Galerkin Approximation Method Research Articles

Existence of Solutions for a Viscoelastic Plate Equation with Variable Exponents and a General Source Term

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions.

Analytical and numerical studies of a cancer invasion model with nonlocal diffusion

Abstract This article investigates the implicit Euler–Galerkin finite element method applied to a cancer invasion model with nonlocal diffusion. This model describes the normal cell density, tumor cell density, excess H + ion concentration, extracellular matrix, and extracellular matrix active metalloproteinases. The primary goal of this work is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. The Faedo–Galerkin approximation method is initially used to establish the existence of a weak solution, followed by deriving the a priori error estimates of optimal order. Additionally, the implicit Euler–Galerkin finite element method is employed as a numerical tool for solving the given cancer invasion parabolic system. Furthermore, a priori error bounds and convergence estimates for the fully discrete problem are derived. Finally, numerical simulations are conducted to validate the theoretical conclusions and enhance the understanding of how cancer spreads within the patient.

Martingale solutions and asymptotic behaviors for a stochastic cross-diffusion three-species food chain model with prey-taxis.

The stochastic food chain model is an important model within the field of ecological research. Since existing models are difficult to describe the influence of cross-diffusion and random factors on the evolution of species populations, this work is concerned with a stochastic cross-diffusion three-species food chain model with prey-taxis, in which the direction of predators' movement is opposite to the gradient of prey, i.e., a higher density of prey. The existence and uniqueness of martingale solutions are established in a Hilbert space by using the stochastic Galerkin approximation method, the tightness criterion, Jakubowski's generalization of the Skorokhod theorem, and the Vitali convergence theorem. Furthermore, asymptotic behaviors around the steady states of the stochastic cross-diffusion three-species food chain model in the time mean sense are investigated. Finally, numerical simulations are carried out to illustrate the results of our analysis.

Oblique Wave Scattering by a Combination of Two Asymmetric Trenches of Finite and Infinite Depths

Abstract The focal point of the current study lies in investigating oblique wave scattering within the framework of linear potential theory, with particular attention to scenarios involving asymmetric trenches of both finite and infinite depths. By employing the eigenfunction expansion method, the physical problem undergoes a transformation into an equivalent boundary value problem. This newly formulated problem is characterized by a system of four weakly singular integral equations, which pertain to the horizontal component of velocity across the gaps situated above the edges of the trenches. The solution to these integral equations is achieved through the utilization of a multi-term Galerkin approximation method. This approach involves expansions using ultraspherical Gegenbauer polynomials as basis functions, coupled with the appropriate weight functions tailored to address the one-third singularity. Graphical representations are employed to depict the numerical evaluations of reflection and transmission coefficients across various non-dimensional parameters. These visualizations offer insight into the behavior and dependencies of these coefficients under different conditions. To validate the accuracy of the current model, it is compared against previously published results available in the literature.

Energy decay analysis for Porous elastic system with thermoelasticity of type III: A second spectrum approach

Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green–Naghdi law of heat conduction into our study. To begin with, we use the Faedo–Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.

Well-posedness for anticipated backward stochastic Schrödinger equations

In this paper, we consider the well-posedness for the anticipated backward stochastic Schrödinger equation in a bounded domain or the whole space R d , which is associated to a stochastic control problem of nonlinear Schrödinger equations with time delay effect. The approach to establish the existence and uniqueness of adapted solutions are mainly based on the complex Itô's formula, the Galerkin's approximation method and the martingale representation theorem.

Boundary contact problems with regard to friction of couple-stress viscoelasticity for inhomogeneous anisotropic bodies (quasi-static cases)

Abstract In this paper, quasi-statical boundary contact problems of couple-stress viscoelasticity for inhomogeneous anisotropic bodies with regard to friction are investigated. We prove the uniqueness theorem of weak solutions using the corresponding Green’s formulas and positive definiteness of the potential energy. To analyze the existence of solutions, we equivalently reduce the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter, and study its solvability by the Galerkin approximate method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure, the existence theorem for the original contact problem with friction is proved.

Stability of systems governed by elliptic partial differential equations

In this paper, we are concerned with global stability and optimal control analysis of systems governed by elliptic PDEs. We first investigate the eigenvalues distribution and the corresponding eigenfunctions of this system by Galerkin approximation method. Then, we show that this method allows us to obtain a new criterion for the global stability for this type of system. Afterward, we describe the structure of optimal controllers. Finally, as applications, we not only obtain the completeness of this system but also prove the existence and exponential stability of the closed-loop system governed by elliptic PDEs.

Dynamics of Fractional Delayed Reaction-Diffusion Equations.

The long-term behavior of the weak solution of a fractional delayed reaction-diffusion equation with a generalized Caputo derivative is investigated. By using the classic Galerkin approximation method and comparison principal, the existence and uniqueness of the solution is proved in the sense of weak solution. In addition, the global attracting set of the considered system is obtained, with the help of the Sobolev embedding theorem and Halanay inequality.

The Galerkin method and hinged beam dynamics

In this paper we prove large-time existence and uniqueness of high regularity weak solutions to some initial/boundary value problems involving a nonlinear fourth order wave equation. These sorts of problems arise naturally in the study of vibrations in beams that are hinged at both ends. The method used to prove large-time existence is the Galerkin approximation method.

Global martingale solutions to stochastic population-toxicant model with cross-diffusion

In this paper, we develop a stochastic population-toxicant model with cross-diffusion and study the local boundedness of strong solutions. The existence of global martingale solution is obtained in a Hilbert space by Galerkin approximation method, the tightness criterion and the energy estimation.

Existence and blow up criterion for strong solutions to the compressible biaxial nematic liquid crystal flow

In recent paper, we study the compressible biaxial nematic liquid crystal flow in a simply connected and smooth bounded domain . An initial vacuum may exist in this flow. We firstly obtain a local existence of a unique strong solution by Galerkin's approximation method. Finally, we get a blow up criterion for such a local strong solution in terms of blow up of and for any .

Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities \(\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,\) involving time-fractional Caputo derivative \(\mathcal {D}_{0|t}^{\alpha }\) and space-fractional p-Laplacian operator \((-\varDelta )^s_p\). We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of \(\gamma ,\mu ,m\) and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.

Global well-posedness and L2 decay estimate of smooth solutions for 3-D incompressible two-phase flows

In this paper, we consider the following Cauchy problem for the 3-D incompressible two-phase flows model{ut+(u⋅∇)u+∇p=μΔu−∇χΔχ,x∈R3,t>0,divu=0,χt+u⋅∇χ−Δχ=0,u|t=0=u0,χ|t=0=χ0. We prove that the Cauchy problem has global weak solutions by means of Galerkin approximation method. Under the assumptions of small initial data, we prove that there exists a global smooth solution by applying the classical energy estimates methods. Moreover, we show the large time behavior of smooth solution to the Cauchy problem through using the Fourier-splitting method.

Weak solutions to the time-fractional g-Bénard equations

The Bénard problem consists in a system that couples the well-known Navier–Stokes equations and an advection-diffusion equation. In thin varying domains this leads to the g-Bénard problem, which turns out to be the classical Bénard problem when g is constant. The main goal of this paper is to, first of all, introduce the g-Bénard problem with time-fractional derivative of order alpha in (0,1). This formulation is new even in the classical Bénard problem, that is with constant g. The second goal of this paper is to prove the existence and uniqueness of a weak solution by means of the Faedo–Galerkin approximation method. Some recent works on time-fractional Navier–Stokes equations have opened new perspectives in studying variational aspects in problems involving time-fractional derivatives.

Numerical Approach on Oblique Wave Scattering by a Wide Rectangular Impediment With a Vent Placed Under a Finite Depth Water Body With Ice-Covered Surface

Abstract In this paper, we investigate the scattering of obliquely incident surface water waves on a hurdle as a form of a thick symmetric wall with a gap immersed in a finite depth water body having a cover of a thin ice sheet. In the context of the linear theory of water waves, this two-dimensional problem is formulated as a first-kind integral equation by splitting the velocity potential into symmetric and antisymmetric parts. The integral equation is tackled by using two numerical methods. The first method is the boundary element method where the range of integration is divided into a finite number of small line elements, and choosing the unknown function of the integral equation as constant in each line interval, we reduce the integral equation into a linear system of an algebraic equation. The second method is the multi-term Galerkin Approximation method where the basis functions are chosen as ultraspherical Gegenbauer polynomials to reduce the integral equation to a system of an algebraic equation. These systems of equations are then solved to obtain the unknown function of the integral equation in both methods. Here, very accurate numerical estimates are obtained for the reflection coefficient by both methods which are depicted graphically against the wave number for different parameters involving this problem. Also, there is a good agreement between the results of the reflection coefficient by the two methods.

Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs

This article analyzes the effect of the penalty parameter used in symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE). DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error. For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html

Conforming virtual element approximations of the two-dimensional Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes. The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the L2 and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.

Existence and blow-up of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity

In this paper, we prove the existence of weak solutions of a pseudo-parabolic equation with logarithmic nonlinearity in an interval [0, T) by employing the Galerkin approximation method and compactness arguments. We show that the solutions become unbounded at a finite time T^{star} and find upper and lower bounds for this time.

Stochastic stability analysis of a fractional viscoelastic plate excited by Gaussian white noise

The fractional viscoelastic model arises naturally in the context of systems where integer order model does not match well with practical needs and finds wide applications in engineering reality. However, the research on stochastic dynamic characteristic of the fractional viscoelastic plate is still limited. In this paper, the stochastic stability of a fractional viscoelastic plate under Gaussian white noise is studied by determining the pth moment Lyapunov exponent. Firstly, by introducing the fractional Kelvin–Voigt model to represent the constitutive relation, the fractional stochastic dynamic equations with two degrees of freedom for the viscoelastic plate are established by piston theory and Galerkin approximate method. Thereafter, the first-order approximate analytic results of the pth moment Lyapunov exponent are calculated through utilizing the singular perturbation method, which agree well with the Monte Carlo simulations. Finally, the effects of noise, viscoelastic factors and system parameters on the stochastic stability of the fractional viscoelastic plate are investigated in detail. We show that the natural frequencies carry significant effects on the stochastic stability of the viscoelastic plate.