The Theory and Application of Rough Integration in Rough Function Model

The definition of the rough integral Pawlak proposed is improved. New notions are proposed, which are respectively the rough integral on a constant interval and the roughly integral upper limit function, and so on. By comparing with the definite integral of real functions, the properties of rough integrals are analyzed. Giving the concept of mean value of discrete functions, the mean value method for rough integration is derived. At the same time, the intermediate value theorem of rough integration is proposed and its geometric significance is analyzed, which provides a dependable theoretical tool for rough integral operation, etc.. Conclusions including the existence theorem of rough primitives and the fundamental formula of rough calculus are proposed. By the representative of discrete functions, the method of computing a primitive is given, by which basic formulas for rough integration in common use are derived, and the method of rough direct integration is obtained. There is also the method of rough integration by parts for rough integrals which is like that of definite integrals. Thus the recurrence formula for rough integration is deduced, in which the integrand of the rough integral is in the shape of the product of a rough power function and a rough exponential function. It is pointed out that integration by substitution is not applicable for rough integral operation.