Unbounded Domains Research Articles

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

In this paper, a lattice Boltzmann method is proposed for solving the linear complementarity problem (LCP) arising in single and multi-asset American put option pricing. The LCP for American option is a variable coefficient parabolic model defined on an unbounded domain. Initially, using the far field estimate method and the penalty method respectively, the LCP could be reformulated into a nonlinear parabolic partial differential equation on a bounded domain. To construct a unified lattice Boltzmann model for the option pricing problems, the above transformation equations are rewritten into an equivalent divergence form. Then, through the incorporation of an amending function into the evolution equation, which assists in recovering the source term and eliminating the error term, the lattice Boltzmann model with spatial second-order accuracy is constructed. Finally, the present model is validated using numerical simulations, and the numerical results agree well with the option values obtained by existing methods, which indicates that the present lattice Boltzmann model is efficient for solving the American put option pricing problem.

Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space RN with N≥3, on R+3 and on Ω⊆R3 exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation vu∈L∞((0,T);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)),∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for s>N2−1 if Ω=RN and s∈(12,1] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.

Approximation theorem for the Kawahara operator and its application in the control theory

AbstractControl properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in . Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.

Complex Ginzburg–Landau equation for time‐varying anisotropic media

AbstractWhen extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time‐varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time‐varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time‐varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time‐varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media.

Asymptotic behavior for Kirchhoff type stochastic plate equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of Kirchhoff type stochastic nonlinear plate equations with dispersive and viscosity dissipative terms driven additive noise defined on the entire space Rn. The existence, uniqueness as well as upper semi-continuity of pullback random attractors are proved in H2(Rn)×H1(Rn).

Convergence of generalized Orlicz norms with lower growth rate tending to infinity

We study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea–Mihăilescu (Orlicz case) and Eleuteri–Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.

Global Classical Large Solutions for the Radiation Hydrodynamics Model in Unbounded Domains

Global Classical Large Solutions for the Radiation Hydrodynamics Model in Unbounded Domains

Plume-scale confinement on thermal convection

Despite the ubiquity of thermal convection in nature and artificial systems, we still lack a unified formulation that integrates the system's geometry, fluid properties, and thermal forcing to characterize the transition from free to confined convective regimes. The latter is broadly relevant to understanding how convection transports energy and drives mixing across a wide range of environments, such as planetary atmospheres/oceans and hydrothermal flows through fractures, as well as engineering heatsinks and microfluidics for the control of mass and heat fluxes. Performing laboratory experiments in Hele-Shaw geometries, we find multiple transitions that are identified as remarkable shifts in flow structures and heat transport scaling, underpinning previous numerical studies. To unveil the mechanisms of the geometrically controlled transition, we focus on the smallest structure of convection, posing the following question: How free is a thermal plume in a closed system? We address this problem by proposing the degree of confinement [Formula: see text]-the ratio of the thermal plume's thickness in an unbounded domain to the lateral extent of the system-as a universal metric encapsulating all the physical parameters. Here, we characterize four convective regimes different in flow dimensionality and time dependency and demonstrate that the transitions across the regimes are well tied with [Formula: see text]. The introduced metric [Formula: see text] offers a unified characterization of convection in closed systems from the plume's standpoint.

Higher-order moments of spline chaos expansion

Spline chaos expansion, referred to as SCE, is a finite series representation of an output random variable in terms of measure-consistent orthonormal splines in input random variables and deterministic coefficients. This paper reports new results from an assessment of SCE’s approximation quality in calculating higher-order moments, if they exist, of the output random variable. A novel mathematical proof is provided to demonstrate that the moment of SCE of an arbitrary order converges to the exact moment for any degree of splines as the largest element size decreases. Complementary numerical analyses have been conducted, producing results consistent with theoretical findings. A collection of simple yet relevant examples is presented to grade the approximation quality of SCE with that of the well-known polynomial chaos expansion (PCE). The results from these examples indicate that higher-order moments calculated using SCE converge for all cases considered in this study. In contrast, the moments of PCE of an order larger than two may or may not converge, depending on the regularity of the output function or the probability measure of input random variables. Moreover, when both SCE- and PCE-generated moments converge, the convergence rate of the former is markedly faster than the latter in the presence of nonsmooth functions or unbounded domains of input random variables.

A novel hybrid framework based on modified continued-fraction for pile-soil dynamic interaction of large-scale nuclear power structures

Accurate unbounded domain radiation damping and ground motion input in the near field are important for analyzing the dynamic safety of large-scale nuclear power structures (NPSs). The scaled boundary finite element method (SBFEM) can automatically satisfy the abovementioned radiation condition. However, owing to the coupling of time and space, conventional SBFEM is not competent to the analysis of large-scale structures. In this investigation, a novel coupled method is developed to improve the computational efficiency of unbounded domain dynamic stiffness using a modified continued-fraction technique. Furthermore, a set of systematic and efficient computational procedures for soil-structure interaction (SSI) are implemented based on parallel programming using a special preprocess. The proposed method is verified using three numerical examples, and another numerical application is analyzed for large-scale NPSs with a pile-raft foundation (PRF). The results show that the innovative method has wide engineering applicability with high numerical accuracy and stability, indicating that this method can be applied to the SSI dynamic seismic analysis of large-scale NPSs.

Dynamical Behavior of the Fractional BBMB Equation on Unbounded Domain

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown.

Random attractors for three-dimensional stochastic globally modified Navier–Stokes equations driven by additive noise on unbounded domains

Abstract This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 𝒪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball’s idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time.

Numerical Reconstruction of Heat Source on a Semi-Infinite Rod

Inverse heat conduction on an unbounded domain has always been the focus and difficulty in engineering and mechanical fields. In this paper, we consider the inverse problem of identifying the heat source by using the terminal measured temperature data. Being different from other ordinary heat conduction problems, our mathematical model is defined in a semi-infinite region, which may lead to many numerical methods that cannot be used directly. Based on the artificial boundary method, we first introduce the truncated boundary and derive the exact boundary conditions on it. Then we use the finite difference method to establish an approximate difference scheme for the forward problem. For the inverse problem, we design the Landweber and conjugate gradient iterative algorithm and do numerical experiments. Numerical results verify the correctness and effectiveness of our algorithm.

A problem with analogue of the Frankl and mixing conditions for the Gellerstedt equation with singular coefficient

For the equation (sign y)| y| muxx + uyy + α0 | y| (m - 2)/2ux + (β0/y)uy = 0, $, considered in some unbounded mixed domain, uniqueness and existence theorems for a solution to the problem with the missing shift condition on the boundary characteristics and an analogue of the Frankl type condition on the interval of degeneracy of the equation are proved.

L p regularity for elliptic transmission problems with two spectral parameters

ABSTRACT We study a family of elliptic transmission problems on heterogeneous unbounded domains posed in UMD spaces. In order to determine the strict solution, this work investigates the effect and relationships between two spectral parameters; the first parameter is in the differential equation, and the second is in the permeability condition on the interface. Necessary and sufficient conditions on the data are established to prove the maximal regularity of the solution by using semigroup theory and the properties of fractional powers of linear operators

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

Monotonicity results of solutions to the uniformly elliptic nonlocal Bellman system

In this paper, we consider the uniformly elliptic nonlocal Bellman problem Fsu(x)=f(u(x),v(x)),Fsv(x)=g(u(x),v(x)).Firstly, we study narrow region principles for the uniformly elliptic nonlocal Bellman operators in bounded and unbounded domains, which play key roles in obtaining the main results by the process of sliding method. Then we deal with monotonicity properties of solutions to the uniformly elliptic nonlocal Bellman system.

Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains

Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains

Positive solutions for a coupled systems involving the fractional p(x)- Laplacian operator in unbounded domains

In this paper, we study the existence of at least one weak solution for a class of nonlocal elliptic systems involving the fractional p(x)-Laplacian. These operators are utilized to solve a system defined within unbounded domain. The right sides of systems treated are closely related to a type of gradient C1-functional which satisfy a sublinear growth conditions. Under some additional assumptions on the nonlinearities and leveraging the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we can use a variational approach to establish the existence result.

A revised condition for harmonic analysis in generalized Orlicz spaces on unbounded domains

AbstractConditions for harmonic analysis in generalized Orlicz spaces have been studied over the past decade. One approach involves the generalized inverse of so‐called weak ‐functions. It featured prominently in the monograph Orlicz Spaces and Generalized Orlicz Spaces [P. Harjulehto and P. Hästö, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019]. While generally successful, the inverse function formulation of the decay condition (A2) in the monograph contains a flaw, which we explain and correct in this note. We also present some new results related to the conditions, including a more general result for the density of smooth functions.