scholarly journals SOME PROPERTIES OF FRACTIONAL BURGERS EQUATION

2002 ◽  
Vol 7 (1) ◽  
pp. 151-158 ◽  
Author(s):  
P. Miškinis
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Conservation Laws ◽  
Traveling Wave ◽  
Wave Solution ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Traveling Wave Solution ◽  
One Dimensional ◽  
Liouville Fractional Derivative

The fractional generalization of a one‐dimensional Burgers equationwith initial conditions ɸ(x, 0) = ɸ0(x); ɸt(x,0) = ψ0 (x), where ɸ = ɸ(x,t) ∈ C2(Ω): ɸt = δɸ/δt; aDx p is the Riemann‐Liouville fractional derivative of the order p; Ω = (x,t) : x ∈ E 1, t > 0; and the explicit form of a particular analytical solution are suggested. Existing of traveling wave solution and conservation laws are considered. The relation with Burgers equation of integer order and properties of fractional generalization of the Hopf‐Cole transformation are discussed.

scholarly journals Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Weiguo Zhang ◽  
Xiang Li
Keyword(s):  
Dynamical Systems ◽  
Error Estimates ◽  
Traveling Wave ◽  
Wave Solution ◽  
Burgers Equation ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Traveling Wave Solution ◽  
Oscillatory Solutions ◽  
Wave Solutions

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical valuesλ1andλ2which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficientα≥λ1, it appears as a monotone kink profile solitary wave solution; that if0<α<λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

scholarly journals On the Solution of Burgers’ Equation with the New Fractional Derivative

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Ali Kurt ◽  
Yücel Çenesiz ◽  
Orkun Tasbozan
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Burgers Equation ◽  
Approximate Analytical Solution ◽  
Initial Conditions ◽  
Conformable Fractional Derivative ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Homotopy Analysis

AbstractFirstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.

scholarly journals Traveling wave solutions for a class of reaction-diffusion system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bingyi Wang ◽  
Yang Zhang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Traveling Wave Solution ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
One Dimensional ◽  
Wave Solutions

AbstractIn this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

scholarly journals ON A MACROSCOPIC LIMIT OF A KINETIC MODEL OF ALIGNMENT

2013 ◽  
Vol 23 (14) ◽  
pp. 2647-2670 ◽  
Author(s):  
JACEK BANASIAK ◽  
MIROSŁAW LACHOWICZ
Keyword(s):  
Kinetic Model ◽  
Diffusion Equation ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solution ◽  
Navier Stokes ◽  
Order Of Approximation ◽  
Linear Diffusion ◽  
One Dimensional ◽  
Position And Orientation

In the present paper we study the macroscopic limits of a kinetic model for interacting entities (individuals, organisms, cells). The kinetic model is one-dimensional and the entities are characterized by their position and orientation (+/-) with swarming interaction controlled by a sensitivity parameter. The macroscopic limits of the model are considered for solutions close either to the diffusive (isotropic) or to the aligned (swarming) equilibrium states for various values of that parameter. In the former case the classical linear diffusion equation results whereas in the latter a traveling wave solution does both in the zeroth ("Euler") and first ("Navier–Stokes") order of approximation.

An Approximate Analytical Solution of Richards’ Equation

2015 ◽  
Vol 16 (5) ◽  
pp. 239-247 ◽  
Author(s):  
Xi Chen ◽  
Ying Dai
Keyword(s):  
Porous Medium ◽  
Analytical Solution ◽  
Traveling Wave ◽  
Wave Solution ◽  
Approximate Analytical Solution ◽  
Traveling Wave Solution ◽  
Richards Equation ◽  
Transform Method ◽  
Differential Transform ◽  
Finite Layer

AbstractAn approximate analytical solution of Richards’ equation (RE) in a semi-finite layer of porous medium is obtained in this article. The basic idea is constructing a function between the Boltzmann variable ϕ and the traveling wave solution of RE, introducing the function into RE and calculating the relevant parameters by differential transform method (DTM). In the end, three examples are prepared to show the accuracy of the presented approximate analytical solution.

scholarly journals Single peaked traveling wave solutions to a generalized μ-Novikov Equation

2020 ◽  
Vol 10 (1) ◽  
pp. 66-75
Author(s):  
Byungsoo Moon
Keyword(s):  
Linear Combination ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Wave Solutions ◽  
Quadratic Nonlinearities ◽  
Novikov Equation

Abstract In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

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