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Register a new userBoundary Conditions for Fractional Diffusion
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This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.
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We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(Omega)$ and $L_1(Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.
We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations.
In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomor-phic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak-$L^1$ theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method, and we finally show the existence of large solutions for some power-like nonlinearities.
In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion $partial_t u = text{div}(k(x) abla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x) abla G(u)cdot u = 0$. Here $xin Bsubset mathbb{R}^d$, a bounded open set with locally Lipchitz boundary, and with $ u$ as the unit outer normal. The function $G$ is Lipschitz continuous and nondecreasing, while $k(x)$ is diagonal matrix. We show that any two weak entropy solutions $u$ and $v$ satisfy $Vert{u(t)-v(t)}Vert_{L^1(B)}le Vert{u|_{t=0}-v|_{t=0}}Vert_{L^1(B)}e^{Ct}$, for almost every $tge 0$, and a constant $C=C(k,G,B)$. If we restrict to the case when the entries $k_i$ of $k$ depend only on the corresponding component, $k_i=k_i(x_i)$, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.
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