Fractional Burgers Equation Research Articles

Density fluctuations for exclusion processes with long jumps

We show that the stationary density fluctuations of exclusion processes with long jumps, whose rates are of the form \(c^\pm |y-x|^{-(1+\alpha )}\) where \(c\pm \) depends on the sign of \(y-x\), are given by a fractional Ornstein–Uhlenbeck process for \(\alpha \in (0,\frac{3}{2})\). When \(\alpha =\frac{3}{2}\) we show that the density fluctuations are tight, in a suitable topology, and that any limit point is an energy solution of the fractional Burgers equation, previously introduced in Gubinelli and Jara (Stoch Partial Differ Equ Anal Comput 1(2):325–350, 2013) in the finite volume setting.

Time–Space Fractional Burger's Equation on Time Scales

This paper deals with a newly born fractional derivative and integral on time scales. A chain rule is derived, and the given indefinite integral is being discussed. Also, an application to the traffic flow problem with a fractional Burger's equation is presented.

Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods

We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf–Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the concept of spectral vanishing viscosity (SVV). For the global spectral method, SVV is very effective and the computational cost is O(N2), which is essentially the same as for the standard Burgers equation. We also develop a local discontinuous Galerkin (LDG) spectral element method to improve the accuracy around discontinuities, and we again stabilize the LDG method with the SVV operator. Finally, we solve numerically the inviscid fractional Burgers equation both with the spectral and the spectral element LDG methods. We study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters.

Approximate solution for burgers equation with local fractional derivative by Yang-Laplace decomposition method

We presented the application of local fractional Yang-Laplace decomposition method to a local fractional Burgers equation. Our results show that the method gives high accuracy series solutions that converge very rapidly.

An analytical solution of fractional burgers equation

Using the fractional complex transform, the fractional partial differential equations can be reduced to ordinary differential equations which can be solved by the auxiliary equation method. Non-linear superposition formulation of Riccati equation is applied, and a complex infinite sequence solution is obtained.

Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs

By using a counterexample, we proved the fractional chain rule appeared in many references does not hold under Riemann–Liouville definition and Caputo definition of fractional derivative. It shows that this chain rule is invalid in investigating exact solutions of nonlinear fractional partial differential equations (PDEs). In this paper, based on the homogenous balanced principle, the function-expansion method of separation variable type are introduced. By using this method, a series of nonlinear time fractional PDEs such as time fractional KdV equation and Burgers equation, time fractional diffusion–convection equations are studied from mathematical viewpoint. The dynamical properties of these exact solutions are discussed and the profiles of several representative exact solutions are illustrated.

Pointwise estimates for solutions of fractal Burgers equation

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of d-dimensional critical fractal Burgers equation ut−Δα/2u+b⋅∇(u|u|q)=0, α∈(1,2), b∈Rd for q=(α−1)/d and u0∈L1(Rd). We consider also q>(α−1)/d under additional condition u0∈L∞(Rd). In both cases we assume u0≥0, which implies that the solution is non-negative. The estimates are given in the terms of the function Ptu0, where Pt denotes the semigroup for the operator ∂t−Δα/2.

Time fractional effect on ion acoustic shock waves in ion-pair plasma

The nonlinear properties of ion acoustic shock waves are studied. The Burgers equation is derived and converted into the time fractional Burgers equation by Agrawal’s method. Using the Adomian decomposition method, shock wave solutions of the time fractional Burgers equation are constructed. The effect of the time fractional parameter on the shock wave properties in ion-pair plasma is investigated. The results obtained may be important in investigating the broadband electrostatic shock noise in D- and F-regions of Earth’s ionosphere.

Lie symmetry analysis, conservation laws and exact solutions of the generalized time fractional Burgers equation

Under investigation in this work are the invariance properties of the generalized time fractional Burgers equation, which can be used to describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Lie group analysis method is applied to consider its vector fields and symmetry reductions. Furthermore, based on the sub-equation method, a new type of explicit solutions for the equation is well constructed with a detailed analysis. By means of the power series theory, exact power series solutions of the equation are also constructed. Finally, by using the new conservation theorem, conservation laws of the equation are well constructed with a detailed derivation.

A linear finite difference scheme for generalized time fractional Burgers equation

This paper is concerned with the numerical solutions of the generalized time fractional burgers equation. We propose a linear implicit finite difference scheme for solving the equation. Iterative methods become dispensable. As a result, the computational cost can be significantly reduced compare to the usual implicit finite difference schemes. Meanwhile, the finite difference method is proved to be unconditional globally stable and convergent. Numerical examples are shown to demonstrate the accuracy and stability of the method.

ECUACIÓN FRACCIONARIA DE BURGERS NO HOMOGÉNEA

En este artículo se estudian diferentes tipos de soluciones de la ecuación fraccionaria unidimensional no lineal de Burgers con un término no homogéneo asociado a fuerzas externas. Esta ecuación es una generalización de la ecuación de difusión no homogénea en la que se incluye una derivada fraccionaria de Caputo que describe una no linealidad no local. Por medio de la transformación de Cole-Hopf generalizada, la ecuación de Burgers fraccionaria no homogénea se convierte en una ecuación diferencial lineal en derivadas parciales, lo que permite obtener soluciones analíticas. Se exploran los efectos asociados al término no homogéneo y al orden de la derivada fraccionaria.

A study of a spatial fractional Burger equation via the optimal homotopy analysis method

In this paper, the optimal homotopy analysis method is applied to obtain the solution of a spatial fractional Burger equation. The fractional derivative is given in Riesz sense. The recurrence relation of the considered method is carried out based on some properties of Riesz fractional derivative. Numerical results illustrate the effects of varying the fractional derivative order $\alpha $ and the coefficient of the kinematics viscosity $\nu >0$ on the solution behavior.

Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order

AbstractThe fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.

On Microscopic Derivation of a Fractional Stochastic Burgers Equation

We derive from a class of microscopic asymmetric interacting particle systems on ${\mathbb Z}$, with long range jump rates of order $|\cdot|^{-(1+\alpha)}$ for $0<\alpha<2$, different continuum fractional SPDEs. More specifically, we show the equilibrium fluctuations of the hydrodynamics mass density field of zero-range processes, depending on the stucture of the asymmetry, and whether the field is translated with process characteristics velocity, is governed in various senses by types of fractional stochastic heat or Burgers equations. The main result: Suppose the jump rate is such that its symmetrization is long range but its (weak) asymmetry is nearest-neighbor. Then, when $\alpha<3/2$, the fluctuation field in space-time scale $1/\alpha:1$, translated with process characteristic velocity, irrespective of the strength of the asymmetry, converges to a fractional stochastic heat equation, the limit also for the symmetric process. However, when $\alpha\geq 3/2$ and the strength of the weak asymmetry is tuned in scale $1-3/2\alpha$, the associated limit points satisfy a martingale formulation of a fractional stochastic Burgers equation.

Numerical solution of time fractional Burgers equation

Abstract In this article, the time fractional order Burgers equation has been solved by quadratic B-spline Galerkin method. This method has been applied to three model problems. The obtained numerical solutions and error norms L2 and L∞ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given.

Stable estimates for source solution of critical fractal Burgers equation

In this paper, we provide two-sided estimates for the source solution of d-dimensional critical fractal Burgers equation ut−Δα/2+b⋅∇(u|u|q)=0, q=(α−1)/d, α∈(1,2), b∈Rd, by the density function of the isotropic α-stable process.

M-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise

This paper is concerned with the essential m-dissipativity of the Kolmogorov operator associated with a fractional stochastic Burgers equation with space-time white noise. Some estimates on the solution and its moments with respect to the invariant measure are given. Moreover we also study the smoothing properties of the transition semigroup and the corresponding fractional Ornstein-Uhlenbeck operator by introducing an auxiliary semigroup and (generalized) Bismut-Elworthy formula. From these results, we prove that the Kolmogorov operator of the problem is m-dissipative and the domain of the infinitesimal generator of the fractional Ornstein-Uhlenbeck operator is a core.

Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power \( \gamma \ge 0\) on the well-posedness of the Camassa–Holm equations with fractional dissipation. It is shown that the zero-filter limit (\(\alpha \rightarrow 0\)) of the Camassa–Holm equation with fractional dissipation is the fractal Burgers equation. It is known that in the supercritical case \(\gamma \in [0, 1)\), the fractal Burgers equation blows up in finite time in \(H^s({\mathbb {R}})\) with \(s>\frac{3}{2}-\gamma \). It is established here that the dissipative Camassa–Holm equation is globally well-posed in the critical Sobolev space \(H^{\frac{3}{2}-\gamma }({\mathbb {R}})\) with the fractional parameter \(\gamma \in [\frac{1}{2}, 1)\). Moreover, it is also demonstrated that the solution of the dissipative Camassa–Holm equation blows up in finite time in the particular supercritical case when \( \gamma = 0\).

An algorithm for the approximate solution of nonlinear total time fractional-order Burger equation

In this paper we have established the simplest equation method to find approximate solutions for total Burgurs equations with time fractional derivative. This method is effective in finding approximate traveling wave solutions of nonlinear, fractional evolution equations (NLEEs) in mathematical physics. The effectiveness of this manageable method has been shown by applying it to several examples in time fractional Burgers equations. The present approach has the potential to apply to other nonlinear fractional differential equations. All numerical calculations in the present study have been performed on a PC, applying programs written in Mathematica and Maple.

Numerical Solution of Time Fractional Burgers Equation by Cubic B-spline Finite Elements

We present some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions obtained by collocation method using cubic B-spline base functions. The aim of this paper is to show that the finite element method based on the cubic B-spline collocation method approach is also suitable for the treatment of the fractional differential equations. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.