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Register a new userSubdiffusion with a time-dependent coefficient: analysis and numerical solution
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In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.
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For the first time, we develop a convergent numerical method for the llinear integral equation derived by M.M. Lavrentev in 1964 with the goal to solve a coefficient inverse problem for a wave-like equation in 3D. The data are non overdetermined. Convergence analysis is presented along with the numerical results. An intriguing feature of the Lavrentev equation is that, without any linearization, it reduces a highly nonlinear coefficient inverse problem to a linear integral equation of the first kind. Nevertheless, numerical results for that equation, which use the data generated for that coefficient inverse problem, show a good reconstruction accuracy. This is similar with the classical Gelfand-Levitan equation derived in 1951, which is valid in the 1D case.
We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-and- time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.
Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications such as in biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of applied mathematics, and provides tools to efficiently simulate them. We show that a sticky Brownian motion arises naturally for a particle diffusing on $mathbb{R}_+$ with a strong, short-ranged potential energy near the origin. This is a limit that accurately models mesoscale particles, those with diameters $approx 100$nm-$10mu$m, which form the building blocks for many common materials. We introduce a simple and intuitive sticky random walk to simulate sticky Brownian motion, that also gives insight into its unusual properties. In parameter regimes of practical interest, we show this sticky random walk is two to five orders of magnitude faster than alternative methods to simulate a sticky Brownian motion. We outline possible steps to extend this method towards simulating multi-dimensional sticky diffusions.
In this work, we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation which involves a fractional derivative of order $alphain(0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational type discretization. With a space mesh size $h$ and time stepsize $tau$, we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(tau^{min({1}/{2}+alpha-epsilon,1)}+h^2)$ in the discrete $L^2(0,T;L^2(Omega))$ norm and $O(tau^{alpha-epsilon}+ell_h^2h^2)$ in the discrete $L^infty(0,T;L^2(Omega))$ norm, with any small $epsilon>0$ and $ell_h=ln(2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.
A posteriori error analysis is a technique to quantify the error in particular simulations of a numerical approximation method. In this article, we use such an approach to analyze how various error components propagate in certain moving boundary problems. We study quasi-steady state simulations where slowly moving boundaries remain in mechanical equilibrium with a surrounding fluid. Such problems can be numerically approximated with the Method of Regularized Stokelets(MRS), a popular method used for studying viscous fluid-structure interactions, especially in biological applications. Our approach to monitoring the regularization error of the MRS is novel, along with the derivation of linearized adjoint equations to the governing equations of the MRS with a elastic elements. Our main numerical results provide a clear illustration of how the error evolves over time in several MRS simulations.
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