Fourier analysis and applications to partial differential equations

• Grant number: 25/08006-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2025
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marta Cilene Gadotti
• Grantee: Étore Ferreira Val
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

One of the existing methods to resolve certain partial differential equations (EDP) is known to be known as Fourier's method or variable separation method. ToApplying this technique is necessary first to analyze what type of function can be re \ -pre \ \ -The by a trigonometric series and know results about the convergence of these series. The topics - derivation and integration of these series; the space of limited variation functions; Identities valid in the space $ {l}^1 $ - are important and must be previously established for such study. Using Fourier's analysis it is possible to reduce the problem - solve EDP - the search for solutions of a set of ordinary differential equations (EDO) associated with EDP. The idea to resolve the EDO is to perform the process of obtaining eigenvalues ¿¿and autofunctions, see references [1] and [2]. Finally the EDP solution is represented by an infinite series involving the set ofAssociated EDO solutions. These solutions must satisfy the initial and boundary conditions of each problem addressed.This EDP resolution technique can be applied to solve the wave, heat and laplace equation. So we want with this project to add to the student's academic backgroundFourier analysis topics and basic concepts of partial differential equations theory - topics of great importance in mathematics and applied mathematics - which will be developed in detail, motivated by problems that portray certain physical phenomena.