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We study renormalized solutions to the problem $$\begin{align*} \begin{cases} -\Delta_pu=0 & \mbox{ in } \left\lbrace x \in \mathbb{R}^N : \left\lvert {x} \right\rvert>1\right\rbrace \\ \left\lvert {\nabla u} \right\rvert^{p-2}u_\nu + g(u) = \mu & \mbox{ on } \left\lbrace x \in \mathbb{R}^N : \left\lvert {x} \right\rvert=1\right\rbrace \end{cases} \end{align*} $$ { − Δ p u = 0 in { x ∈ R N : | x | > 1 } | ∇ u | p − 2 u ν + g (u) = μ on { x ∈ R N : | x | = 1 } where 1

This work, aims to study and examine the description of the gradient reduced order-strategic sensors of type boundary exponential (γΘ* EGRO-strategic sensors) for completion gradient reduced order-detectability of type boundary exponential (γΘ* EGRO-detectability). Thus, this concept is linked to an estimator in distributed parameter systems (DPSS) in Neumann problem. So, we present numerous consequences regarding to diverse kinds of information, region Θ* and conditions of boundary region to allow existence of γΘ* EGRO-detectable systems. In addition, we have estimated at the junction interface that the interior solution is harmonized with the exterior solution for γΘ* EGRO-detectable and, we give the relationship between this concept and sensors structures. Finally, we demonstrate some applications with many circumstances of sensor positions. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into the stability and control aspects of delayed partial differential equation (PDE) systems utilizing the Lyapunov method. PDEs serve as powerful mathematical tools for modeling diverse and intricate systems such as heat transfer processes, chemical reactors, flexible arms, and population dynamics. However, the presence of delays within the feedback loop of such systems can introduce significant challenges, as even minor delays can potentially trigger system instability. To address this issue, the Lyapunov method, renowned for its efficacy in stability analysis, is employed to assess the exponential stability of a specific cohort of delayed PDE systems. By adopting Dirichlet boundary conditions and incorporating delay-dependent techniques such as the Galerkin method and Halanay inequality, the inherent stability properties of these systems are rigorously examined. Notably, the utilization of Dirichlet boundary conditions in this study allows for simplified analysis, and it is worth mentioning that the stability analysis outcomes under Neumann conditions and combined boundary conditions align with those of the Dirichlet boundary conditions discussed herein. Furthermore, this research endeavor delves into the implications of the obtained results in terms of control considerations and convergence rates. The integration of the Galerkin method aids in approximating the behavior of dominant modes within the system, thereby enabling a more comprehensive understanding of stability and control. The exploration of convergence rates provides valuable insights into the speed at which stability is achieved in practice, thus enhancing the practical applicability of the findings. The outcomes of this study contribute significantly to the broader comprehension and effective control of delayed PDE systems. The elucidation of stability behaviors not only provides a comprehensive understanding of the impact of delays but also offers practical insights for the design and implementation of control strategies in various domains. Ultimately, this research strives to enhance the stability and reliability of complex systems represented by PDEs, thereby facilitating their effective utilization across numerous scientific and engineering applications.

In this paper, using the notion of ( I q )–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [ a , b ] have been established for four different types of boundary conditions for a singularly perturbed differential equation ε y ″ = f (x , y , y ′) ,   a ≤ x ≤ b . In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f (x , y , y ′) = 0 . [ABSTRACT FROM AUTHOR]

In this document we will study and solve the nonlinear partial differential equation, with initial conditions for vehicle entry that serves to model the dynamics of traffic flow. To find a numerical solution to the dynamics that govern the behavior of traffic flow, the Finite Element Method in a spatial dimension was used. In accordance with the temporal dynamics, simulations were developed to know the flow in terms of time. The numerical solution is interesting for predicting the number of vehicles at the entrance to a high-flow road. Some theorems are enunciated that guarantee the existence of the solution and the uniqueness is given by the boundary conditions. [ABSTRACT FROM AUTHOR]

Several important problems in Mechanics can be efficiently solved using Raviart–Thomas mixed finite element methods. Whenever the domain of interest has a curved boundary the methods of this family for N-simplexes are the natural choice. But in this case the question arises on the best way to prescribe normal flux conditions across the boundary, if any. It is generally acknowledged that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polytope in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in Bertrand and Starke (2016). In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this work an alternative with straight-edged triangles for two-dimensional problems is examined. The key feature of this approach is a Petrov–Galerkin formulation, in which the test-flux space is a little different from the shape-flux space. Based on previous author’s experience with this technique, as applied to Lagrange finite elements, it would lead to an overall accuracy improvement here as well. The experimentation reported hereafter provides examples and counterexamples confirming or not such an expectation, depending on the unknown field of the mixed problem at hand.

This manuscript explores the flow features of the Modified Cross Model in a channel with symmetric trapezoidal cavities in the presence of a circular obstacle. The non-dimensional governing equations and model for different parameters are evaluated via a Galerkin Finite Element Method The system of non-linear algebraic equations is computed by adopting the Newton method. A space involving the quadratic polynomials (P2) has been selected to compute for the velocity profile while the pressure profile is approximated by a linear (P1) finite element space of functions. Simulations are performed for a wide range of physical parameters such as modified parameter (from 0.0 to 0.5), power-law index (from 0.5 to 1.5), relaxation parameter (from 1 to 3), and Reynolds number (from 10 to 40). For the case of a modified parameter (b) and relaxation parameter (λ), it is observed that the drag coefficient (CD) shows an increasing trend while the lift coefficient (CL) is changing sign at lower values of (λ), and then becomes positive at λ=3.

In this paper, using the notion of (Iq)–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [a,b] have been established for four different types of boundary conditions for a singularly perturbed differential equation εy″=f(x,y,y′), a≤x≤b. In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f(x,y,y′)=0.

In this article, we study the well-posedness of the Master Equation of Mean Field Games in a framework of Neumann boundary condition. The definition of solution is closely related to the classical one of the Mean Field Games system, but the boundary condition here leads to two Neumann conditions in the Master Equation formulation, for both space and measure. The global regularity of the linearized system, which is crucial in order to prove the existence of solutions, is obtained with a deep study of the boundary conditions and the global regularity at the boundary of a suitable class of parabolic equations. [ABSTRACT FROM AUTHOR]

This work reviews some basic features on the second (first nontrivial) eigenvalue λ2 to the Neumann problem ... where Ω is a bounded Lipschitz domain of RN, is the outer unit normal, and ∆pu = div(∇) is the p-Laplacian operator. We are mainly concerned with the variational characterization of &#955;2 and place emphasis on the range 1 < p < 2, where the nonlinearity |∆| becomes non smooth. We also address the corresponding result for the p-Laplacian in graphs. [ABSTRACT FROM AUTHOR]