With the aid of computer algebraic system and symbolic computation,are obtained some explicit and exact solutions of a(2+1)-dimensional nonlinear integrable generalization of the Kaup equation by the truncated adjunct function method,involving periodic solutions and solitary wave solutions.

Discussion on the relationship between auxiliary function method and the solution in higher mathematics;

Parameter alternating method to construct the auxiliary function;

Several ways of establishing auxiliary function to testi fy median proposition;

On the construction of auxiliary function when using the calculous intermediate value theorem;

A supplementary function can be given.

This paper, by means of Rolle theorem, introduces constant method to demonstrate mean value theorem for differential calculus and attains the corresponding supplementary function of the demonstrating method.

In this paper,a new structural method of supplementary function is introduced, and the method is used to prove differential identical equation or integral inequality.

By constructing auxiliary functions and ordinary differential equations, we have obtained some new exact traveling wave solutions of the nonlinear coupled KdV equations which include soliton wave solutions and periodic solutions.

By constructing auxiliary functions,some new exact traveling wave solutions of the nonlinear coupled VB equations are obtained,which include solitary wave solutions,trigonometric function solutions,Jacobian elliptic function solutions and rational solutions,and some solutions are complex linear solutions.

By constructing auxiliary functions, we have obtained some new exact traveling wave solutions of the nonlinear coupled KdV equations which include solitary wave solutions, trigonometric function solutions,Jacobian elliptic function solutions and rational solutions.