Licenciatura em Matemática (Sede)
URI permanente desta comunidadehttps://arandu.ufrpe.br/handle/123456789/24
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APP - Artigo Publicado em Periódico
TAE - Trabalho Apresentado em Evento
TCC - Trabalho de Conclusão de Curso
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Resultados da Pesquisa
Item A Transformada de Fourier: da motivação à equação do calor numa barra infinita(2024-10-02) Basilio, Evellyn Karoline Alves Freitas; Freitas, Lorena Brizza Soares; http://lattes.cnpq.br/2302580820419163; http://lattes.cnpq.br/8020476628705052This paper presents an overview of the Fourier Transform, its properties and applications, with a particular emphasis on the space of rapidly decreasing functions, which is known as the Schwartz space. The Fourier transform is a fundamental tool in mathematical analysis, employed for the resolution of partial differential equations. This enables differential equations to be converted into more readily manageable algebraic equations. The methodology adopted is based on an analysis of the Schwartz space and its properties, which are essential to ensure the proper behavior of functions in the context of the Fourier transform. Subsequently, the principal properties of the Fourier Transform are examined, including its linearity, differentiability and applicability within the context of Schwartz space, as well as its inverse transform. This work was developed based on an investigation of bibliographical references and theoretical materials listed at the end of this paper. The results obtained demonstrate the significance of the Fourier Transform in determining the solution to the heat equation in an infinite bar, thereby facilitating the identification of a potential solution for the associated partial differential equation. It is our intention that this work provides a clear and comprehensive overview of the Fourier Transform, its properties and its theoretical applications, thus establishing it as an essential tool in analysis.Item Existência e unicidade de solução para problemas envolvendo o operador Laplaciano(2019-12-17) Nunes, Thays Ingrid dos Santos; Araújo, Yane Lísley Ramos; http://lattes.cnpq.br/6642941380570085; http://lattes.cnpq.br/3740642465035306In this work we approach some basic concepts related to the theory of partial differential equations guaranteeing the existence of solution for problems involving the Laplacian operator. Initially, we use the method of variable separation and Fourier Analysis tools to ensure the existence of classical solutions to Dirichlet problems in the rectangle and unit disk involving the Laplace equation, as well as a maximum principle to ensure the uniqueness of the solution. Then, we use results from Functional Analysis and Sobolev spaces to ensure under certain conditions that there is only one weak solution to the Dirichlet problem involving the Poisson equation.Item Equação da onda: soluções de problemas de valores iniciais e de fronteira a partir da análise de Fourier(2020-12-19) Lopes, Daniel César Pereira; Silva, Clessius; http://lattes.cnpq.br/2401078773322406; http://lattes.cnpq.br/0084883368794758In this work, we will study the Wave Equation, an important equation in the study of Partial Differential Equations. We will work on problems involving the finite-string wave equation with fixed ends, the infinite string, and the semi-infinite string. However, to get to such problems, we will need to do a study about the Fourier Series, studying the convergence of such series, so we will need some Real Analysis. In addition, we will study some important inequalities such as: Bessel Inequality, Cauchy-Schwarz Inequality and Minkowski Inequality. For that, we have a basis for solving the EDP’s in question.