Ambarzumyan-Type Theorem for a Conformable Fractional Diffusion Operator

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $\Omega \subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $\Omega$. In $\Omega\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(\Omega)$. We also prove that tensorization of a $P_1$-FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_\Omega$, the number of degrees of freedom in the domain $\Omega$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $\Omega$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_\Omega$. Finally, under the stronger assumption that the data is analytic in $\overline{\Omega}$, and without compatibility at $\partial \Omega$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

We examine the fractional heat diffusion operators L γ , a := ( − Δ a ) γ 2 + ∂ t , where Δ a is the Laplace- or the Bessel-Laplace operator. We give conditions for removability which are sufficient and which are necessary, by L p -capacities. Introducing a spherical modulus of smoothness we can treat the Laplace and Bessel-Laplace cases together.

Radial basis functions method for solving the fractional diffusion equations

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are respectively verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.

Several models based on discrete and continuous fields have been proposed to comprehend residential criminal dynamics. This study introduces a two-dimensional model to describe residential burglaries diffusion, employing Lèvy flights dynamics. A continuous model is presented, introducing bidimensional fractional operator diffusion and its differences with the 1-dimensional case. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even fractional derivative order is modified. We also provide qualitative evidence that steady-state approximation in one dimension by series expansion is insufficient to capture similar original system behavior. At least for the case where series coefficients have a linear relationship with derivative order. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even if fractional derivative order is modified. Two dynamic regimes emerge in maximum and total attractiveness magnitude as a result of fractional derivative changes, these regimes can be understood as considerations about different urban environments. Finally, we add a Law enforcement component, embodying the "Cops on dots" strategy; in the Laplacian diffusion dynamic, global attractiveness levels are significantly reduced by Cops on dots policy but lose efficacy in Lèvy flight-based diffusion regimen. The four-step Preditor-Corrector method is used for numerical integration, and the fractional operator is approximated, getting the advantage of the spectral methods to approximate spatial derivatives in two dimensions.

Fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second-order differential equations. There are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second-order differential equations. These are (i) fractional differential operators are nonlocal operators, and (ii) the adjoint of a fractional differential operator is not the negative of itself. The wellposedness of a Galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously. Many subsequent works were carried out to extend the analysis to other numerical methods. A constant diffusivity coefficient has been assumed in all these works. In this paper we present a counterexample which shows that the Galerkin weak formulation loses coercivity in the context of variable-coefficient conservative fractional elliptic differential equations. Hence, the previous results cannot be extended to variable-coefficient conservative fractional elliptic differential equations. We adopt an alternative approach to prove the existence and uniqueness of the classical solution to the variable-coefficient conservative fractional elliptic differential equation and characterize the solution in terms of the classical solutions to second-order elliptic differential equations. Furthermore, we derive a Petrov--Galerkin weak formulation to the fractional elliptic differential equation. We prove that the bilinear form of the Petrov--Galerkin weak formulation is weakly coercive and so the weak formulation has a unique weak solution and is well posed. Finally, we outline potential application of these results in the development of numerical methods for variable-coefficient conservative fractional elliptic differential equations.

A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\\alpha$ ($0 \\lt \\alpha \\lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

Let L be an elliptic differential operator on a complete connected Riemannian manifold M such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let L ( α ) ${L^{(\alpha )}}$ be the α-stable subordination of L for α ∈ ( 1 , 2 ) ${\alpha \in (1,2)}$ . We found some classes 𝕂 α γ , β ( β , γ ∈ [ 0 , α ) ) ${\mathbb {K}_{\alpha }^{\gamma ,\beta } (\beta ,\gamma \in [0,\alpha ))}$ of time-space functions containing the Kato class, such that for any measurable functions b : [ 0 , ∞ ) × M → T M ${b: [0,\infty )\times M\rightarrow TM}$ and c : [ 0 , ∞ ) × M → M ${c: [0,\infty )\times M\rightarrow M}$ with | b | , c ∈ 𝕂 α 1 , 1 ${|b|, c\in \mathbb {K}_{\alpha }^{1,1}}$ , the operator L b , c ( α ) ( t , x ) : = L ( α ) ( x ) + 〈 b ( t , x ) , ∇ · 〉 + c ( t , x ) , ( t , x ) ∈ [ 0 , ∞ ) × M , $ L_{b,c}^{(\alpha )}(t,x):= L^{(\alpha )}(x)+ \langle b(t,x),\nabla \cdot \,\rangle +c(t,x),\quad (t,x)\in [0,\infty )\times M,\vspace*{-1.42262pt} $ has a unique heat kernel p b , c ( α ) ( t , x ; s , y ) ${p_{b,c}^{(\alpha )}(t,x;s,y)}$ , 0 ≤ s < t ${0\le s<t}$ , x , y ∈ M ${x,y\in M}$ , which is jointly continuous and satisfies t - s C { ρ ( x , y ) ∨ ( t - s ) 1 α } d + α ≤ p b , c ( α ) ( t , x ; s , y ) ≤ C ( t - s ) { ρ ( x , y ) ∨ ( t - s ) 1 α } d + α $ \frac{t-s}{C\lbrace \rho (x,y)\vee (t-s)^{\frac{1}{\alpha }}\rbrace ^{d+\alpha }}\le p_{b,c}^{(\alpha )}(t,x;s,y)\le \frac{C(t-s)}{\lbrace \rho (x,y)\vee (t-s)^{\frac{1}{\alpha }}\rbrace ^{d+\alpha }}\vspace*{-1.42262pt} $ and | ∇ x p b , c ( α ) ( t , x ; s , y ) | ≤ C ( t - s ) α - 1 α { ρ ( x , y ) ∨ ( t - s ) 1 α } d + α , 0 ≤ s < t , x , y ∈ M , $ \big |\nabla _x p_{b,c}^{(\alpha )}(t,x; s,y)\big |\le \frac{C(t-s)^{\frac{\alpha -1}{\alpha }}}{\lbrace \rho (x,y)\vee (t-s)^{\frac{1}{\alpha }}\rbrace ^{d+\alpha }},\quad 0\le s<t,\; x,y\in M,\vspace*{-1.42262pt} $ for some constant C > 1, where ρ is the Riemannian distance. The estimate of ∇ y p b , c ( α ) ${\nabla _yp^{(\alpha )}_{b,c}}$ and the Hölder continuity of ∇ x p b , c ( α ) ${\nabla _x p_{b,c}^{(\alpha )}}$ are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where L = Δ ${L=\Delta }$ on ℝ d and b, c are time-independent.

In this paper we analyze the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population super-diffusion. First, we study how cross super-diffusion influences the formation of spatial patterns: a linear stability analysis is carried out, showing that cross super-diffusion triggers Turing instabilities, whereas classical (self) super-diffusion does not. In addition we perform a weakly nonlinear analysis yielding a system of amplitude equations, whose study shows the stability of Turing steady states. A second goal of this contribution is to propose a fully adaptive multiresolution finite volume method that employs shifted Grünwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators. The scheme is aimed at efficient dynamic mesh adaptation and substantial savings in computational burden. A numerical simulation of the model was performed near the instability boundaries, confirming the behavior predicted by our analysis.

Data-driven learning of nonlocal physics from high-fidelity synthetic data

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.

Let us consider the equation 𝒜α u = f , 0 < α < 1, where 𝒜 is a selfadjoint positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω ∈ Rd, d ∈ {1, 2, 3}. We assume that the non-local fractional diffusion operator 𝒜α is defined through the spectral decomposition of 𝒜. The current advances in numerical methods for such spectral spatial-fractional diffusion problems have been analyzed in [6]. The presented results are in the spirit of BURA methods. This class of methods is based on the best uniform rational approximation of a scalar function in the interval [0, 1] (see [8] and the references therein). The focus of our study is on some challenges beyond the scalar elliptic case.

We propose and analyze new numerical methods to evaluate fractional norms and apply fractional powers of elliptic operators. By means of a reduced basis method, we project to a small dimensional subspace where explicit diagonalization via the eigensystem is feasible. The method relies on several independent evaluations of ({{,mathrm{I},}}-t_i^2Delta )^{-1}f, which can be computed in parallel. We prove exponential convergence rates for the optimal choice of sampling points t_i, provided by the so-called Zolotarëv points. Numerical experiments confirm the analysis and demonstrate the efficiency of our algorithm.

A two-dimensional fractional Laplacian operator is derived and used to model nonlocal, nondiffusive transport. This integro-differential operator appears in the long-wavelength, fluid description of quantities undergoing non-Brownian random walks without characteristic length scale. To study bounded domains, a mask function is introduced that modifies the kernel in the fractional Laplacian and removes singularities at the boundary. Green's function solutions to the fractional diffusion equation are presented for the unbounded domain and compared to the one-dimensional Cartesian approximations. A time-implicit numerical integration scheme is presented to study fractional diffusion in a circular disk with azimuthal symmetry. Numerical studies of steady-state reveal temperature profiles in which the heat flux and temperature gradient are in the same direction, i.e., uphill transport. The response to off-axis heating, scaling of confinement time with system size, and propagation of cold pulses are investigated.