Kinds Of Approximate Solutions Research Articles

Study of weak solutions to a nonlinear fourth‐order parabolic equation with boundary degeneracy

A double degenerate fourth‐order parabolic equation with a nonlinear second‐order diffusion from thin film theory is introduced, and the existence of weak solutions for the boundary degeneracy problem is studied. For this purpose, a non‐degenerate fourth‐order elliptic problem is constructed, and the corresponding existence and uniqueness are given by the variational method. From it, two kinds of approximate solutions are defined and then the existence and uniqueness of weak solutions for the related non‐degenerate parabolic problem are shown by the iterative estimate, energy method, and compactness argument. Finally, the parabolic problem with boundary degeneracy is solved by applying energy estimate and several limit procedures.

Concepts for Approximate Solutions of Vector Optimization Problems with Variable Order Structures

In this paper, we introduce the concepts for approximately minimal, approximately nondominated solutions, and approximate minimizers of vector optimization problems with respect to variable order structures. In order to describe solution concepts, we use a set-valued map and this map is not necessarily a (pointed, convex) cone-valued map. We illustrate the different concepts for approximate solutions by several examples. Important properties of these three different kinds of approximate solutions of vector optimization problems with respect to variable order structures are discussed. Finally, we give some necessary conditions for approximate solutions of vector optimization problems with variable order structures using a vector-valued variant of Ekeland’s variational principle.

Approximate Solutions of Generalized Fifth-Order Korteweg-De Vries (KdV) Equation by the Standard Truncated Expansion Method

It is difficult to obtain exact solutions of the nonlinear partial differential equations (PDEs) due to the complexity and nonlinearity, especially for non-integrable systems. In this case, some reasonable approximations of real physics are considered, by means of the standard truncated expansion approach to solve real nonlinear system is proposed. In this paper, a simple standard truncated expansion approach with a quite universal pseudopotential is used for generalized fifth-order Korteweg-de Vries (KdV) equation, we can get two kinds of approximate solutions of the above equation, in some special cases, the approximate solutions may become exact. The same idea can also used to find approximate solutions of other well known nonlinear equations. We find a quite universal expansion approach which is valid for various nonlinear partial differential equations (PDEs).

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.

양시등급 항공기 동력학의 근사 궤환 제어기 설계

본 논문에서는 양시등급 항공기 동력학에서 빠른 종속시스템의 극점들을 그대로 둔 채 느린 종속시스템의 극점만을 재배치함으로써 페루프의 근사치 해를 획득하는 방법이 제안되었다. 행렬대각화를 통하여 얻어지는 이러한 근사치 해는 수정된 것과 수정되지 않은 것 두 종류로 구분된다. 이들의 차이는 전자의 경우 오차가 <TEX>$O({\varepsilon})$</TEX>이며 후자의 경우는 오차가 <TEX>$O({\varepsilon}^2)$</TEX>이다. 두 가지 해는 모두 감소해이지만 충분한 견실성을 보여준다. 제안된 기법의 우수성은 항공기 종방향 운동 모델의 시뮬레이션을 통하여 확인되었다. A new method to obtain approximate solutions by placing the only poles of the slow subsystem for the two-time-scale aircraft dynamic systems. The two kinds of approximate solutions are obtained by a matrix block diagonalization. One is called the uncorrected solution, and the other is called the corrected solution. The former has an error of <TEX>$O({\varepsilon})$</TEX>, and the latter has an error of <TEX>$O({\varepsilon}^2)$</TEX>. Of course, both solutions are robust enough even though they are reduced solutions. The excellence of the proposed method is illustrated by an numerical example of an aircraft longitudinal dynamics.

On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the function and subgradient values involves solving exactly a certain optimization problem, which can be relatively costly (and as we show, in any case unnecessary). We show that approximation with some finite precision is sufficient in this setting and give an explicit characterization of this precision. Alternatively, our result can be viewed as a stability analysis of standard proximal bundle methods, as it answers the following question: for a given approximation error, what kind of approximate solution can be obtained and how does it depend on the magnitude of the perturbation?

Periodic solutions and the stability in a non-linear parametric excitation system (2nd report, Consideration of solutions in the neighborhood of resonance of order 1)

Behaviors of a parametrically excited system with a self-excitation are investigated by analyzing a differential equation of van der Pol-Mathieu type. Beat phenomena are observed in the neighborhood of parametric resonance of order 1 and their two kinds of approximate solutions are obtained. The comparison between them and a solution by Runge-Kutta-Gill method is performed and it is found that the existence of a solution by two components is limited to a narrow region of frequency. The character on the waveforms and frequency components is considered in detail.

Periodic Solutions and the Stability in a Non-linear Parametric Excitation System : 2nd Report, Consideration of Solutions in the Neighborhood of Resonance of Order 1

Behaviors of a parametrically excited system with a self-excitation are investigated by analyzing a differential equation of van der Pol-Mathieu type. Beat phenomena are observed in the neighborhood of parametric resonance of order land their two kinds of approximate solutions are obtained. The comparison between them and a solution by Runge-Kutta-Gill method are performed and it is found that the existence of a solution approximated by two components is limited to a narrow region of frequencies. The character on the waveforms and frequency components is considered in detail.