Travelling waves and instability in a Fisher-KPP problem with a non-linear advection and a high order diffusion.

pre-print594 K

Non-Lipschitz heterogeneous reaction with a p-Laplacian operator.

The intention along this work is to provide analytical approaches for a degenerate parabolic equation formulated with a p-Laplacian operator and heterogeneous non-Lipschitz reaction. Firstly, some results are discussed and presented in relation with uniqueness, existence and regularity of solutions. Due to the degenerate diffusivity induced by the p-Laplacian operator (specially when ∇u=0, or close zero), solutions are studied in a weak sense upon definition of an appropriate test function. The p-Laplacian operator is positive for positive solutions. This positivity condition is employed to show the regularity results along propagation. Afterwards, profiles of solutions are explored specially to characterize the propagating front that exhibits the property known as finite propagation speed. Finally, conditions are shown to the loss of compact support and, hence, to the existence of blow up phenomena in finite time.post-print288 K

Travelling Waves Approach in a Parabolic Coupled System for Modelling the Behaviour of Substances in a Fuel Tank.

The aim of this work was to provide a formulation of a non-linear diffusion model with forced convection in the form of a reaction–absorption system. The model was studied with analytical and numerical approaches in the frame of the parabolic operators theory. In addition, the solutions are applied to a gas interaction phenomenon with the intention of producing an inerted ullage in an Airbus A320 aircraft centre fuel tank. We made use of the travelling wave (TW) solutions approach to study the existence of solutions, stability and the precise evolution of profiles. The application exercise sought to answer a key question for aerospace sciences which can be formulated as the time required to ensure an aircraft fuel tank is safe (inerted) to prevent explosion due to the presence of oxygen in the tank ullage.post-print721 K

Characterization of Traveling Waves Solutions to an Heterogeneous Diffusion Coupled System with Weak Advection.

The aim of this work is to characterize Traveling Waves (TW) solutions for a coupled system with KPP-Fisher nonlinearity and weak advection. The heterogeneous diffusion introduces certain instabilities in the TW heteroclinic connections that are explored. In addition, a weak advection reflects the existence of a critical combined TW speed for which solutions are purely monotone. This study follows purely analytical techniques together with numerical exercises used to validate or extent the contents of the analytical principles. The main concepts treated are related to positivity conditions, TW propagation speed and homotopy representations to characterize the TW asymptotic behaviour.post-print637 K

Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion.

It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.post-print287 K

A mathematical analysis of an extended MHD Darcy–Forchheimer type fluid.

The presented analysis has the aim of introducing general properties of solutions to an Extended Darcy–Forchheimer flow. The Extended Darcy–Forchheimer set of equations are introduced based on mathematical principles. Firstly, the diffusion is formulated with a non-homogeneous operator, and is supported by the addition of a non-linear advection together with a non-uniform reaction term. The involved analysis is given in generalized Hilbert–Sobolev spaces to account for regularity, existence and uniqueness of solutions supported by the semi-group theory. Afterwards, oscillating patterns of Travelling wave solutions are analyzed inspired by a set of Lemmas focused on solutions instability. Based on this, the Geometric Perturbation Theory provides linearized flows for which the eigenvalues are provided in an homotopy representation, and hence, any exponential bundles of solutions by direct linear combination. In addition, a numerical exploration is developed to find exact Travelling waves profiles and to study zones where solutions are positive. It is shown that, in general, solutions are oscillating in the proximity of the null critical state. In addition, an inner region (inner as a contrast to an outer region where solutions oscillate) of positive solutions is shown to hold locally in time.post-print2749 K

Invasive-invaded system of non-Lipschitz porous medium equations with advection.

This work provides analytical results towards applications in the field of invasive-invaded systems modelled with non-linear diffusion and with advection. The results focus on showing regularity, existence and uniqueness of weak solutions using the condition of a non-linear slightly positive parabolic operator and the reaction-absorption monotone properties. The coupling in the reaction-absorption terms, that characterizes the species interaction, impedes the formulation of a global comparison principle that is shown to exist locally. Additionally, the present work provides analytical solutions obtained as selfsimilar minimal and maximal profiles. A propagating diffusive front is shown to exist until the invaded specie notes the existence of the invasive. When the desertion of the invaded starts, the diffusive front vanishes globally and the non-linear diffusion concentrates only on the propagating tail which exhibits finite speed. Finally, the invaded specie is shown to exhibit an exponential decay along a characteristic curve. Such exponential decay is not trivial in the non-linear diffusion case and confirms that the invasive continues to feed on the invaded during the desertion.post-print379 K

Geometric Perturbation Theory and Travelling Waves profiles analysis in a Darcy–Forchheimer fluid model.

The intention along the presented analysis is to develop existence, uniqueness and asymptotic analysis of solutions to a magnetohydrodynamic (MHD) flow saturating porous medium. The influence of a porous medium is provided by the Darcy–Forchheimer conditions. Firstly, the existence and uniqueness topics are developed making used of a weak formulation. Once solutions are shown to exist regularly, the problem is converted into the Travelling Waves (TW) domain to study the asymptotic behaviour supported by the Geometric Perturbation Theory (GPT). Based on this, analytical expressions are constructed to the velocity profile for the mentioned Darcy–Forchheimer flow. Afterwards, the approximated solutions based on the GPT approach are shown to be sufficiently accurate for a range of travelling waves speeds in the interval [2.5, 2.8].post-print1640 K

Los Modelos Docentes en la Matemáticas Preuniversitarias y su Relación con los Diferentes Grupos de Edad de los Docentes en Formación

This paper addresses the mathematical education received during the pre-university stage based on the teaching-learning processes experienced by 225 students from the master’s degree in Teacher Training of Secondary, Baccalaureate, and FP and the Degree of Teacher of Primary Education of the Madrid Open University (UDIMA). For collecting the required information, a computerized questionnaire designed by the authors of this work and validated by the Ethics Committee of the Madrid Open University (UDIMA), has been used. The results of our study reveal the preservice teachers' memories about mathematics during the Primary and Secondary stages. Traditional teaching models, based on the repetition of calculation procedures, are the majority compared to other active teaching models. It is observed that a progressive increase in the methodologies supported by solving complex problems, detecting a moderate influence of the legislative changes produced in Spain in 2006. The mastery of classical teaching models and the moderate work around complex problems detected in pre-university education can be major constraints when developing new competency-based legislative approaches.En este trabajo se aborda la formación matemática recibida en la etapa de formación preuniversitaria a través del recuerdo de los procesos de enseñanza-aprendizaje vividos, de 225 estudiantes procedentes del Máster de Formación del Profesorado de Secundaria, Bachillerato y FP y del Grado de Maestro de Educación Primaria de la Universidad XXXX. Para la recogida de información se ha empleado un cuestionario informatizado diseñado por los autores del trabajo y que ha sido validado por el Comité de Ëtica de la XXXX. Mediante estadística descriptiva y el análisis de correlaciones entre las diferentes variables se ha dado respuesta a cuatro hipótesis de investigación. Los resultados del estudio revelan unos recuerdos escolares en relación a la materia de matemáticas de la etapa de Primaria y Secundaria donde los modelos docentes tradicionales, basados en la repetición de procedimientos de cálculo, son mayoritarios frente a otros modelos docentes de corte activo. Adicionalmente se observa un aumento progresivo de la importancia de resolución de problemas complejos detectándose una influencia moderada de los cambios legislativos producidos en España en el año 2006. El dominio de los modelos docentes clásicos y el moderado trabajo en torno a problemas complejos detectados en la formación recibida puede suponer una restricción importante a la hora de desarrollar los nuevos enfoques legislativos basados en competencias

Front propagation in the interaction of gases to model a fuel tank inerting process with a nonlinear parabolic operator

Purpose: The objective of this study is to model the propagating front in the interaction of gases in an aircraft fuel tank. To this end, we introduce a nonlinear parabolic operator, for which solutions are shown to be regular. Design/methodology/approach: The authors provide an analytical expression for the propagating front, that shifts any combination of oxygen and nitrogen, in the tank airspace, into a safe condition to avoid potential explosions. The analytical exercise is validated with a real flight. Findings: According to the flight test data, the safe condition, of maximum 7% of oxygen, is given for a time t = 45.2 min since the beginning of the flight, while according to our analysis, such a safe level is obtained for t = 41.42 min. For other safe levels of oxygen, the error between the analytical assessment and the flight data was observed to be below 10%. Originality/value: The interaction of gases in a fuel tank has been little explored in the literature. Our value consists of introducing a set of nonlinear partial differential equations to increase the accuracy in modeling the interaction of gasses, which has been typically done via algebraic equations.2022-2