A NOVEL VARIATIONAL PERSPECTIVE TO FRACTAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS

Fractals ◽  
2021 ◽  
Author(s):  
XUE-FENG HAN ◽  
KANG-LE WANG
Keyword(s):  
Wave Equations ◽  
Variational Principles ◽  
Variable Coefficients ◽  
Transform Method ◽  
Numerical Examples ◽  
Fractal Models ◽  
Inverse Transform ◽  
Fractal Calculus ◽  
Different Types ◽  
Wave Models

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.

scholarly journals Frequency formula for a class of fractal vibration system

2022 ◽  
Vol 3 (1) ◽  
pp. 55-61
Author(s):  
Yi Tian ◽  
Keyword(s):  
Variational Principles ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Vibration System ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

scholarly journals Supply Chain Coordination Contracts under Double Sided Disruptions Simultaneously

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Huan Zhang ◽  
Yang Liu ◽  
Jingsi Huang
Keyword(s):  
Supply Chain ◽  
Supply Chain Coordination ◽  
Optimal Price ◽  
Supply Chain System ◽  
Pricing Policy ◽  
Product Cost ◽  
Numerical Examples ◽  
Supply Chain Contracts ◽  
Different Types ◽  
Key Factor

Supply chain coordination models are developed in a two-echelon supply chain with double sided disruptions. In a supply chain system, the supplier may suffer from the product cost disruption and the retailer suffers from the demand disruption simultaneously. The purpose of this study is to design proper supply chain contracts, under which the supply chain with double sided disruption can be coordinated. Firstly, the centralized decision-making models are applied to find the optimal price and quantity under three cases as the baseline. The different cases are divided by the different relationship between the product cost disruption and the demand disruption. Secondly, two different types of contracts are introduced to coordinate the whole supply chain. One is all-unit wholesale quantity discount policy (AQDP) contract, and the other one is capacitated linear pricing policy (CLPP) contract. And it is found out that the gap between the demand disruption and the product cost disruption is the key factor to influence the supply chain coordination. Some numerical examples and sensitivity analysis are given to illustrate the models. The AQDP contracts are listed out under different cases to show how to use it under double sided disruptions.

ANALYTICAL APPROXIMATE SOLUTIONS OF (N + 1)-DIMENSIONAL FRACTAL HARRY DYM EQUATIONS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850094
Author(s):  
JIANSHE SUN
Keyword(s):  
Approximate Solutions ◽  
Travelling Wave ◽  
Cantor Sets ◽  
Travelling Wave Solutions ◽  
Fractional Partial Differential Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractal Models ◽  
Differential Transform ◽  
Harry Dym Equation

The new fractal models of the [Formula: see text]-dimensional and [Formula: see text]-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of [Formula: see text]-dimensional variables is generalized and the theorems of [Formula: see text]-dimensional LFRDTM are supplementary extended. The travelling wave solutions of the fractal HDE show that the proposed LFRDTM is effective and simple for obtaining approximate solutions of nonlinear local fractional partial differential equations.

scholarly journals Variational theory for (2+1)-dimensional fractional dispersive long wave equations

2021 ◽  
pp. 23-23
Author(s):  
Xiao-Qun Cao ◽  
Cheng-Zhuo Zhang ◽  
Shi-Cheng Hou ◽  
Ya-Nan Guo ◽  
Ke-Cheng Peng
Keyword(s):  
Porous Media ◽  
Nonlinear Waves ◽  
Inverse Method ◽  
Wave Equations ◽  
Continuous Medium ◽  
Variational Principles ◽  
Variational Theory ◽  
Long Wave ◽  
Potential Applications ◽  
Fractional System

This paper extends the (2+1)-dimensional Eckhaus-type dispersive long wave equations in continuous medium to their fractional partner, which is a model of nonlinear waves in fractal porous media. The derivation is shown briefly using He?s fractional derivative. Using the semi-inverse method, the variational principles are established for the fractional system, which up to now are not discovered. The obtained fractal variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical modelling.

scholarly journals Probability Transform Based on the Ordered Weighted Averaging and Entropy Difference

2020 ◽  
Vol 15 (4) ◽  
Author(s):  
Lipeng Pan ◽  
Yong Deng
Keyword(s):  
Probability Distribution ◽  
Evidence Theory ◽  
Mass Function ◽  
New Method ◽  
Weighted Averaging ◽  
Minimum Entropy ◽  
Transform Method ◽  
Numerical Examples ◽  
Ordered Weighted Averaging ◽  
Open Issue

Dempster-Shafer evidence theory can handle imprecise and unknown information, which has attracted many people. In most cases, the mass function can be translated into the probability, which is useful to expand the applications of the D-S evidence theory. However, how to reasonably transfer the mass function to the probability distribution is still an open issue. Hence, the paper proposed a new probability transform method based on the ordered weighted averaging and entropy difference. The new method calculates weights by ordered weighted averaging, and adds entropy difference as one of the measurement indicators. Then achieved the transformation of the minimum entropy difference by adjusting the parameter r of the weight function. Finally, some numerical examples are given to prove that new method is more reasonable and effective.

Consistent Hashin-Shtrikman Bounds on the Effective Properties of Periodic Composite Materials

1999 ◽  
Vol 66 (4) ◽  
pp. 858-866
Author(s):  
P. Bisegna ◽  
R. Luciano
Keyword(s):  
Composite Materials ◽  
Saddle Point ◽  
Variational Principles ◽  
Effective Properties ◽  
The Other ◽  
Constitutive Behavior ◽  
Homogenization Problem ◽  
Numerical Examples ◽  
Periodic Composites ◽  
Periodic Composite

In this paper the four classical Hashin-Shtrikman variational principles, applied to the homogenization problem for periodic composites with a nonlinear hyperelastic constitutive behavior, are analyzed. It is proved that two of them are indeed minimum principles while the other two are saddle point principles. As a consequence, every approximation of the former ones provide bounds on the effective properties of composite bodies, while approximations of the latter ones may supply inconsistent bounds, as it is shown by two numerical examples. Nevertheless, the approximations of the saddle point principles are expected to provide better estimates than the approximations of the minimum principles.

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