The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the method of Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certa...

The real number system is a fundamental tool for rigorous demonstrations of the differential and integral calculus results. Even after a century of formalization on solid foundations, discussions about the construction of this field are generally omitted in advanced courses such as Real Analysis. In the present work, we present a comprehensive review on the construction and characterization of the real numbers field. The presentation focuses on the construction through Cauchy sequences of rational numbers. The notion of completeness is delimited differently from completeness when Dedekind’s cut construction is used. The results indicate Q and R Archimedean as a necessary condition for these two notions of completeness to be equivalent. To illustrate this, inspired by the work of Leon W. Cohen and Gertrude Ehrlich, we present an example of a Cauchy-complete non-Archimedean ordered ...

Demuestra que un campo de Morse - Smale, definida en una variedad compacta de dimensión dos, es estructuralmente estable, esto es, que el comportamiento topológico de sus órbitas no se altera mediante pequeñas perturbaciones del campo.

Let, (V, a(u, v)), (H, (u, v)) Hilbert spaces and the immersion V H is dense and compact. We consider the abstract problem: { Bu" + M(|u|)Au = f                    (*) Where B : H → H is a linear simetric and positive operator, (W, (W,|u|0), is a C1class Banach space, the non linear M is  class and strictively positive. In this article we prove the existence and unicity of the local solution of the problem (*).

Let, (V, a(u, v)), (H, (u, v)) Hilbert spaces and the immersion V H is dense and compact. We consider the abstract problem: { Bu" + M(|u|)Au = f                    (*) Where B : H → H is a linear simetric and positive operator, (W, (W,|u|0), is a C1class Banach space, the non linear M is  class and strictively positive. In this article we prove the existence and unicity of the local solution of the problem (*).

In this paper we are concerned with the stability of solutions for the wave equation with a viscoelastic Boundary condition and source term by using the potential well method, the multiplier technique and unique continuation theorem for the wave equation with variable coefficient.

En este artículo estamos interesados en la estabilidad de las soluciones de una ecuación de onda con una condición de frontera viscoelástica y un término fuente, usaremos el método potencial, la técnica de multiplicadores y el teorema de unicidad para una ecuación de onda con coeficientes variables.

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