Faedo-Galerkin Method Research Articles

Faedo-Galerkin method for the time-fractional reaction-diffusion model

This paper investigates a time-fractional reaction-diffusion equation characterized by a Caputo derivative of fractional order , incorporates non-local and memory effects that are essential for accurately modeling complex diffusion processes in such environments. These effects arise from the fractal nature of the media, which leads to anomalous diffusion behavior that cannot be adequately described by classical diffusion models. We start by giving a definition of a weak solution for this problem, ensuring that the solution satisfies the equation in a generalized sense. Subsequently, the Faedo-Galerkin method combined with compactness arguments is employed to establish the existence and uniqueness of the weak solution. This approach involves approximating the solution using a sequence of finite-dimensional functions and then passing to the limit to obtain the desired solution . Moreover, we establish the stability of these solutions, which is fundamental for understanding their behavior under varying conditions and ensuring the reliability of the model . Our findings contribute to a deeper understanding of anomalous diffusion phenomena in fractal media and provide a solid foundation for further research in this area.

A Three-Dimensional Model of a Spherically Symmetric, Compressible Micropolar Fluid Flow with a Real Gas Equation of State

In this work, we analyze a spherically symmetric 3D flow of a micropolar, viscous, polytropic, and heat-conducting real gas. In particular, we take as a domain the subset of R3 bounded by two concentric spheres that present solid thermoinsulated walls. Also, here, we consider the generalized equation of state for the pressure in the sense that the pressure depends, as a power function, on the mass density. The model is based on the conservation laws for mass, momentum, momentum moment, and energy, as well as the equation of state for a real gas, and it is derived first in the Eulerian and then in the Lagrangian description. Through the application of the Faedo–Galerkin method, a numerical solution to a corresponding problem is obtained, and numerical simulations are performed to demonstrate the behavior of the solutions under various parameters and initial conditions in order to validate the method. The results of the simulations are discussed in detail.

Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone.

Mathematical study of a new coupled electro-thermo radiofrequency model of cardiac tissue

This paper presents a nonlinear reaction–diffusion-fluid system that simulates radiofrequency ablation within cardiac tissue. The model conveys the dynamic evolution of temperature and electric potential in both the fluid and solid regions, along with the evolution of velocity within the solid region. By formulating the system that describes the phenomena across the entire domain, encompassing both solid and fluid phases, we proceed to an analysis of well-posedness, considering a broad class of right-hand side terms. The system involves parameters such as heat conductivity, kinematic viscosity, and electrical conductivity, all of which exhibit nonlinearity contingent upon the temperature variable. The mathematical analysis extends to establishing the existence of a global solution, employing the Faedo–Galerkin method in a three-dimensional space. To enhance the practical applicability of our theoretical results, we complement our study with a series of numerical experiments. We implement the discrete system using the finite element method for spatial discretization and an Euler scheme for temporal discretization. Nonlinear parameters are linearized through decoupling systems, as introduced in our continuous analysis. These experiments are conducted to demonstrate and validate the theoretical findings we have established.

Well-posedness of the Optimal Control Problem Related to Degenerate Chemo-attraction Models

This paper delves into the mathematical analysis of optimal control for a nonlinear degenerate chemotaxis model with volume-filling effects. The control is applied in a bilinear form specifically within the chemical equation. We establish the well-posedness (existence and uniqueness) of the weak solution for the direct problem using the Faedo Galerkin method (for existence), and the duality method (for uniqueness). Additionally, we demonstrate the existence of minimizers and establish first-order necessary conditions for the adjoint problem. The main novelty of this work concerns the degeneracy of the diffusive term and the presence of control over the concentration in our nonlinear degenerate chemotaxis model. Furthermore, the state, consisting of cell density and chemical concentration, remains in a weak setting, which is uncommon in the literature for solving optimal control problems involving chemotaxis models.

Existence and asymptotic behavior of a nonlinear axially moving string with variable tension and subject to disturbances

In this paper, we consider the stabilization question for a nonlinear model of an axially moving string. The model is assumed to undergo the variable tension and variable disturbances. The Hamilton principle is used to describe the dynamic of transverse vibrations. We establish the well-posedness by means of the Faedo-Galerkin method. A boundary control with a time-varying delay is designed to stabilize uniformly the string. Then, we derive a decay rate of the solution assuming that the retarded term be dominated by the damping one. Some examples are given to clarify when the rate is exponential or polynomial.

Stochastic electromechanical bidomain model *

We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.

Boundary feedback control of Navier–Stokes equations around arbitrary bluff body with prescribed drag and lift coefficients

This paper presents a boundary feedback control method for the two- and three-dimensional Navier–Stokes equations around an arbitrary bluff body, with prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier–Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. We utilize the artificial compressibility method, which approximates the Navier–Stokes equations in the limit, along with a priori estimation techniques, to establish a boundary control. Such a control law ensures that the prescriptions of the drag and lift coefficients are attained. Subsequently, a compactness result allows to let the compressibility parameter tend to zero. The application of the Faedo-Galerkin method is essential in reaching this result.

Blowing‐up solutions for the Moorea–Gibson–Thompson equation with a viscoelastic memory and an external force

The Moore–Gibson–Thompson equation with a viscoelastic memory and a forcing term is considered. The existence and uniqueness of a local solution are obtained via Faedo–Galerkin's method. Furthermore, it is shown that solutions with or without a positive initial energy experience blowing‐up due to the nonlinear forcing term.

Analysis on nonlinear differential equation with a deviating argument via Faedo–Galerkin method

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

Longtime dynamics of solutions for higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff models with higher-order rotational inertia and nonlocal damping

The Kirchhoff model is derived from the vibration problem of stretchable strings. This paper focuses on the longtime dynamics of a higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff system with higher-order rotational inertia and nonlocal damping. We first obtain the state of the model’s solutions in different spaces through prior estimation. After that, we immediately prove the existence and uniqueness of their solutions in different spaces through the Faedo-Galerkin method. Subsequently, we prove their family of global attractors using the compactness theorem. Finally, we reflect on the subsequent research of the model and point out relevant directions for further research on the model. In this way, we systematically study the longtime dynamics of the higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff model with higher-order rotational inertia, thus enriching the relevant findings of higher-order coupled Kirchhoff models and laying a theoretical foundation for future practical applications.

On the solvability of a boundary value problem for a two-dimensional system of Navier-Stokes equations in a cone

Due to the fact that the Navier-Stokes equations are involved in the formulation of a large number of interesting problems that are important from an applied point of view, these equations have been the object of attention of mechanics, mathematicians and other scientists for several decades in a row. But despite this, many problems for the Navier-Stokes equation remain unexplored to this day. In this work, we are exploring the solvability of a boundary value problem for a two-dimensional Navier-Stokes system in a non-cylindrical degenerating domain, namely, in a cone with its vertex at the origin. Previously, we studied cases of the linearized Navier-Stokes system or non-degenerating cylindrical domains, so this work is a logical continuation of our previous research in this direction. To the above-mentioned degenerate domain we associate a family of non-degenerate truncated cones, which, in turn, are formed by a oneto-one transformation into cylindrical domains, where for the problem under consideration we established uniform a priori estimates with respect to changes in the index of the domains. Further, using a priori estimates and the Faedo-Galerkin method, we established the existence, uniqueness of solution in Sobolev classes, and its regularity as the smoothness of the given functions increases.

A pseudo‐parabolic equation with logarithmic nonlinearity: Global existence and blowup of solutions

This article deals with a fourth‐order pseudo‐parabolic partial differential equation with logarithmic nonlinearity. We adopt the Faedo–Galerkin method to analyze the global existence of weak solutions and discuss the existence of a solution in subcritical and critical initial energy situations. Further, applying the concavity approach, we have shown that the solution blows up when the initial energy is subcritical, critical, and supercritical. Moreover, we have provided an upper and lower bound for blow‐up time and a decay estimate for the global solutions.

Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

On the exponential decay of a Balakrishnan-Taylor plate with strong damping

<abstract><p>In this manuscript, we study a thin and narrow plate equation that models the deck of a suspension bridge that is subject to a Balakrishnan-Taylor damping and a strong damping. First, by using the Faedo Galerkin method, we prove the existence of both global weak and regular solutions. Second, we prove the exponential stability of the energy for regular solutions by combining the multiplier method and a well-known result of Komornik.</p></abstract>

Well-posedness and Exponential Decay of Energy for the Solution of a Wave Equation with Nonlinear Source and Localized Damping Termes

We consider the wave equation with a locally damping and a nonlinear source term in a bounded domain. ytt − Δy + a(x)g(yt) = |y|p−2y, where p > 2. The damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary. We show, for certain initial data and suitable conditions on g,a and p that this solution is global we use the Faedo-Galerkin method. Also we established the exponential decay of the energy when the nonlinear damping grows linearly by introducing a suitable Lyapunov functional.

Existence and stability results of nonlinear swelling equations with logarithmic source terms

<abstract><p>We considered a swelling porous-elastic system characterized by two nonlinear variable exponent damping and logarithmic source terms. Employing the Faedo-Galerkin method, we established the local existence of weak solutions under suitable assumptions on the variable exponents functions. Furthermore, we proved the global existence utilizing the well-depth method. Finally, we established several decay results by employing the multiplier method and the Logarithmic Sobolev inequality. To the best of our knowledge, this represents the first study addressing swelling systems with logarithmic source terms.</p></abstract>

Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping

Abstract In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - ( ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ ( ∇ ⁡ z , ∇ ⁡ z t ) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ ( x ) ⁢ f ⁢ ( z t ) + g ⁢ ( z ) = 0 |z_{t}|^{\alpha}z_{tt}-\Delta z_{tt}-\bigl{(}\xi_{1}+\xi_{2}\|\nabla z\|^{2}+% \sigma(\nabla z,\nabla z_{t})\bigr{)}\Delta z-\Delta z_{t}+\beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {\mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.

On a logarithmic wave equation with nonlinear dynamical boundary conditions: local existence and blow-up

This paper deals with a hyperbolic-type equation with a logarithmic source term and dynamic boundary condition. Given convenient initial data, we obtained the local existence of a weak solution. Local existence results of solutions are obtained using the Faedo-Galerkin method and the Schauder fixed-point theorem. Additionally, under suitable assumptions on initial data, the lower bound time of the blow-up result is investigated.

Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.