ABSTRACT This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces H k ( ℝ d ) × H l ( ℝ d ) × H l - 1 ( ℝ d ). We recall the well-posedness and ill-posedness results known to date and establish new ill-posedness results. We prove C 2 ill-posedness for some new indices ( k , l ) ∈ ℝ 2. Moreover, our results are valid in arbitrary dimension. We believe that our detailed proofs are built on a methodical approach and can be adapted to obtain similar results for other systems and equations. . wellposedness well posedness illposedness ill Moreover dimension equations

Resumen En este artículo introducimos un operador que puede ser pensado como una derivada de orden variable, i.e. el orden de la derivada es una función. Demostramos varias propiedades de este operador, por ejemplo, obtenemos una fórmula generalizada de Leibniz, teoremas de valor medio de Rolle y Cauchy y un polinomio de tipo Taylor. Más aún, obtenemos su operador inverso. También con esta derivada analizamos la existencia de soluciones de un problema no lineal de valor en la frontera de tres puntos de “orden variable”.

Abstract In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz’s formula, Rolle and Cauchy’s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

Abstract The Hankel transformation method was used in this work to determine the normal and shear stress distributions due to point, line and distributed loads applied to the surface of an elastic media. The elastic media considered in this study was assumed to be inextensible in the horizontal directions, and the only non-vanishing displacement component is the vertical component. Such materials were first considered by Westergaard as models of elastic half-space with alternating layers of soft and stiff materials with the stiff materials of negligible thickness and so closely spaced that the composite material characteristics is idealised as isotropic and homogeneous. The study adopted a displacement formulation. The Hankel transformation was applied to the governing Cauchy - Navier differential equation of equilibrium to reduce the problem to a second order ordinary differential equation (ODE) in terms of the deflection function in the Hankel transform space. Solution of the ODE subject to the boundedness condition yielded bounded deflection function in the Hankel transform space. Equilibrium of the internal vertical forces and the external applied load, was used to obtain the constant of integration, for the deflection function in the Hankel transform space. Inversion yielded the deflection in the physical domain variable. The stress displacement equations were then used to determine the Cauchy stresses. The vertical stress distributions due to line and distributed load over a rectangular area were also determined by using the point load solution for vertical stresses as Green functions, and then performing integration along the line and over the rectangular area of the load. The results obtained for vertical stresses due to point, line and distributed loads were determined in terms of dimensionless influence coefficients which were presented. The results obtained for the deflection, normal and shear stresses due to point, line and distributed load agreed with the solutions originally presented by Westergaard who used a stress function method.

Resumen [12] En el presente artículo analizaremos el problema de Cauchy local asociado a la ecuación de Korteweg-De Vries (KdV) en H s con s > 3/2. El objetivo de este trabajo, consiste en establecer la buena formulación local del problema cuando u 0 ∈ H s , s > 3/2, para ello aplicaremos la teoría cuasi-lineal de Kato, el cual consta de (06) hipótesis, en el caso lineal y (08) hipótesis en el caso no-lineal. En la solución del problema de Cauchy para la ecuación de evolución cuasi-lineal, nos basaremos en el teorema del punto fijo de Banach.

Abstract [16] In the present paper we will analyze the local Cauchy problem associated with the Korteweg-De Vries (KdV) equation in H s with s > 3/2. The objective of this work is to establish the good local formulation of the problem when u 0 ∈ H s , s > 3/2, for this we apply the quasi-linear theory of Kato, which consists of (06) hypotheses, in the linear case and (08) hypotheses in the non-linear case. In the solution of Cauchy’s problem for the quasi-linear equation of evolution, we will rely on Banach’s fixed-point theorem. [17] Keywords: local existence and uniqueness theorems; existence of generalized solutions; applications of PDE in areas other than physics.

Resumo Após uma revisão na literatura vimos que o Problema Restrito de Três Corpos (PRTC) é um tema com algumas publicações voltadas, principalmente, para a resolução do problema matemático, com muito poucas aplicações em situações reais e contextualizadas. Além disso, ao analisarmos alguns livros textos de mecânica clássica utilizados nos cursos de física, observamos que o PRTC não é desenvolvido e explorado didaticamente. Apoiados nessas análises o nosso objetivo neste trabalho é apresentar, desenvolver e aplicar o PRTC para simular as órbitas de dois astros do sistema solar: a do asteroide troiano da Terra, 2010 TK7, e a libração de Plutão devido a Netuno. Para resolver as equações diferencias do PRTC utilizamos o método de Cauchy em um código escrito na linguagem de programação Python. Nos dois casos estudados conseguimos verificar as trajetórias, valores e comportamentos orbitais dos astros. Por isso, entendemos que o PRTC pode ser aplicado para simular o movimento de diversos astros do sistema solar e, consequentemente, ser utilizado como um recurso pedagógico para contextualizar conceitos de mecânica celeste e dinâmica orbital. Uma vez que ele se relaciona com o estudo das equações diferencias, com a gravitação universal e com os métodos numéricos.

Abstract After a review in the literature we have seen that the Restricted Three-Body Problem (RTBP) is a topic with some publications focused, primarily, on resolution of the mathematical problem, with very few application in real and contextualized situations. In addition, in analyzing textbooks of classical mechanics we observed that the RTBP is not developed and didactically explored. Based on this facts, our objective in this work is to present, develop and apply the RTBP to simulate the orbits of two objects in the solar system : motion of Earth-trojan asteroid 2010 TK7 and libration of pluto due to Neptune. To solve differential equations of RTBP we uses the cauchy method in a code written in the Python programming language. With this work we hope to demonstrate that the study of the RTBP can be used to contextualize some concepts of celestial mechanics and orbital dynamics.Since it is related to the study of differential equations, with universal gravitation and numerical methods.

Abstract Introduction A monodomain model of electrical activity for an isolated ventricle is formulated. This model is written as a reaction diffusion PDE coupled to an ODE, The Rogers-Mculloch model is used to represent the electrical activity through the cell membrane. Method We give a definition of weak and strong solution of the variational Cauchy problem associated to the monodomain model. A sequence of approximate solutions of Faedo-Galerkin type is proposed. Results It is shown that the sequence of approximate solutions converge to a weak solution according to the proposed definition. Finally, we have that this weak solution is also a strong solution. Conclusion The monodomain model of electrical activity in an isolated ventricle that is proposed has a weak solution in an appropriate sense. In addition, this weak solution is also a strong solution.

Resumen Introducción Se formula un modelo de monodominio de actividad eléctrica en un ventrículo aislado. Este modelo se escribe como una EDP de tipo reacción difusión acoplada a una EDO, se utiliza el modelo de Rogers-Mculloch para representar la actividad eléctrica a través de la membrana celular. Método Se proponen definiciones de solución débil y fuerte respectivamente para el problema de Cauchy variacional asociado al modelo de monodominio. Se propone una sucesión de soluciones aproximadas de tipo Faedo-Galerkin. Resultados Se demuestra que la sucesión de soluciones aproximadas converge a una solución débil según la definición que se propone. Finalmente, se obtiene que la solución débil es también una solución fuerte. Conclusión El modelo de monodominio de actividad eléctrica en un ventrículo aislado que se propone tiene solución débil en un sentido apropiado. Además, esta solución débil también es una solución fuerte.

Resumen En este artículo se considera el problema de Cauchy para un sistema no simétrico de tipo Keyfitz-Kranzer y utilizando argumentos de viscosidad nula junto con el método de compacidad compensada se obtiene soluciones débiles entrópicas globales. La principal dificultad es obtener estimaciones uniformemente acotadas en el método de viscosidad y en este trabajo se estudia.

Abstract In this paper we consider the Cauchy problem for a particular non-symmetric Keyfitz-Kranzer type system, by using the vanishing viscosity method coupled with the compensated compactness method we get global bounded entropy weak solutions. The main difficulty is to get uniformly bounded estimates on the viscosity method and in this paper is studied.

Abstract An accurate modeling of materials is essential to obtain reliable results in fields calculation. The Jiles-Atherton approach is widely used for modeling the magnetic hysteresis and depends on its set of five parameters to properly represent material. In this article is proposed an original methodology for obtaining this set of parameters avoiding the derivatives rough calculation and using the calculation of integrals. From the model equations, a new methodology with two nonlinear algebraic systems of five equations in five unknowns is obtained. The initial magnetization curve, the anhysteretic curve and filtering data are not necessary. The proposed methodology also does not restrict the search space of parameters. The parameters assume values in the interval (0,∞). Calculated data were compared with experimental data to validate the methodology. The simulations showed that the proposed method can obtain an accurate set of parameters from a single experimental hysteresis loop and with low computational effort.

Abstract The theoretical analysis for the Brazilian test is a classical plane stress problem of elasticity theory, where a vertical force is applied to a horizontal plane, the boundary of a semi-infinite medium. Hypothesizing a normal radial stress field, the results of that model are correct. Nevertheless, the superposition of three stress fields, with two being based on prior results and the third based on a hydrostatic stress field, is incorrect. Indeed, this work shows that the Cauchy vectors (tractions) are non-vanishing in the parallel planes in which the two opposing vertical forces are applied. The aim of this work is to detail the process used in the construction of the theoretical model for the three stress fields used, with the objective being to demonstrate the inconsistency often stated in the literature.

Resumen Probamos que el problema de valor inicial asociado a una perturbación de la ecuación de Benjamín-Ono o ecuación de Chen-Lee u t + uu x + β H u xx + η (H u x - u xx) = 0, donde x ∈ T, t > 0, η > 0 y H denota la transformada de Hilbert usual, es localmente y globalmente bien planteado en espacios de Sobolev Hs(T) para cualquier s > -½. También probamos un tipo de mal planteamiento cuando s < -1.

Abstract We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation u t + uu x + β H u xx + η (H u x - u xx) = 0, where x ∈ T, t > 0, η > 0 and H denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces Hs(T) for any s > -½. We also prove some ill-posedness issues when s < -1.

Resumen Estudiamos la convergencia de las formulaciones semidiscreta y completamente discreta de un método espectral Fourier-Galerkin para aproximar las soluciones de una ecuación no lineal Benney-Luke-Paumond que modela ondas largas con pequeña amplitud que se propagan sobre un canal raso con fondo plano. La precisión del método numérico se verifica usando algunas soluciones de onda solitaria exactas. A fin de aplicar el esquema Fourier-espectral en un contexto no periódio, aproximamos el problema de valor inicial con x ∈ ℝ por el correspondiente problema de Cauchy periódico para x ∈ [0,L], con un periodo espacial L grande.

Abstract We study convergence of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a nonlinear Benney-Luke-Paumond equation that models long water waves with small amplitude propagating over a shallow channel with flat bottom. The accuracy of the numerical solver is checked using some exact solitary wave solutions. In order to apply the Fourier-spectral scheme in a non periodic setting, we approximate the initial value problem with x ∈ ℝ by the corresponding periodic Cauchy problem for x ∈ [0,L], with a large spatial period L.

In this note we establish a result of differentiability for the mild solution of the inhomogeneous abstract Cauchy problem when the underlying space is reflexive