由相容性求任意阶非线性偏微分方程的决定方程

研究了求任意阶的非线性偏微分方程的决定方程,运用相容性方法,和运用传统的向量场及其延拓的方法求得的结果相同.但运用相容性方法不用再计算复杂的无穷小生成元的延拓的系数,这样在计算过程中既能提高计算的速度,又能提高计算的准确率,因此,这种方法比运用向量场及其延拓的方法更简便、快捷,并举例验证了这一事实.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.     

Remarks on the paper “Controllability of second order differential inclusion in Banach spaces” [J. Math. Anal. Appl. 285 (2003) 537-550]

An error is pointed out in the paper [J.R. Kang, Y.C. Kwun, J.Y. Park, Controllability of second order differential inclusion in Banach spaces, J. Math. Anal. Appl. 285 (2003) 537-550]. By an additional condition the considered system is controllable.