Partial Differential Equations and the Existence and Uniqueness Theorem for the Cauchy Problem, linear case

Authors

Keywords:

Initial value problem, Variable changes, characteristic curve

Abstract

If Ω of R2 is a open region, I an open interval and the linear equation of first order non homogeneous in its most general forma (x,y)ux + b(x,y)uy = c(x,y) and the initial condition u(σ(t),ρ(t))=f(t) that a, b and c are C1 class in Ω that contains the smoth curve γ=(σ(t),ρ(t)), t∈I, initial curve of problem. This type of problem is called a Cauchy Problem. To solve this Cauchy problem is fundamental the concept of characteristic curves of the equation, since these represent the starting point in the search for solution to the problem. Also, we will see that the way in which the characteristic curves intersect the initial curve (σ(t),ρ(t)) given determines whether the problem will have unique solution, endless solutions or if the solution does not exist. We will see the Existence and Uniqueness Theorem that gives us the necessary conditions for the existence and uniqueness of solution of Cauchy Problem cited. It is important to note that the Existence and Uniqueness Theorem guarantees just local results, since the behavior of the characteristic curves away from the initial curve can become to complex.

 

Author Biography

Cristiane C. F. Cintra, Universidade Federal de Alfenas

Doutor em Estatística e Experimentação Agropecuária com Pós-doutorado em Estatística Multivariada

References

BASSANEZI, R. C.; JÚNIOR, W. C. F. Equações Diferenciais com Aplicações. São Paulo: Ed. Harbra Ltda., 1988.

BIEZUNER, R. J. Introdução às Equações Diferenciais Parciais. Notas de aula. Disponível em: http://www.mat.ufmg.br/~rodney/notas_de_aula/iedp.pdf. Acesso em: 05 out 2012.

CULLEN, M. R.; ZILL, D.G. Equações Diferenciais. Vol. 2, 3. ed., São Paulo: Ed. Pearson, 2001.

FERREIRA, A.P. Problema de Cauchy para Equações Diferenciais Ordinárias. In: XI Encontro Anual de Iniciação Científica, Maringá, PR: Universidade Estadual de Maringá, 2002.

IÓRIO, V. EDP - Um curso de Graduação. Coleção Matemática Universitária, 2.ed.,Rio de Janeiro: IMPA, 2007.

Published

31-12-2012

How to Cite

Cintra, C. C. F., & Goecking, L. B. (2012). Partial Differential Equations and the Existence and Uniqueness Theorem for the Cauchy Problem, linear case. Sigmae, 1(1), 44–56. Retrieved from https://publicacoes.unifal-mg.edu.br/revistas/index.php/sigmae/article/view/92