β-uyumlu kesirli türev ile modellenen bazı fiziksel ve biyolojik problemlerin tam çözümleri

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Thesis Type: Postgraduate

Institution Of The Thesis: Ataturk University, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, Turkey

Approval Date: 2021

Thesis Language: Turkish

Student: SADULLAH BULUT

Supervisor: Mesut Karabacak

Open Archive Collection: AVESIS Open Access Collection

Abstract:

Purpose: In this thesis, it has been presented reviewing and validating of existing solutions with a different approach nonlinear partial differential equations modeling some physical and biological problems that has been found their exact solutions before. Method: First of all, derivative definitions in the literature were given and the difference of β-conformable fractional derivative from other derivative definitions was explained. Then (G'/G)-expansion method was explained and the exact solutions of equations discussed were obtained by redefining with 𝛽-conformable fractional derivative by (G'/G)-expansion method. Findings: The exact solutions of the space-time fractional Fitzhugh-Nagumo, Dodd-Bullough Mikhailov, (2 + 1) Dimensional Cubic Klein-Gordon and Phi-Four equations defined with the help of 𝛽-conformable fractional derivative of Atangana were obtained depending on 𝛽. It was seen that these solutions were compatible with the exact solutions found in the literature before for 𝛽 = 1. In addition, exact solutions for different fractional 𝛽 values were plotted. Results: Since the 𝛽-conformable fractional derivative of Atangana does not require fractional integrals as in other fractional derivative definitions, it has been observed that the mathematical operations performed shorten and become easier. Keywords: β- conformable Fractional Derivative of Atangana, (G'/G)-expansion Method, Neural transmission, Space-Time Fractional Differential Equations