The aim of this thesis is to develop discrete fractional models of tumor growth for a given data and to estimate parameters of these models in order to have better data tting. We use discrete nabla fractional calculus because we believe the discrete counterpart of this mathematical theory will give us a better and more accurate outcome.
This thesis consists of ve chapters. In the rst chapter, we give the history of the fractional calculus, and we present some basic de nitions and properties that are used in this theory. We de ne nabla fractional exponential and then nabla fractional trigonometric functions. In the second chapter, we concentrate on completely monotonic functions on R, and we introduce completely monotonic functions on discrete domain. The third chapter presents discrete Laplace N-transform table which is a great tool to nd solutions of -th order nabla fractional di erence equations. Furthermore, we nd the solution of nonhomogeneous up to rst order nabla fractional di erence equation using N-transform. In the fourth chapter, rst we give the de nition of Casoration for the set of solutions up to n-th order nabla fractional equation. Then, we state and prove some basic theorems about linear independence of the set of solutions. We focus on the solutions of up to second order nabla fractional di erence equation. We examine these solutions case by case namely, for the real and distinct characteristic roots, real and same, and complex ones. The fth chapter emphasizes the aim of this thesis. First, we give a vi brief introduction to parameter estimation with Gomperts and Logistic curves. In addition, we recall a statistical method called cross-validation for prediction. We state continuous, discrete, continuous fractional and discrete fractional forms of Gompertz and Logistic curves. We use the tumor growth data for twenty-eight mice for the comparison. These control mice were inoculated with tumors but did not receive any succeeding treatment. We claim that the discrete fractional type of sigmoidal curves have the best data tting results when they are compared to the other types of models
