2D Unbounded Domains Research Articles

Existence and regularity of pullback attractors for a non-Newtonian fluid with delays in 2D unbounded domains

Existence and regularity of pullback attractors for a non-Newtonian fluid with delays in 2D unbounded domains

Extraction of soliton for the confirmable time-fractional nonlinear Sobolev-type equations in semiconductor by [formula omitted]-modal expansion method

The current study deals with the exact solutions of nonlinear confirmable time fractional Sobolev type equations. Such equations have applications in thermodynamics, the flow of fluid through fractured rock. The underlying models are 2D equation of a semi-conductor with heating and Sobolev equation in 2D unbounded domain. These equation are used to describe the different aspects in semi-conductor. The analytical solutions of underlying models is not addressed yet or it is difficult to find. We gain the exact solutions of such models with help of analytical technique namely ϕ6-model expansion method. The abundant families of solutions are obtained by the Jacobi elliptic function and it will give us soliton and solitary wave solutions. So, we extract the different types of solutions such as, dark, bright, singular, combine, periodic and mixed periodic. The unique physical problems are selected from a variety of the solutions that will help the reader for the verification and data experiment. The graphical behavior of the underlying models is represented in the form of 3D, line graphs and their corresponding contours for the various values of the parameters.

Partially pivoted ACA based acceleration of the energetic BEM for time-domain acoustic and elastic waves exterior problems

We consider acoustic and elastic wave propagation problems in 2D unbounded domains, reformulated in terms of space-time Boundary Integral Equations (BIEs). For their numerical solution, we employ a weak formulation related to the energy of the system and we discretize the weak problems by a Galerkin-type Boundary Element Method (BEM): this approach, called Energetic BEM, has revealed accurate and stable even on large time intervals of analysis. In particular, it results that, when standard Lagrangian basis functions are considered, the BEM matrices have Toeplitz lower triangular block structure, where blocks for growing time become fully populated; hence the overall memory cost of the energetic BEM is O(M2N), M and N being the number of the space degrees of freedom and the total number of performed time steps, respectively. This drawback prevents the application of such method to large scale problems. As a possible remedy, we propose a fast technique based on the Adaptive Cross Approximation (ACA). The core of this procedure is the approximation of sufficiently large time blocks of the energetic BEM matrix through the partially pivoted ACA algorithm, which allows to compute only few of the original entries. This leads to reduced assembly time, which for the energetic BEM is generally relevant, coupled with reduced memory storage requirements. Additionally, the consequent acceleration of the matrix/vector multiplication together with a marching on time procedure, leads to remarkable reduction of the computational solution time. The effectiveness of the proposed method is theoretically proved and several numerical results are presented and discussed.

On the existence and uniqueness of solution of boundary‐domain integral equations for the Dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2D unbounded domain

A system of boundary‐domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in nonhomogeneous media defined on an exterior two‐dimensional domain. We use a parametrix different from the one employed in Dufera and Mikhailov (2019). The system of BDIEs is formulated in terms of parametrix‐based surface and volume potentials whose mapping properties are analyzed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for unbounded domains.

A steepest descent algorithm for the computation of traveling dissipative solitons

An algorithm is proposed to calculate traveling dissipative solitons for the FitzHugh–Nagumo equations. It is based on the application of the steepest descent method to a certain functional. This approach can be used to find solitons whenever the problem has a variational structure. Since the method seeks the lowest energy configuration, it has robust performance qualities. It is global in nature, so that initial guesses for both the pulse profile and the wave speed can be quite different from the correct solution. Also, bifurcations have a minimal effect on the performance. In the literature, there is a conjecture that no stable traveling pulse exists for a 2-component system in 2D unbounded domains. In many instances, such numerical studies investigate only solutions with a small speed, as they rely on good initial guesses based on stable standing pulse profiles. Studying a modified problem with a 2D strip domain $${\mathbb {R}} \times [-L,L]$$ with zero Dirichlet boundary conditions at $$y=\pm L$$, by using our algorithm we establish the existence of fast-moving solitons. With an appropriate set of physical parameters in this unbounded rectangular strip domain, we observe the co-existence of single-soliton and 2-soliton solutions together with additional unstable traveling pulses. The algorithm automatically calculates these various pulses as the energy minimizers at different wave speeds. In addition to finding individual solutions, we anticipate that this approach could be used to augment or initiate continuation algorithms. We also note that the rectangular strip domain can serve as a first step to investigating waves in the whole of $${\mathbb {R}}^2$$.

A reduced-order extrapolated natural boundary element method based on POD for the parabolic equation in the 2D unbounded domain

In this article, we devote ourselves to the research of order reduction of natural boundary element (NBE) based on a proper orthogonal decomposition (POD) for the parabolic equation in the two-dimensional (2D) unbounded domain. For this purpose, we first build a NBE format for the parabolic equation in the 2D unbounded domain and discuss the existence, stability, and convergence of the NBE solutions. And then, we build a reduced-order NBE extrapolated (RONBEE) format based on POD, analyze the errors between the classical NBE and RONBEE solutions, and supply the implementation procedure for the RONBEE format. Finally, we utilize some numerical experiments to validate that the numerical computational consequences are consistent with the theoretical ones such that the effectiveness and feasibility of the RONBEE format are further verified.

Pullback Attractor for the 2D Micropolar Fluid Flows with Delay on Unbounded Domains

In this paper, we investigate the pullback asymptotic behavior of micropolar fluid flows with delay on 2D unbounded domains. Firstly, the existence of pullback attractor for the universe given by a tempered condition is established. Then we obtain the consistency of the pullback attractor with that for the universe of fixed bounded sets. Furthermore, the tempered behavior of the pullback attractor is given. Here we develop the energy method with the technique of decomposition of spatial domain to overcome the lack of compactness due to unbounded domains.

Finite energy of generalized suitable weak solutions to the Navier–Stokes equations and Liouville-type theorems in two dimensional domains

Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

MICROPOLAR FLUID FLOWS WITH DELAY ON 2D UNBOUNDED DOMAINS

In this paper, we investigate the incompressible micropolar fluid flows on 2D unbounded domains with external force containing some hereditary characteristics. Since Sobolev embeddings are not compact on unbounded domains, first, we investigate the existence and uniqueness of the stationary solution, and further verify its exponential stability under appropriate conditions { essentially the viscosity δ₁:=min{<i>ν,c_a</i> + <i>c</i>_<i>d</i>} is asked to be large enough. Then, we establish the global well-posedness of the weak solutions via the Galerkin method combined with the technique of truncation functions and decomposition of spatial domain.

A natural boundary element method for the Sobolev equation in the 2D unbounded domain

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE format based on the natural integral equation and the Poisson integral formula of this problem and analyze the errors between the exact solution and the fully discretized NBE solutions. Finally, we use some numerical experiments to verify that the NBE method is effective and feasible for solving the Sobolev equation in the 2D unbounded domain.

A Natural BEM on Non-Uniform Grids and its Coupling Method with their Moving Mesh Applications

For solving Laplace equation in 2D unbounded domains, a natural BEM on non-uniform grid is derived and its convergence theorem is proved. The moving mesh methods for the NBEM and the NBE-FE coupling method are also studied. Numerical results confirm the efficiency of the methods.

Dynamics of non-autonomous equations of non-Newtonian fluid on 2D unbounded domains

Dynamics of non-autonomous equations of non-Newtonian fluid on 2D unbounded domains

Cone exchange transformations and boundedness of orbits

AbstractWe introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.

Exterior problem of magnetic field in Darwin model and its numerical solution

AbstractIn this paper, we consider the solutions of magnetic field in the Darwin model to the Maxwell's equations in 2D unbounded domain. We first deduce the variational formulation and prove the well‐posedness of the weak solution, and then prove the existence and uniqueness of the infinite element solution. Error estimate and the numerical examples are provided. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Formula omitted]-Uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains

This paper discusses the long time behavior of solutions for a two-dimensional (2D) nonautonomous micropolar fluid flow in 2D unbounded domains in which the Poincaré inequality holds. We use the energy method to obtain the so-called asymptotic compactness of the family of processes associated with the fluid flow and establish the existence of H 1 -uniform attractor. Then we prove that an L 2 -uniform attractor belongs to the H 1 -uniform attractor, which implies the asymptotic smoothing effect for the fluid flow in the sense that the solutions become eventually more regular than the initial data.