From the complex variables to the Integral Transforms applied to the Fractional Calculus.

• Grant number: 23/03164-6
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): April 01, 2023
• Effective date (End): March 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Rubens de Figueiredo Camargo
• Grantee: Dennis Muniz de Souza
• Host Institution: Faculdade de Ciências (FC). Universidade Estadual Paulista (UNESP). Campus de Bauru. Bauru , SP, Brazil

Abstract

The aim of this project is to study, at first, complex analysis, in particular Cauchy's integral theorem, its consequences, and applications. As a consequence of Cauchy's integral theorem we have the residue theorem, a key element for the development of this work.In a second moment, the focus will be on the study of integrals and derivatives of non-integer order, i.e., the so-called Fractional Calculus, its origin, main definitions, properties, and special functions, aiming to work with modeling done with differential equations of non-integer order, in order to refine the description given by the respective integer order equation.With regard to applications, it is intended to unify the two previous parts, that is, to emphasize the calculation of real integrals via complex variables, and inversion of Laplace and Fourier transforms, with the aim of using this knowledge in the study of integral transforms as a methodology for solving both ordinary and fractional differential equations. In addition, to study, in an original way, the modeling with fractional differential equations, for specific engineering problems and compare their solutions with the respective integer order solutions, in order to verify which order of the fractional derivative makes the description of the problem original more realistic. Finally, it should be noted that the LaTeX, MatLab, and Mathematica programs will be used in the development of this project, which will be of great value for this and future scientific works by the candidate, who has academic pretensions.