Thesis (M.S.)--University of Wisconsin--Madison, 1967. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Šiame darbe išnagrinėta išsigimstanti keturių pirmos eilės dalinių išvestinių diferencialinių lygčių sistema. Dalinių išvestinių diferencialinių lygčių sistemai spręsti pritaikytas apibendrintų laipsninių eilučių metodas. Rasti analiziniai šios sistemos sprendiniai ir ištirtos jų savybės išsigimimo daugdaros taškų aplinkoje. Apibendrintų laipsninių eilučių metodas gali būti pritaikytas sprendžiant panašios struktūros dalinių išvestinių diferencialines lygtis, kurių eilė išsigimsta. Darbe gauti rezultatai gali būti pritaikomi modeliuojant ir tiriant realius procesus. / In this work the system of four degenerating differential equations with partial derivatives of first order was studied. For the solution of system of differential equations with partial derivatives the method of generalized power series was applied. Analytical solutions of this system were found and properties of solutions on neighbourhood of points of degeneration manifold were investigated. The method of generalized power series can be applied to the solution of systems of differential equations with partial derivatives of similar structure, which order is degenerating. The results, which were obtained in this work, can be applied to modelling and studying the real processes.

En esta tesis, se utilizan t ecnicas matem atico-epidemiol ogicas para modelar el rendimiento acad emico en Espa~na (prestando especial atenci on en el fracaso escolar) para comprender mejor los mecanismos detr as de esta importante cuesti on, as como para predecir c omo evolucionar an los resultados acad emicos en el Bachillerato espa~nol en los pr oximos a~nos. El nivel educativo de Bachillerato en Espa~na est a formado por los dos ultimos cursos antes de acceder a la universidad o al mercado de trabajo y corresponde a los estudiantes de 16 18 a~nos. Este nivel educativo es muy importante para la formaci on de los estudiantes ya que representa un periodo en el que deber an tomar importantes decisiones sobre el futuro acad emico y profesional. En primer lugar, en el Cap tulo 2, se presenta un modelo determinista donde se analiza el rendimiento acad emico asumiendo que la actitud negativa de los alumnos de Bachillerato puede ser debida a su comportamiento aut onomo y la in uencia de compa~neros con malos resultados acad emicos. Luego, en el Cap tulo 3, se mejora el modelo basado en la idea de que no s olo los malos h abitos acad emicos se transmiten socialmente sino tambi en los buenos h abitos de estudio. Adem as, descomponemos los par ametros de transmisi on de h abitos acad emicos con el n de analizar con m as detalle qu e grupos de estudiantes son m as susceptibles a ser in uenciados por compa~neros con buenos o malos h abitos acad emicos. El abandono escolar tambi en han sido incluido en este modelo. El enfoque adoptado permite proporcionar predicciones deterministas y con intervalos de con anza de la evoluci on del rendimiento escolar (incluyendo las tasas de abandono) en Bachillerato en Espa~na en los pr oximos a~nos. Este enfoque, adem as, nos permite modelar el rendimiento acad emico en otros niveles educativos del sistema acad emico espa~nol o de fuera de Espa~na tal y como se muestra en el Cap tulo 4, donde el modelo se aplica satisfactoriamente al sistema acad emico actual de la regi on alemana de Renania del Norte-Westfalia. Para concluir esta tesis, proporcionamos una estimaci on de los costes relacionados con el rendimiento acad emico espa~nol basado en nuestras predicciones. Esta estimaci on representa la inversi on en Bachillerato por parte del Gobierno espa~nol y las familias en los pr oximos a~nos, con especial atenci on en los grupos de estudiantes que no promocionan y abandonan en los diferentes cursos acad emicos. / In this dissertation, we use epidemiologic-mathematical techniques to model the academic performance in Spain (paying special attention on the academic underachievement) to understand better the mechanisms behind this important issue as well as to predict how academic results will evolve in the Spanish Bachillerato over the next few years. The Spanish Bachillerato educational level is made up of the last courses before accessing to the university or to the work market and corresponds to students of 16¿18 years old. This educational level is a milestone in the career training of students because it represents a period to make important decisions about academic and professional future. In a rst step, in the Chapter 2 we will present a deterministic model where academic performance is analyzed assuming the negative attitude of Bachillerato students may be due to their autonomous behavior and the in uence of classmates with bad academic results. Then, in the Chapter 3, the model is improved based on the idea that not only the bad academic habits are socially transmitted but also the good study habits. Besides, we decompose the transmission academic habits into good and bad academic habits, in order to analyze with more detail which group of students are more susceptible to be in uenced by good or bad academic students. The consideration of quantifying the abandon rates is also a new issue dealt with in it. The adopted approach allow to provide both punctual and con dence intervals predictions to the evolution of academic performance (including the abandon rates) in Bachillerato in Spain over the next few years. The adopted approach allows us to model academic performance in academic levels other than Bachillerato and/or beyond the Spanish academic system. This issue is assessed in Chapter 4, where the model is satisfactorily applied to the current academic system of the German region of North Rhine-Westphalia. To conclude this dissertation, we provide an estimation of the cost related to the Spanish academic underachievement based on our predictions. This estimation represents the investment in the Spanish Bachillerato from the Spanish Government and families over the next few years, paying special attention on the groups of students who do not promote and abandon during their corresponding academic year. / Sánchez Sánchez, A. (2013). Modelling the evolution dynamics of the academic performance in high school in Spain. Probabilistic predictions of future trends and their economical consequences [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/32280 / TESIS

This master's thesis deals with qualitative analysis of nonlinear differential equations of second order. For autonomous equations some basic notions of Hamiltonian systems (mainly construction of phase portrait) are presented. For non-autonomous equations the method of lower and upper functions for periodic boundary value problem is used. These notions are then applied to a model of mechanical oscillator, a question of existence of solutions to autonomous and non-autonomous nonlinear differential equations is studied.

We consider a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales. Based on sharp estimates for first order derivatives, Linß [T. Linß, Computing 79 (2007) 23–32.] analyzed the upwind finite-difference method on a Shishkin mesh. We derive such sharp bounds for second order derivatives which show that the coupling generates additional weak layers. Finally, we prove the first robust convergence result for the Galerkin finite element method for this class of problems on modified Shishkin meshes introducing a mesh grading to cope with the weak layers. Numerical experiments support our theory.