Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations

Yang, Xiao‐Jun; Gao, Feng; Srivastava, H. M.

doi:10.1016/j.camwa.2016.11.012

Cited by 276 publications

(118 citation statements)

References 24 publications

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“…13 There are several alternative approaches for describing the complex and fractal behaviors in nature. [1][2][3][4][5][6][7][8] The theory of the local fractional derivative (LFD) is a mathematical tool for describing fractals, that was used to model the fractal complexity in shallow water surfaces, 14 LC-electric circuit, 15 traveling-wave solution of the Burgerstype equation, 16 PDEs, [17][18][19][20] ODEs, 21 and inequalities. 22,23 The useful models for the LFD were considered [24][25][26][27][28][29] and discussed.…”

Section: Introductionmentioning

confidence: 99%

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

2017

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The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable ¶ Corresponding author. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. X. -J. Yang, J. A. T. Machado & D. Baleanu graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains. 1740006-1

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Section: Introductionmentioning

confidence: 99%

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

2017

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“…Many analytic techniques [10,26] are pretty reasonable methods to understand some nonlinear differential equation. But for models whose exact solutions hardly are found, the numerical method is an alternative choice.…”

Section: Introductionmentioning

confidence: 99%

A high-accuracy conservative difference approximation for Rosenau-KdV equation

Hu¹,

Jun²,

Zhuo³

2017

J. Nonlinear Sci. Appl.

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In this paper, we study the initial-boundary value problem of Rosenau-KdV equation. A conservative two level nonlinear Crank-Nicolson difference scheme, which has the theoretical accuracy O(τ 2 + h 4 ), is proposed. The scheme simulates two conservative properties of the initial boundary value problem. Existence, uniqueness, and priori estimates of difference solution are obtained. Furthermore, we analyze the convergence and unconditional stability of the scheme by the energy method. Numerical experiments demonstrate the theoretical results.

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“…Figure 5 shows the hand image covered by boxes with size G of 8. The fractional calculus methods [43][44][45] shall be considered in the future. …”

Section: Minkowski-bouligand Dimensionmentioning

confidence: 99%

Synthetic Minority Oversampling Technique and Fractal Dimension for Identifying Multiple Sclerosis

Zhang

Phillips

et al. 2017

Fractals

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Multiple sclerosis (MS) is a severe brain disease. Early detection can provide timely treatment. Fractal dimension can provide statistical index of pattern changes with scale at a given brain image. In this study, our team used susceptibility weighted imaging technique to obtain 676 MS slices and 880 healthy slices. We used synthetic minority oversampling technique to process the unbalanced dataset. Then, we used Canny edge detector to extract distinguishing edges. The Minkowski-Bouligand dimension was a fractal dimension estimation method and used to extract features from edges. Single hidden layer neural network was used as the classifier. Finally, we proposed a three-segment representation biogeography-based optimization to train the classifier. Our method achieved a sensitivity of 97.78±1.29%, a specificity of 97.82±1.60% and an accuracy of 97.80±1.40%. The proposed method is superior to seven state-of-the-art methods in terms of sensitivity and accuracy.

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Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations

Cited by 276 publications

References 24 publications

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

A high-accuracy conservative difference approximation for Rosenau-KdV equation

Synthetic Minority Oversampling Technique and Fractal Dimension for Identifying Multiple Sclerosis

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