Boundary regularity and a priori estimates for fractional equations on unbounded domains

The numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.

We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.

An averaging principle for slow–fast fractional stochastic parabolic equations on unbounded domains

An efficient numerical scheme is developed to solve a linearized time fractional KdV equation on unbounded spatial domains. First, the exact absorbing boundary conditions (ABCs) are derived which reduces the pure initial value problem into an equivalent initial boundary value problem on a finite interval that contains the compact support of the initial data and the inhomogeneous term. Second, the stability of the reduced initial-boundary value problem is studied in detail. Third, an efficient unconditionally stable finite difference scheme is constructed to solve the initial-boundary value problem where the nonlocal fractional derivative is evaluated via a sum-of-exponentials approximation for the convolution kernel. As compared with the direct method, the resulting algorithm reduces the storage requirement from O ( M N ) O(MN) to O ( M log d ⁡ N ) O(M\log ^dN) and the overall computational cost from O ( M N 2 ) O(MN^2) to O ( M N log d ⁡ N ) O(MN\log ^dN) with M M the total number of spatial grid points and N N the total number of time steps. Here d = 1 d=1 if the final time T T is much greater than 1 1 and d = 2 d=2 if T ≈ 1 T\approx 1 . Numerical examples are given to demonstrate the performance of the proposed numerical method.

In this study, the Caputo fractional derivative is used to define time fractional Fokker-Planck equation in an unbounded domain. To solve this equation, the Jacobi polynomials together with the tanh-Jacobi functions are employed. The operational matrices of the classical and fractional derivatives of these basis functions are obtained to use them in constructing a numerical method for the expressed equation. In the proposed method, the introduced basis functions are used simultaneously to approximate the equation’s unknown solution. More precisely, the shifted Jacobi polynomials are applied to approximate the solution in the temporal direction and the tanh-Jacobi functions are utilized to approximate the solution in the spatial direction. By substitute the expressed approximation into the equation and employing the introduced operational matrix, solving the problem under consideration transforms into solving an algebraic system of equations, which can be solved easily. The accuracy and efficiency of the presented method are investigated numerically by solving some numerical examples. The reported results confirms the high accuracy of the established method.

A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions

Due to the nonlocality of fractional derivatives, the numerical methods for solving nonlinear fractional Whitham–Broer–Kaup (WBK) equations are time-consuming and tedious. Therefore, it is a research hotspot to explore the numerical solution of fractional-order WBK equation. The main goal of this study is to provide an efficient method for the fractional-in-space coupled WBK equations on unbounded domain and discover some novel anomalous transmission behaviors. First, the numerical solution is compared with the exact solution to determine the validity of the proposed method on large time-spatial domain. Then, anomalous transmission of waves propagation of the fractional WBK equation is numerically simulated, and the influence of different fractional-order derivatives on wave propagation of the WBK equation is researched. Some novel anomalous transmission behaviors of wave propagation of the fractional WBK equation on unbounded domain are shown.

<p style='text-indent:20px;'>We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{3} $\end{document}</tex-math></inline-formula> are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method.</p>

This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of [32] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space.

In this paper, we investigate the long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains with a polynomial drift term of arbitrary order driven by nonlinear noise. We first define a mean random dynamical system in a Hilbert space for the solutions of the equation and prove the existence and uniqueness of weak pullback mean random attractors. We then establish the existence and regularity of invariant measures of the system under further conditions on the nonlinear delay and diffusion terms. We also prove the tightness of the set of all invariant measures of the equation when the time delay varies in a bounded interval. We finally show that every limit of a sequence of invariant measures of the delay equation must be an invariant measure of the limiting system as delay approaches zero. The uniform tail-estimates and the Ascoli–Arzelà theorem are used to derive the tightness of distribution laws of solutions in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.

Large deviations of fractional stochastic equations with non-Lipschitz drift and multiplicative noise on unbounded domains

In this note we discuss a slight generalization of the following result by Alt and Caffarelli: if the logarithm of the Poisson kernel of a Reifenberg flat chord arc domain is Hölder continuous, then the domain can be locally represented as the area above the graph of a function whose gradient is Hölder continuous. In this note we show that if the Poisson kernel of an unbounded Reifenberg flat chord arc domain is 1 a.e. on the boundary then the domain is (modulo rotation and translation) the upper half plane. This result plays a key role in the study of regularity of the free boundary below the continuous threshold.

Two efficient spectral methods for the nonlinear fractional wave equation in unbounded domain

In this paper, we first prove a uniform contraction principle for verifying the uniform large deviation principles of locally Hölder continuous maps in Banach spaces. We then show the local Hölder continuity of the solutions of a class of fractional parabolic equations with polynomial drift of any order defined on [Formula: see text]. We finally establish the large deviation principle of the fractional stochastic equations uniformly with respect to bounded initial data, despite the solution operators are not compact due to the non-compactness of Sobolev embeddings on unbounded domains.

In this article, we study a non-local Love problem on unbounded domains where the non-locality in the main equation is interpreted as a fractional Laplacian operator. With various assumptions on the initial conditions, we derive several estimates for mild solutions for the homogenous source scenario. For the nonlinear problem, we show the existence and uniqueness of a global mild solution. In two cases, we obtain convergence results. The first one states that the solution to the fractional Love equation converges to the mild solution of the fractional wave equation according to a cross-section radius parameter. The second result shows that solutions of the fractional Love equation incorporating the fractional Laplacian operator converge to those of the classical problem, involving the usual Laplacian, as the fractional orders approach $1$. This work is the first that we are aware of that deals with mild solutions of Love equations on unbounded domains. For more information see https://ejde.math.txstate.edu/Volumes/2025/76/abstr.html