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Register a new userLie group classifications and exact solutions for time-fractional Burgers equation
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Guo-cheng Wu
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Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
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Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method.
We carry out extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, which enhances and generalizes many results existing in the literature. The complete point-symmetry group of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one- and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti-Leon-Pempinelli system via differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended using Laplace and Darboux transformations.
Symmetries of a differential equations is one of the most important concepts in theory of differential equations and physics. One of the most prominent equations is KdV (Kortwege-de Vries) equation with application in shallow water theory. In this paper we are going to explain a particular method for finding symmetries of KdV equation, which is called Harrison method. Our tools in this method are Lie derivatives and differential forms, which will be discussed in the first section more precisely. In second chapter we will have some analysis on the solutions of KdV equation and we give a method, which is called first integral method for finding the solutions of KdV equation.
this paper we show some new exact solutions for the generalized modified Degasperis$-$Procesi equation (mDP equation)
In this paper we continue the formal analysis of the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation that we began in [18]. They have the form $fleft( x,v,tright) =gleft(v-Lleft( tright) x,tright) $ where $Lleft( tright) =Aleft(I+tAright) ^{-1}$ where $A$ is a constant matrix. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. Depending on the properties of the collision kernel the collision and the hyperbolic terms might be of the same order of magnitude as $ttoinfty$, or the collision term could be the dominant one for large times, or the hyperbolic term could be the largest. The first case has been rigorously studied in [17]. Formal asymptotic expansions in the second case have been obtained in [18]. All the solutions obtained in this case can be approximated by Maxwellian distributions with changing temperature. In this paper we focus in the case where the hyperbolic terms are much larger than the collision term for large times (hyperbolic-dominated behavior). In the hyperbolic-dominated case it does not seem to be possible to describe in a simple way all the long time asymptotics of the solutions, but we discuss several physical situations and formulate precise conjectures. We give explicit formulas for the relationship between density, temperature and entropy for these solutions. These formulas differ greatly from the ones at equilibrium.
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