A Study on Subjectivities of Type 1 and 2 in Parameters of Differential Equations

Abstract

<span style="color: #000000;"> This study presents the </span><span style="color: #000000;">Verhulst's</span><span style="color: #000000;"> model for the analysis of population growth with the rate of </span><span style="color: #000000;">reproductivity</span><span style="color: #000000;"> depending on the fertility rate and the country economic development. These linguistic variables are defined through Fuzzy Rule-Based Systems (</span><span style="color: #000000;">FRBS</span><span style="color: #000000;">). The analysis is made for </span><span style="color: #000000;">FRBS</span><span style="color: #000000;"> types 1 and 2 where in the first case, the inference method used is </span><span style="color: #000000;">Mamdani's</span><span style="color: #000000;"> and the </span><span style="color: #000000;">defuzzification</span><span style="color: #000000;"> is the center of gravity. For type 2 </span><span style="color: #000000;">FRBS</span><span style="color: #000000;"> is used the inference method by </span><span style="color: #000000;">Karnik</span><span style="color: #000000;"> Mendel (KM) where the output is </span><span style="color: #000000;">defuzzificated</span><span style="color: #000000;"> by the </span><span style="color: #000000;"> Type Reducer method. A comparative study of the solutions of the </span><span style="color: #000000;">Verhulst's</span><span style="color: #000000;"> model for both techniques is performed. I has been noticed that the region determined by the solutions corresponding to the minimum and maximum rate resulting from </span><span style="color: #000000;">FRBS</span><span style="color: #000000;"> type 2 is contained in the region built similarly from type 1, showing a higher accuracy in the response.</span>

A Tour of Type-1 and Interval Type-2 Fuzzy Sets Theory

“…Depicted in Fig. 2.24b are the membership functions of input x 2 : upper μ X 21 and lower μ X 21 , corresponding to the fuzzy set X 21 ; and the membership functions: upper μ X 22 , and lower, μ X 22 , for the fuzzy set X 22 [20,49]. The universes for the variables x 1 and…”