scholarly journals New Exact Travelling Wave Solutions for Fractional Differential Equations in a Shallow Water Waves

2021 ◽  
Author(s):  
Adem Cevikel ◽  
Esin Aksoy
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Water Waves ◽  
Travelling Wave ◽  
Space Time ◽  
Shallow Water Waves ◽  
Trigonometric Function Solutions ◽  
Fractional Differential ◽  
Modified Simple Equation Method ◽  
Pkp Equation

In this article, the modified simple equation method is proposed to solve nonlinear space-time fractional differential equations. This method is applied to solve space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation, the space-time fractional generalized reaction duffing model and the space-time fractional potential Kadomtsev-Petviashvili (pKP) equation. The solutions found are hyperbolic and trigonometric function solutions. Some of these solutions are new solutions that are not available in the literature.

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

scholarly journals Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Emad A.-B. Abdel-Salam ◽  
Zeid I. A. Al-Muhiameed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Mapping Method ◽  
Space Time ◽  
Fractional Differential ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
First Time

The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.

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