Approximate Series Solution of Nonlinear, Fractional Klein-Gordon Equations Using Fractional Reduced Differential Transform Method

Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems

Modern technology makes it easier to store datasets, but extracting and isolating useful information with its full meaning from this data is crucial and hard. Recently, several algorithms for clustering data have used complex fuzzy sets (CFS) to improve clustering performance. Thus, adding a second dimension (phase term) to the range of membership avoids the problem of losing the full meaning of complicated information during the decision-making process. In this research, the notion of the complex shadowed set (CSHS) was introduced and considered as an example of the three region approximations method simplifying processing with the support of CFS and improving the representation of results attained within. This notion can be founded by extending the shadowed set codomain from $ \left\{0, \left[0, 1\right], 1\right\} $ into $ \left\{0{e}^{i\theta }, \left[0, 1\right]{e}^{i\theta }, 1{e}^{i\theta }\right\} $. The significance of CSHS was illustrated by giving an example. Additionally, some properties of the CSHS were examined. The basic CSHS operations, complement, union, and intersection were investigated with their properties. Finally, an application in decision-making was illuminated to support the present notion.

This article offers an analytical solution for the fractional Riccati differential equation in three distinct cases. These cases are determined by the discriminant and the analytical solution based on the properties of the Caputo-Fabrizio fractional derivative and integral. Several examples were tested using this analytical solution. It is noteworthy that various methods have yielded related results as indicated in the literature.