THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

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Sheng Zhang

,

Sen Zhao

,

Yue Zhang

EXTENDING OPERATOR METHOD TO LOCAL FRACTIONAL EVOLUTION EQUATIONS IN FLUIDS

ABSTRACT
This paper is aimed to solve non-linear local fractional evolution equations in fluids by extending the operator method proposed by Zenonas Navickas. Firstly, we give the definitions of the generalized operator of local fractional differentiation and the multiplicative local fractional operator. Secondly, some properties of the defined operators are proved. Thirdly, a solution in the form of operator representation of a local fractional ordinary differential equation is obtained by the extended operator method. Finally, with the help of the obtained solution in the form of operator representation and the fractional complex transform, the local fractional Kadomtsev-Petviashvili (KP) equation and the fractional Benjamin-Bona-Mahoney (BBM) equation are solved. It is shown that the extended operator method can be used for solving some other non-linear local fractional evolution equations in fluids.
KEYWORDS
Local fractional evolution equation, operator method, the generalized operator of local fractional differentiation, the multiplicative local fractional operator, the fractional KP equation, the fractional BBM equation
PAPER SUBMITTED: 2018-08-20
PAPER REVISED: 2018-11-23
PAPER ACCEPTED: 2019-01-18
PUBLISHED ONLINE: 2019-06-08
DOI REFERENCE: https://doi.org/10.2298/TSCI180820261Z
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Issue 6, PAGES [3759 - 3766]
REFERENCES
  1. Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Elsevier, London, UK, 2015
  2. Yang, X. J., et al., Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation, Abstract and Applied Analysis, 2014 (2014), ID 278672
  3. Yang, X. J., et al., An Asymptotic Perturbation Solution for a Linear Oscillator of Free Damped Vibrations in Fractal Medium Described by Local Fractional Derivatives, Communications in Nonlinear Science and Numerical Simulation, 29 (2015), 1-3, pp. 499-504
  4. Yang, X. J., et al., On Exact Traveling-Wave Solutions for Local Fractional Korteweg-de Vries Equation, Chaos, 26 (2016), 8, ID 084312
  5. Yang, X. J., et al., Exact Travelling Wave Solutions for the Local Fractional Two-Dimensional Burgers-Type Equations, Computers and Mathematics with Applications, 73 (2017), 2, 203-210
  6. Yang, X. J., et al., On a Fractal LC-Electric Circuit Modeled by Local Fractional Calculus, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 6, 200-206
  7. Yang, X. J., et al., New Rheological Models within Local Fractional Derivative, Romanian Reports in Physics, 69(2017), 3, Article ID 113
  8. Yang, X. J., et al., New Family of the Local Fractional PDEs, Fundamenta Informaticae, 151 (2017), 1-4, pp. 63-75
  9. Yang, X. J., et al., Non-Differentiable Exact Solutions for the Non-Linear ODEs Defined on Fractal Sets, Fractals, 25 (2017), 4, ID 1740002
  10. He, J. H., Variational Iteration Methoda Kind of Non-Linear Analytical Technique: Some Examples, International Journal of Nonlinear Mechanics, 34 (1999), 4 , pp. 699-708
  11. He, J. H., A New Approach to Non-Linear Partial Differential Equations, Communications in Nonlinear Science and Numerical Simulation, 2 (1997), 4, pp. 203-205
  12. Momani, S., et al., Variational Iteration Method for Solving the Space- and Time-Fractional KdV Equation, Numerical Methods for Partial Differential Equations, 24 (2008), 1, pp. 262-271
  13. Yang, Y. J., et al., A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators, Abstract and Applied Analysis, 2014 (2014), ID 202650
  14. Hemeda, A. A., et al., Local Fractional Analytical Methods for Solving Wave Equations with Local Fractional Derivative, Mathematical Methods in the Applied Sciences, 41(2018), 6, pp.2515-2529
  15. Navichkas, Z., Operator Method of Solving Non-Linear Differential Equations, Lithuanian Mathematical Journal, 42 (2002), 4, pp. 387-393
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Extending operator method to local fractional evolution equations in fluids

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