一类分数阶微分方程的李对称分析和守恒定律
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摘要
利用李对称方法研究一类分数阶微分方程的约化和守恒定律.根据给出的方程李对称分析,得到无穷小生成元,求解出方程的精确解.通过相似变换与相似变量,将方程约化为带有Erdélyi-Kober分数阶算子的非线性常微分方程.进一步在李代数的基础上,讨论分数阶微分方程的守恒定律,并分别求得x分量和t分量的守恒向量.
The reduction and conservation law of a class of fractional differential equations are solved by the Lie symmetry analysis.According to the Lie symmetry analysis of the given equation,we can get the infinitesimal generator,and find out the exact solution of the system.Then,the equation is reduced to a nonlinear ordinary differential equation with Erdélyi-Kober fractional operator through similarity transformation and similarity variables.And,on the basis of Lie algebra,the conservation law of fractional differential equation is discussed,and the conservation vector of x and t components is given respectively.
The reduction and conservation law of a class of fractional differential equations are solved by the Lie symmetry analysis.According to the Lie symmetry analysis of the given equation,we can get the infinitesimal generator,and find out the exact solution of the system.Then,the equation is reduced to a nonlinear ordinary differential equation with Erdélyi-Kober fractional operator through similarity transformation and similarity variables.And,on the basis of Lie algebra,the conservation law of fractional differential equation is discussed,and the conservation vector of x and t components is given respectively.
引文
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[3]KOMAL S,GUPTA R K.Space-time fractional nonlinear partial differential equations:symmetry analysis and conservation laws[J]. Nonlinear Dyn,2017,89:321-331.
[4]PASHAYI S,HASHEMI M S,SHAHMORAD S.Analytical Lie group approach for solving fractional integro-differential equations[J]. Commun Nonlinear Sci Numer Simulat,2017,51:66-77.
[5]PODLUBNY I.Fractional diferential equations[M]. America:Mathematics in Science and Engineering,1999:41-149.
[6]KIRYAKOVA V.Generalised fractional calculus and applications[M]. Harlow:Longman Scientific&Technical,1994,22-56.
[7]GEORGE W,BLUMAN A F,CHEVIAKOV S C,et al. Application of symmetry methods to partial differential equation[M].Beijing:World Book Publishing House,2014:22-56.
[8]SAHOO S,SAHA R S.Analysis of Lie symmetries with conservation laws for the(3+1)dimensional time-fractional mKdvZk equation in ion-acoustic waves[J]. Nonlinear Dyn,2017,90:1105-1113.
[9]RUI Wenjuan,ZHANG Xiangzhi.Invariant analysis and conservation laws for the time fractional foam drainage equation[J]. Eur Phys J Plus,2015,130:15192-15193.
[10]OLVER PETER J.Application of Lie group to differential equation[M]. Beijing:World Book Publishing House,1986:242-281.
[11]KOMAL S,GUPTA R K.Conservation laws for certain time fractional nonlinear systems of partial differential equations[J]. Commun Nonlinear Sci Numer Simulat,2017,53:10-21.
[12]IBRAGIMOV N H.Nonlinear self-adjointness in constructing conservation laws[J]. Arch Alga,2011,7:1-39.