A solution of fractional-order mathematical model

This thesis consists of five parts. In the first part, definitions of fractional derivative and integral are given in chronological order. Also the history of fractional derivative and integral is explained. In the second part, the definition, theorem and the lemmas that used in the thesis related to derivative and are given integral operators of fractional order under the heading of the theoretical base. In addition, the Keller-Segel model to be used in the thesis is introduced and necessary information is given about the model. In the third chapter, Caputo, Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu fractional derivative and integral operators are introduced and their properties are shown. In addition to these, Laplace transforms of operators Caputo-Fabrizio and Atangana-Baleanu are given. In the fourth chapter, the Keller-Segel model is modified for each operator and the existence and uniqueness solutions are examined. By using different numerical solutions, coupled solutions of the system have been obtained. Detailed analyzes and comparisons are made using Mathematica program for obtained coupled solutions. The fifth section is devoted to discussion and conclusion.