The Existence And Uniqueness Of The Solution Of Linear Differential Equations With Uncertainty

This paper mainly studies the existence and uniqueness of the solution of lineardifferential equations with uncertainty. Firstly, we briefly introduced thedevelopment and practical significance of the fuzzy set theory, research status oflinear differnential equations with uncertainty, and some relevant knowledge offuzzy space.Secondly, this paper use three parts to proved the existence and uniqueness ofthe solution of linear differential equations with uncertainty.The first part mainly studied the distinct linear differential equations withinitial conditions. Given the conversion matrix and the solution of specificexpression. It has laid a good foundation for solving linear differential equationswith fuzzy initial value.The second part, we fuzzified the initial conditions studied in the first part. Thedefinition of transformation matrix and Weierstra standard elements are given. Anda method is proposed which transformed the differential equations with fuzzy initialvalue into distinct linear differential equations. The solution of distinct lineardifferential equations is given according to the conclusion in the first part. We alsoproved the relationship between the solution of differential equations with fuzzyinitial value and the solution of distinct linear differential equations, constructed thesolution of differential equations with fuzzy initial value, and proved the uniquenessof the solution.The third part, examples are given to verify the theory. The third part, wefuzzified the right side function which was studied in the second part. Then wetransformed the linear differential equations with uncertainty into distinct lineardifferential equations, and added the restrictive conditions, then the solutions ofequations were obtained, which were the solutions of the linear differentialequations with uncertainty. By the theorem proved in the second part, weconstructed the solution of differential equations with uncertainty, and proved theuniqueness of the solution.