From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.

Cross-Section Configuration Coefficient in the Zhurkov’S Generalized Equation

Постановка проблемы. Исследование долговечности строительных материалов с термофлуктуационной позиции является наиболее сложным, но и в тоже время наиболее адекватным методом. Ввиду того, что данная концепция является нечувствительной к изменениям физической структуры, возникает необходимость учета не только материала, но и конфигурации конструкции. Необходимо провести сравнение механизма разрушения при различных вариантах сечения для двух структурно отличающихся элементов - поливинилхлорида (ПВХ) и древесины. Результаты. Для элементов ПВХ и дерева составного сечения в два слоя без специальных связей получен одинаковый коэффициент k = 2. Для элементов составного сечения в три слоя без специальных связей получен коэффициент с интервалом k = 3,5…5,5, что требует дальнейшего уточнения. Выводы. На основе проведенного исследования теоретически обоснованы и экспериментально выявлены закономерности деформирования и разрушения ПВХ-элементов цельного сечения и составного сечения без специального соединения в два и три слоя. Определение термофлуктуационных зависимостей позволяет приблизить теоретические представления о работоспособности строительного материала в конструкции к реальным условиям. Statement of the problem. Studying the durability of construction materials in the aspect of thermal fluctuations is the most complicated, yet the most appropriate method. Considering that this concept does not take into account the changes of the physical structure, it becomes necessary to consider not the material alone but also the configuration of the structure. Therefore it is necessary to make a comparison of two structurally different elements - PVC and wood. Results. The coefficient of the 2-layer composite cross-section (no special connection) is the same for the PVC and wood elements: k = 2. The derived coefficient of the 3-layer composite cross-section (no special connection) is within the following range: k = 3,5…5,5, which requires a more precise definition. Conclusions. Based on the above experiment, we have theoretically established and experimentally confirmed the regularities of deformation and destruction of PVC elements of the 2 and 3-layer solid and composite cross-section without a special connection. Determining the thermal fluctuation relations allows us to bring theoretical concepts of the capacity of the construction material in a structure closer to actual conditions.

Seismic wave equation formulated by generalized viscoelasticity in fluid-saturated porous media

Biot’s theory of poroelasticity describes seismic waves propagating through fluid-saturated porous media, so-called two-phase media. The classic Biot’s theory of poroelasticity considers the wave dissipation mechanism being the friction of relative motion between the fluid in the pores and the solid rock skeleton. However, within the seismic frequency band, the friction has a major influence only on the slow P-wave and has an insignificant influence on the fast P-wave. In order to represent the intrinsic viscoelasticity of the solid skeleton, we incorporate a generalized viscoelastic wave equation into Biot’s theory for the fluid-saturated porous media. The generalized equation which unifies the pure elastic and viscoelastic cases is constituted by a single viscoelastic parameter, presented as the fractional order of the wavefield derivative in the compact form of the wave equation. The generalized equation that includes the viscoelasticity appropriately describes the dissipation characteristics of the fast P-wave. Plane-wave analysis and numerical solutions of the proposed wave equation reveal that (1) the viscoelasticity in the solid skeleton causes the energy attenuation on the fast P-wave and the slow P-wave at the same order of magnitude, and (2) the generalized viscoelastic wave equation effectively describes the dissipation effect of the waves propagating through the fluid-saturated porous media.

Study of the residual imperfections using methods of probability theory

Discontinuities in the products that occur during manufacture, mounting or upon operation can be missed during non-destructive testing which do not provide their complete detectability at a current level of the technology. Therefore, it is necessary to take into account that certain structural elements may have discontinuities of significant dimensions. We present the results of using the methods of probability theory in studying the residual imperfections that remains in the structure after non-destructive control and repair of the previously identified defects. We used the results of operational control of units carried out by ultrasonic and radiographic methods. We present a method for determining a multifactorial coefficient that takes into account the detectability of defects, the number of control procedures and the errors in the instrumentation and methodological support, as well as a generalized equation for the probability distribution of detecting discontinuities. The developed approach provides assessing of the level of damage to the studied objects, their classification proceeding from the quantitative data and determination of the values of postulated discontinuities for deterministic calculations. The results obtained can be used to improve the methods of monitoring NPP facilities.

Lie Algebra Classification, Conservation Laws, and Invariant Solutions for a Generalization of the Levinson–Smith Equation

We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.

A boundary value problem for a partial differential equation with fractional derivative

In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved.

An accelerated differential equation system for generalized equations

<p style='text-indent:20px;'>An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>to the solution set <inline-formula><tex-math id="M2">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> of a generalized equation <inline-formula><tex-math id="M3">\begin{document}$ 0 \in T(x) $\end{document}</tex-math></inline-formula>. With smart choices of parameters <inline-formula><tex-math id="M4">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \alpha(t) $\end{document}</tex-math></inline-formula>, we prove the weak convergence of the trajectory to some point of <inline-formula><tex-math id="M6">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ \|\dot{x}(t)\|\leq {\rm O}(1/t) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ t\rightarrow +\infty $\end{document}</tex-math></inline-formula>. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory <inline-formula><tex-math id="M9">\begin{document}$ x^{h}(t) \rightarrow x(t) $\end{document}</tex-math></inline-formula> on interval <inline-formula><tex-math id="M10">\begin{document}$ [0,+\infty) $\end{document}</tex-math></inline-formula> is demonstrated when the step size <inline-formula><tex-math id="M11">\begin{document}$ h \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>