M-linear differential equation systems and applications

In this thesis, systems of linear differential equations with local derivatives are analysed and new approaches to the solution of these systems are developed. The aim of the study is to analyse a general form of ℳ-derivative differential equation systems and apply these methods to real world problems. In this study, which consist of six chapters, the ℳ- ℒaplace transform is used as the basic tool for solving ℳ-derivative systems of linear differential equations. In addition, Cramer's methods is integrated with the local derivative and more comprehensive and general solution methods are obtained with the help of the Mittag-Leffler function. The effectiveness of these methods is supported by theoretical analyses and the advantages of the proposed methods are emphasised. Various real world problems such as SIR model, tumour growth model and beam applications are investigated. For these problems, solutions are obtained using both the classical Laplace transform and the ℳ- ℒaplace transform and these solutions are presented visually with graphs. This study aims to make an important contribution to the field of ℳ-derivative differential equations and demonstrates the potential of the ℳ- ℒaplace transform in mathematical modeling and engineering applications. It is thought that the methods and analyses obtained as a results of the study will provide a new perspective to the literature in the field of differential equations and fractional analysis.