Analytical approach to Boussinesq equation with space- and time-fractional derivatives

2010 ◽  
Vol 66 (10) ◽  
pp. 1315-1324 ◽  
Author(s):  
Ahmet Yıldırım ◽  
Sefa Anıl Sezer ◽  
Yasemin Kaplan
Keyword(s):  
Analytical Approach ◽  
Fractional Derivatives ◽  
Boussinesq Equation ◽  
Space And Time

THE $W^{(2)}_3$ CONFORMAL ALGEBRA AND THE BOUSSINESQ HIERARCHY

1991 ◽  
Vol 06 (26) ◽  
pp. 2397-2409 ◽  
Author(s):  
P. MATHIEU ◽  
W. OEVEL
Keyword(s):  
Nonlinear Equations ◽  
Poisson Structure ◽  
Boussinesq Equation ◽  
Free Field ◽  
Conformal Algebra ◽  
Integrable Hierarchy ◽  
Classical Formula ◽  
Field Representation ◽  
Space And Time ◽  
Time Variables

The classical [Formula: see text] algebra Polyakov is shown to be equivalent to the second Poisson structure of a new integrable hierarchy of nonlinear equations. The hierarchy is related to the Boussinesq hierarchy by interhcanging the roles of the space and time variables x and t in the Boussinesq equation. From this relation the Miura map, relating the new hierarchy to its modified version, can be derived systematically. It is found to be equivalent to the known free field representation of the [Formula: see text] algebra.

Analytical Approach for the Space-time Nonlinear Partial Differential Fractional Equation

2014 ◽  
Vol 15 (7-8) ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner
Keyword(s):  
Function Method ◽  
Analytical Approach ◽  
Fractional Derivatives ◽  
Space Time ◽  
Variable Method ◽  
Partial Differential ◽  
Functional Variable ◽  
Liouville Derivative ◽  
Functional Variable Method ◽  
Fractional Equation

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the functional variable method, exp-function method and

A Novel Approximate Analytical approach to solve the Boussinesq equation

2020 ◽  
Vol 33 (02) ◽  
Author(s):  
Dr. Amruta Daga Bhandari ◽  
◽  
Dr. Vikas H Pradhan ◽  
Keyword(s):  
Analytical Approach ◽  
Boussinesq Equation

ANALYSIS OF REDISTRIBUTION OF RADIATION DEFECTS TAKING INTO ACCOUNT DIFFUSION AND SEVERAL SECONDARY PROCESSES

2008 ◽  
Vol 22 (28) ◽  
pp. 2779-2791 ◽  
Author(s):  
E. L. PANKRATOV
Keyword(s):  
Ion Implantation ◽  
Point Defects ◽  
Analytical Approach ◽  
Irradiation Time ◽  
Defect Concentration ◽  
Radiation Defects ◽  
Space And Time ◽  
Dose Irradiation ◽  
Semiconductor Sample

In this paper, we analyzed the evolution of concentration of radiative defects, which is generated in a semiconductor sample during ion implantation. Approximate analytical approach for the description of the evolution of concentration of radiative defects with account diffusion and some secondary processes (recombination of the point defects and generation of divacancies) has been used. Discontinuity of the ions in space and time has been also accounted. The main results are: (i) the estimation of dependencies of the defect concentration from depth at different values of dose (irradiation time), (ii) the different amorphization doses from the density of current of the ions for the more common case in comparison with those considered in literature. As an example, we consider the implantation of ions of neon into a sample of the silicon.

Analytical Approach to (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional Kadomtsev-Petviashvili Equation

2010 ◽  
Vol 65 (5) ◽  
pp. 411-417 ◽  
Author(s):  
Selin Sarıaydın ◽  
Ahmet Yıldırım
Keyword(s):  
Analytical Approach ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Perturbation Series ◽  
Solitary Wave Solutions ◽  
Convergent Power Series ◽  
Kp Equation ◽  
Homotopy Perturbation ◽  
Wave Solutions ◽  
Petviashvili Equation

In this paper, we studied the solitary wave solutions of the (2+1)-dimensional Boussinesq equation utt −uxx−uyy−(u2)xx−uxxxx = 0 and the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation uxt −6ux 2 +6uuxx −uxxxx −uyy −uzz = 0. By using this method, an explicit numerical solution is calculated in the form of a convergent power series with easily computable components. To illustrate the application of this method numerical results are derived by using the calculated components of the homotopy perturbation series. The numerical solutions are compared with the known analytical solutions. Results derived from our method are shown graphically.

scholarly journals The parabolic p-Laplacian with fractional differentiability

2020 ◽  
Author(s):  
Dominic Breit ◽  
Lars Diening ◽  
Johannes Storn ◽  
Jörn Wichmann
Keyword(s):  
Numerical Experiments ◽  
Fractional Derivatives ◽  
Convergence Rates ◽  
Time Discretization ◽  
Coupling Condition ◽  
Implicit Euler Scheme ◽  
Space And Time ◽  
Optimal Convergence Rates ◽  
Fractional Differentiability ◽  
Nikolskii Spaces

Abstract We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space–time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskiǐ spaces and therefore cover situations when the (gradient of the) solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolutions $h$ and $\tau $. For this we show that the $L^2$-projection is compatible with the quasi-norm. The theoretical error analysis is complemented by numerical experiments.

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