Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

Abdel‐Gawad, H. I.; Tantawy, M.; Băleanu, Dumitru

doi:10.1002/mma.6178

Cited by 7 publications

(7 citation statements)

References 39 publications

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“…The unified, extended and generalized unified methods have been proposed in [42][43][44][45][46][47][48][49]. The unified method asserts that , the solutions of a nonlinear evolution equation can be written in the forms of polynomial or rational functions in auxiliary functions, that satisfy appropriate auxiliary equations.…”

Section: Solutions Of the Fractal And Fractional Time-derivative Bnementioning

confidence: 99%

Analytic Solutions of Fractal and Fractional Time Derivative-Burgers–Nagumo Equation

Abdel‐Gawad

Tantawy

Abdel-Aziz

et al. 2021

Int. J. Appl. Comput. Math

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The Nagumo equation describes a reaction-diffusion system in biology. Here, it is coupled to Burgers equation, via including convection, which is the Burgers-Nagumo equation (BNE). The first objective of this work is to present a theorem to reduce, approximately, the different versions of the fractional time derivatives (FTD) to an ordinary derivative (OD) with time dependent coefficients (non autonomous). The second objective is to find the exact solutions of the fractal and FTD-BNE is reduced to BNE with time dependent coefficient. Further similarity transformations are introduced. The unified and extended unified method are used to find the exact traveling waves solutions. Also, self-similar solutions are obtained. The novelties in this work are (i) reducing, via an analytic approximation, the different versions of FTD to non autonomous OD. (ii) Traveling and self-similar waves solutions of the FTD-BNE are derived. (iii) The effect of the order of fractional and fractal derivatives, on waves structure, are investigated. It is found that significant fractal effects hold for smaller order derivatives. While significant fractional effects hold for higher-order derivatives. It is found that, the solutions obtained show solitary wave, wrinkle soliton, solitons with double kinks or with spikes and undulated wave. Further It is shown that wrinkle soliton, with double kink configuration holds for smaller fractal order. While in the case of fractional derivative, this holds for higher orders. We mention that the results found here are completely new. Symbolic computations are carried by using Mathematica.

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Section: Solutions Of the Fractal And Fractional Time-derivative Bnementioning

confidence: 99%

Analytic Solutions of Fractal and Fractional Time Derivative-Burgers–Nagumo Equation

Abdel‐Gawad

Tantawy

Abdel-Aziz

et al. 2021

Int. J. Appl. Comput. Math

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“…Here, we shall reduce the fractional derivatives to non autonomous ordinary derivatives, that is with time dependent coefficients [22][23][24].…”

Section: Reduction Of the Fractional Derivativesmentioning

confidence: 99%

“…By inserting (22) and ( 23) into (21), and by using (24), we get a system of nonlinear algebraic equations in a i (v), b i (v) and d i (v), i = 1, 2... . We found that the calculations are not straight forward due to two facts.…”

Section: Case Of Linear Auxiliary Equationsmentioning

confidence: 99%

Exact Solutions of the Time Derivative Fokker-Planck Equation: A Novel Approach

Abdel-Gawad,

Sweilam,

AL-Mekhlafi

et al. 2020

Preprint

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In the present article, an approach to find the exact solution of the fractional Fokker-Planck equation is presented. It is based on transforming it to a system of first-order partial differential equation via Hopf transformation, together with implementing the extended unified method.On the other hand, reduction of the fractional derivatives to non autonomous ordinary derivative.Thus the fractional Fokker-Planck equation is reduced to non autonomous classical ones. Some explicit solutions of the classical, fractional time derivative Fokker-Planck equation, are obtained .It is shown that the solution of the Fokker-Planck equation is bi-Gaussian's. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found the fractionality has stronger effect than fractality. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning. Further, when varying the order of the fractional time derivatives, results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly.

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“…Osman et al, 43 obtained double‐wave solutions of coupled Burgers' equation by employing the generalized unified method (GUM). Abdel‐Gawad et al, 44 investigated both of fractional KdV and Boussenisq–Burger's equations by using the unified method to attain semi‐self‐similar wave solutions. Sweilam et al, 45 presented a numerical technique to investigate complex order of fractional nonlinear Burgers' equation.…”

Section: Introductionmentioning

confidence: 99%

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

Mohamed

Rashed

Melaibari

et al. 2021

Math Methods in App Sciences

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A numerical scheme based on backward differentiation formula (BDF) and generalized differential quadrature method (GDQM) has been developed. The proposed scheme has been employed to investigate three cases of Burgers' equation, one‐dimensional, two‐dimensional, and two‐dimensional coupled models. The results showed effectiveness accuracy in absolute error and error norms L2 and L∞ compared to other methods. The results were evaluated at different values of Reynold's number, Re, and viscosity, ν.

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Fractional KdV and Boussenisq‐Burger's equations, reduction to PDE and stability approaches

Cited by 7 publications

References 39 publications

Analytic Solutions of Fractal and Fractional Time Derivative-Burgers–Nagumo Equation

Analytic Solutions of Fractal and Fractional Time Derivative-Burgers–Nagumo Equation

Exact Solutions of the Time Derivative Fokker-Planck Equation: A Novel Approach

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

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