118 research outputs found
Aproximación de ecuaciones diferenciales mediante una nueva técnica variacional y aplicaciones
[SPA] En esta Tesis presentamos el estudio teórico y numérico de sistemas de ecuaciones
diferenciales basado en el análisis de un funcional asociado de forma natural al
problema original. Probamos que cuando se utiliza métodos del descenso para
minimizar dicho funcional, el algoritmo decrece el error hasta obtener la convergencia
dada la no existencia de mínimos locales diferentes a la solución original. En cierto
sentido el algoritmo puede considerarse un método tipo Newton globalmente
convergente al estar basado en una linearización del problema. Se han estudiado la
aproximación de ecuaciones diferenciales rígidas, de ecuaciones rígidas con retardo,
de ecuaciones algebraico‐diferenciales y de problemas hamiltonianos. Esperamos que
esta nueva técnica variacional pueda usarse en otro tipo de problemas diferenciales. [ENG] This thesis is devoted to the study and approximation of systems of differential
equations based on an analysis of a certain error functional associated, in a natural
way, with the original problem. We prove that in seeking to minimize the error by
using standard descent schemes, the procedure can never get stuck in local minima,
but will always and steadily decrease the error until getting to the original solution.
One main step in the procedure relies on a very particular linearization of the problem,
in some sense it is like a globally convergent Newton type method. We concentrate on
the approximation of stiff systems of ODEs, DDEs, DAEs and Hamiltonian systems. In all
these problems we need to use implicit schemes. We believe that this approach can be
used in a systematic way to examine other situations and other types of equations.Universidad Politécnica de Cartagen
A study of optimization problems and fixed point iterations in Banach spaces.
Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF
The Geometry of Monotone Operator Splitting Methods
We propose a geometric framework to describe and analyze a wide array of
operator splitting methods for solving monotone inclusion problems. The initial
inclusion problem, which typically involves several operators combined through
monotonicity-preserving operations, is seldom solvable in its original form. We
embed it in an auxiliary space, where it is associated with a surrogate
monotone inclusion problem with a more tractable structure and which allows for
easy recovery of solutions to the initial problem. The surrogate problem is
solved by successive projections onto half-spaces containing its solution set.
The outer approximation half-spaces are constructed by using the individual
operators present in the model separately. This geometric framework is shown to
encompass traditional methods as well as state-of-the-art asynchronous
block-iterative algorithms, and its flexible structure provides a pattern to
design new ones
Applied Mathematics and Fractional Calculus
In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia
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