scholarly journals Integral Balance Methods for Stokes’ First Equation Described by the Left Generalized Fractional Derivative

Physics ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 154-166 ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Approximate Solution ◽  
Fractional Derivative ◽  
Heat Balance ◽  
Time Derivative ◽  
Double Integral ◽  
Derivative Operator ◽  
Integral Methods ◽  
Fractional Time Derivative ◽  
The Heat Balance ◽  
Generalized Fractional Derivative

In this paper, the integral balance methods of the Stokes’ first equation have been presented. The approximate solution of the fractional Stokes’ first equation using the heat balance integral method has been proposed. The approximate solution of the fractional Stokes’ first equation using the double integral methods has been proposed. The generalized fractional time derivative operator has been used. The graphical representations of the cubic profile and the quadratic profile for the Stokes’ first problem have been provided. The impacts of the orders of the generalized fractional derivative in the Stokes’ first problem have been investigated. The exponent of the assumed profile for the Stokes’ first equation has been discussed.

scholarly journals Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution

2021 ◽  
Vol 2 (1) ◽  
pp. 60-75
Author(s):  
Ndolane Sene
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Integral Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Double Integral ◽  
Uniqueness Of The Solution ◽  
The Impact ◽  
The Heat Balance

In this paper, we propose the approximate solution of the fractional diffusion equation described by a non-singular fractional derivative. We use the Atangana-Baleanu-Caputo fractional derivative in our studies. The integral balance methods as the heat balance integral method introduced by Goodman and the double integral method developed by Hristov have been used for getting the approximate solution. In this paper, the existence and uniqueness of the solution of the fractional diffusion equation have been provided. We analyze the impact of the fractional operator in the diffusion process. We represent graphically the approximate solution of the fractional diffusion equation.

scholarly journals Heat-balance integral to fractional (half-time) heat diffusion sub-model

2010 ◽  
Vol 14 (2) ◽  
pp. 291-316 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Diffusion Equation ◽  
Heat Balance ◽  
Integral Method ◽  
Time Derivative ◽  
Heat Diffusion ◽  
Half Time ◽  
Heat Diffusion Equation ◽  
Parabolic Profile ◽  
Heat Balance Integral Method ◽  
The Heat Balance

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

Universal fractional Euler-Lagrange equation from a generalized fractional derivate operator

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Rami El-Nabulsi
Keyword(s):  
Fractional Derivative ◽  
Variational Approach ◽  
Lagrange Equation ◽  
Derivative Operator ◽  
Hamilton Equations ◽  
Generalized Fractional Derivative

AbstractThe purpose of this paper is to extend the fractional actionlike variational approach by introducing a generalized fractional derivative operator. The generalized fractional formalism introduced through this work includes some interesting features concerning the fractional Euler-Lagrange and Hamilton equations. Additional attractive features are explored in some details.

scholarly journals Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Majid Bagheri ◽  
Ali Khani
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Kdv Equation ◽  
Time Derivative ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Fractional Time Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

scholarly journals Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Salih Djilali ◽  
Behzad Ghanbari
Keyword(s):  
Fractional Derivative ◽  
Dynamical Behavior ◽  
Time Derivative ◽  
Interaction Model ◽  
Integration Rule ◽  
Existence And Stability ◽  
Natural Result ◽  
Biological Environment ◽  
Fractional Time Derivative ◽  
The Impact

AbstractThe behavior of any complex dynamic system is a natural result of the interaction between the components of that system. Important examples of these systems are biological models that describe the characteristics of complex interactions between certain organisms in a biological environment. The study of these systems requires the use of precise and advanced computational methods in mathematics. In this paper, we discuss a prey–predator interaction model that includes two competitive predators and one prey with a generalized interaction functional. The primary presumption in the model construction is the competition between two predators on the only prey, which gives a strong implication of the real-world situation. We successfully establish the existence and stability of the equilibria. Further, we investigate the impact of the memory measured by fractional time derivative on the temporal behavior. We test the obtained mathematical results numerically by a proper numerical scheme built using the Caputo fractional-derivative operator and the trapezoidal product-integration rule.

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