure was designed. The generation mechanism and recurrence formula of the Hilbert fractal were expressed by Lin’s language, and the second-order Hilbert sandwich structure was constructed from thermoplastic polyurethane. The constitutive model of the hyperelastic body was established by using the finite element method. With the unit mass energy absorption as the optimization goal, the fractal sandwich structure was optimized, and the best result was obtained when the order was…
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Open Access
ARTICLE
The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique
Hassan Khan, Poom Kumam, Asif Nawaz, Qasim Khan, Shahbaz Khan
Computer Modeling in Engineering & Sciences, Vol.135, No.1, pp. 259-273, 2023, DOI:10.32604/cmes.2022.021332
（This article belongs to this Special Issue:
Fractal-Fractional Models for Engineering & Sciences)
Abstract In the last few decades, it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences. For this reason, fractional partial differential equations (FPDEs) are of more importance to model the different physical processes in nature more accurately. Therefore, the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution. In the current work, the idea of fractional calculus has been used, and fractional Fornberg Whitham equation (FFWE) is represented in its fractional view analysis. A well-known method…
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