The Analysis Of Integrability Of Two Classes Of Differential Equations

In this paper, we consider a class of generalized Black-Scholes equation and a class of Klein-Gordon-Zakharov equation, and the main content is the following aspects.(1) The admissible transformation problem of a class of Black-Scholes equations is considered. Based on the theory of symmetry group, a class of symmetry group and invariant solutions of the Black-Scholes equation are obtained.(2) Based on the Lie group theory, the two single parameter Lie groups which are accepted by the traveling wave solutions equation of the Black-Scholes equation are found and proved. Thus, the first integrals and integrating factor of the corresponding traveling wave solutions equation are obtained, and then a class of traveling wave solutions of the Black-Scholes equation is obtained. By using the qualitative theory of differential equations, the traveling wave solutions equation of the Black-Scholes equation are analyzed and a class of traveling wave solutions for the Black-Scholes equation with certain conditions is obtained.(3) Based on the Lie group theory, the two single parameter Lie groups which are accepted by the traveling wave solution equation of a class of Klein-Gordon-Zakharov equation are found and proved. Thus, the first integrals and integral factor of the corresponding traveling wave solutions equation are obtained, and then a class of traveling wave solutions of the Klein-Gordon-Zakharov equation is obtained. By using the qualitative theory of differential equations, the traveling wave solutions equation of the Klein-Gordon-Zakharov equation are analyzed, and a class of traveling wave solutions for the Klein-Gordon-Zakharov equation with certain conditions is obtained.