Theory Of Nonlinear Problems Research Articles

SOLVABILITY OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE QUASILINEAR EQUATION OF HEAT CONDUCTIVITY IN DOMAINS THAT CAN BE TRANSFORMED INTO RECTANGLES

In this paper, we study the initial-boundary-value problem for the quasilinear heat equation in regions that are reduced to rectangular. Mathematical modeling of many processes taking place in the real world leads to the study of the problems of equations of mathematical physics, when the areas are not rectangular. The theory of nonlinear problems is an actively developing section of the theory of modern differential equations. In the theory of nonlinear equations, a special place is occupied by the study of unbounded solutions or, in other words, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally unsolvable: solutions grow unlimitedly over a finite period of time. In this paper, the initial-boundary-value problem for the quasilinear heat equation in regions that can be reduced to rectangular ones, the existence of a solution is proved by the Galerkin method. The uniqueness of the solution was proved by the obtained a priori estimates. Sufficient conditions for the destruction of the solution in a finite time in a bounded domain are obtained. The exponential decay of the solution with an infinite increase in time is proved. In the final time, it was proved that the solution is localized, i.e. disappears (nullifies).

Localization and multiplicity in the homogenization of nonlinear problems

AbstractWe propose a method for the localization of solutions for a class of nonlinear problems arising in the homogenization theory. The method combines concepts and results from the linear theory of PDEs, linear periodic homogenization theory, and nonlinear functional analysis. Particularly, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland's variational principle. A significant gain in the homogenization theory of nonlinear problems is that our method makes possible the emergence of finitely or infinitely many solutions.

Common fixed points for generalized ψ-contractions in weak non-Archimedean fuzzy metric spaces

<p>Fixed point theory in fuzzy metric spaces plays very important role in theory of nonlinear problems in applied science. In this paper, we prove an existence result of common fixed point of four nonlinear mappings satisfying a new type of contractive condition in a generalized fuzzy metric space, called weak non-Archimedean fuzzy metric space. Our main results can be applied to solve the existence of solutions of non-linear equations in fuzzy metric spaces. Some examples supporting our main theorem are also given. Our results improve and generalize some recent results contained in Vetro (2011)[16]to generalized contractive conditions under some suitable conditions and many known results in the literature.</p>

Nonlinear Hydrodynamic Forces on Flexible Structures: A Combined-Force Model

A new combined-force method is suggested to approximate the second-order difference frequency forces from diffraction theory (Φ2 theory) with less computational effort. The new method is formulated by combining two limiting cases of the Φ2 theory; i.e., Newman’s approximation and the slender Φ2 theory. Numerical results show that the new method reproduces the individual nonlinear effects of the Φ2 theory better than the existing approximations. Results of this work also show the limitations of Morison’s equation as the slender-body counterpart of the diffraction theory for nonlinear problems.