In this paper, we have developed a new method called Generalized Taylor collocation method (GTCM), which is based on the Taylor collocation method, to give approximate solution of linear fractional differential equations with variable coefficients. Using the collocation points, this method transforms fractional differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Generalized Taylor coefficients. Generally, the method is based on computing the Generalized Taylor coefficients by means of the collocation points. This method does not require any intensive computation. Moreover, It is easy to write computer codes in any symbolic language. Hence, the proposed method can be used as effective alternative method for obtaining analytic and approximate solutions for fractional differential equations. The effectiveness of the proposed method is illustrated with some examples. The results show that the method is very effective and convenient in predicting the solutions of such problems.

For decades, the researchers have developed many ways as optimization procedures with the aim of find the best solution in short time for many problems under certain conditions in the field of engineering, medicine and banking. These ways also used for parameter updating of algorithms. The most popular Optimization algorithms methods known are mining classification and clustering. In this article, the clustering used to identify the most important point in the best cluster centers of set data. Artificial Algae Algorithm (AAA) optimization algorithm used in the clustering process and implemented on UCI datasets. Balance, Breast Cancer Wisconsin Diagnostic, Breast Cancer Wisconsin original, Pima Diabetes, Glass, Iris, Wine, Urban Land Cover and Hill Valley UCI datasets used to assess the performing of the Algae Algorithm-based clustering algorithm. Euclides method used to calculate the distance between the data. The performance of the AAA based clustering algorithm, Total square distance values in different iteration numbers calculated for each data set. The total square error rate value calculated for each iteration and as the number of iterations progresses, the total square error rate value decreases smoothly. The obtained results compared with k-means, Differential Evolution (DE), Genetic Algorithm (GA), Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA) clustering algorithms. According to the experimental results in this study, the proposed AAA-based clustering algorithm achieved better results in iris and wine data sets compared to other clustering algorithms, while it obtained close to good results in other data sets. As a result, the Artificial Algae Algorithm-based clustering algorithm showed that the method showed a stable appearance and the performance of the clusters also increased, which shows that this study successfully achieved its purpose.

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