Ne integral transform and applications

This thesis comprises both theoretical and applied studies in the field of fractional derivatives and integral transforms, and it consists of six chapters. In the first chapter, the aim, scope, and position of the study within the existing literature are presented in a general framework. The second chapter provides the theoretical foundation of the study by introducing integral transforms, fractional derivatives, and related fundamental concepts. In the third chapter, the newly introduced NE integral transform is examined in detail; its definition, basic properties, and mathematical structure are thoroughly investigated. The fourth chapter demonstrates the applicability of the NE transform to linear fractional differential equations; various models involving fractional derivatives such as the gross domestic product model, various economic models, the falling body problem, non-homogeneous fractional differential equations, and mathematical models corresponding to electrical circuits are discussed. In the fifth chapter, approximate solutions of nonlinear differential equations are obtained using the NE integral transform. Finally, in the last chapter, the findings of the study are summarized, the contributions are evaluated, and suggestions for future research are provided.