Study Of Mathematical Problems Research Articles

Existence and Uniqueness of Renormalized Solution to Nonlinear Anisotropic Elliptic Problems with Variable Exponent and L 1 -Data

Nonlinear partial differential equations are considered as an essential tool for describing the behavior of many natural phenomena. The modeling of some phenomena requires to work in Sobolev spaces with constant exponent. But for others, such as electrorheological fluids, the properties of classical spaces are not sufficient to have precision. To overcome this difficulty, we work in the appropriate spaces called Lebesgue and Sobolev spaces with variable exponent. In recent works, researchers are attracted by the study of mathematical problems in the context of variable exponent. This great interest is motivated by their applications in many fields such as elastic mechanics, fluid dynamics, and image restoration. In this paper, we combine the technic of monotone operators in Banach spaces and approximation methods to prove the existence of renormalized solutions of a class of nonlinear anisotropic problem involving p ⟶ . − Leray–Lions operator, a graph, and L 1 data. In particular, we establish the uniqueness of the solution when the graph data are considered a strictly increasing function.

On the exterior Dirichlet-Neumann problem for the Biharmonic equation and its application in mechanics

We study the unique solvability of the mixed Dirichlet-Neumann biharmonic problem in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet (energy) integral with weight |x|a. Using the variational principle and depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of the Dirichlet-Neumann problem or present exact formulas for the dimension of the space of solutions. The results of this paper are used in the study of mathematical problems in mechanical models, in particular, in transport models and procedures.

Оптимальное управление с обратной связью для одной модели движения нелинейно-вязкой жидкости

В работе рассматривается задача оптимального управления с обратной связью для начально-краевой задачи, описывающей движение нелинейно-вязкой жидкости. Доказывается существование оптимального решения, дающего минимум заданному функционалу качества. Для доказательства существования оптимального решения используется аппроксимационно-топологический метод исследования задач гидродинамики.

La integración del momento de la técnica en el proceso de estudio de campos de problemas de matemáticas

This article provides a model of mathematical activity which describes it in terms of the study of fields of problems. This model helps us to analyze some didactical phenomena which are characteristic of the present systems of mathematics teaching. Special emphasis is devoted to the relation between the absence of so-called technique work, and some paradoxes about «mathematical creativity». A new didactical device is proposed, the workshop of mathematical practice, in order to bring the technique work alive in the classrooms as an integrating force in the study of mathematical problems. Finally, as an example, one of the workshops designed and tested is described.

Denkprozesse beim Lösen mathematischer Deduktionsaufgaben — Bericht über eine exploratorische Untersuchung

An exploratory study of mathematical problem solving processes was carried out. Think-aloud sessions of students solving problems from incidence geometry were videotaped. A category system was developed to form data from the observations. Correlation-methods and sequence-analysis techniques were used to analyze the data. The hypotheses generated deal with systematic behavior (strategies), problem solving success, modes of representation, problem solving types and mistakes. Some prospects are made concerning future research.