On numerical methods and error estimates for degenerate fractional convection-diffusion equations

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods - even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations

Numerical approximation of statistical solutions of scalar conservation laws

We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multi-level Monte Carlo discretizations of the probability space. Both sets of methods are proved to converge to the entropy statistical solution. We also prove that there is a considerable gain in efficiency resulting from the multi-level Monte Carlo method over the standard Monte Carlo method. Numerical experiments illustrating the ability of both methods to accurately compute multi-point statistical quantities of interest are also presented

On the spectral vanishing viscosity method for periodic fractional conservation laws

We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump L\'{e}vy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kru\v{z}kov-Alibaud entropy formulation and prove well-posedness. Then we introduce the numerical method, which is a non-linear Fourier Galerkin method with an additional spectral viscosity term. This type of approximation was first introduced by Tadmor for pure conservation laws. We prove that this {\em non-monotone} method converges to the entropy solution of the problem, that it retains the spectral accuracy of the Fourier method, and that it diagonalizes the fractional term reducing dramatically the computational cost induced by this term. We also derive a robust $L^1$-error estimate, and provide numerical experiments for the fractional Burgers' equation

Stability of Entropy Solutions for Levy Mixed Hyperbolic-Parabolic Equations

We analyze entropy solutions for a class of Levy mixed hyperbolicparabolic equations containing a non-local (or fractional) diffusion operator originating from a pure jump Levy process. For these solutions we establish uniqueness (L1 contraction property) and continuous dependence results.Comment: 20 pages, no figures. Manuscript submitted to Elseive

A note on models for anomalous phase-change processes

We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier's law leads to a natural generalization of the classical Stefan problem in which time derivatives are replaced by fractional ones

A generalization of HH-measures and application on purely fractional scalar conservation laws

We extend the notion of $H$-measures on test functions defined on $\R^d\times P$, where $P\subset \R^d$ is an arbitrary compact simply connected Lipschitz manifold such that there exists a family of regular nonintersecting curves issuing from the manifold and fibrating $\R^d$. We introduce a concept of quasi-solutions to purely fractional scalar conservation laws and apply our extension of the $H$-measures to prove strong $L^1_{loc}$ precompactness of such quasi-solutions.Comment: 11 pages, 1 figure; to appear in CPA

A Central Difference Numerical Scheme for Fractional Optimal Control Problems

This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a fractional derivative (FDs) at a time node point is approximated using a modified Gr\"{u}nwald-Letnikov approach. For the first order derivative, the proposed modified Gr\"{u}nwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.Comment: 16 pages, 12 figure

Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates

We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller type such as the fractal Burgers equation. The existence of traveling wave solutions with monotone decreasing profile has been established recently (in special cases). We show the local asymptotic stability of these traveling wave solutions in a Sobolev space setting by constructing a Lyapunov functional. Most importantly, we derive the algebraic-in-time decay of the norm of such perturbations with explicit algebraic-in-time decay rates

A variational approach to the analysis of non-conservative mechatronic systems

We develop a method for systematically constructing Lagrangian functions for dissipative mechanical, electrical and, mechatronic systems. We derive the equations of motion for some typical mechatronic systems using deterministic principles that are strictly variational. We do not use any ad hoc features that are added on after the analysis has been completed, such as the Rayleigh dissipation function. We generalise the concept of potential, and define generalised potentials for dissipative lumped system elements. Our innovation offers a unified approach to the analysis of mechatronic systems where there are energy and power terms in both the mechanical and electrical parts of the system. Using our novel technique, we can take advantage of the analytic approach from mechanics, and we can apply these pow- erful analytical methods to electrical and to mechatronic systems. We can analyse systems that include non-conservative forces. Our methodology is deterministic and does does require any special intuition, and is thus suitable for automation via a computer-based algebra package

A note on the nonlinear Schr\"odinger equation in a general domain

We consider the Cauchy problem for nonlinear Schr\"odinger equations in a general domain $\Omega\subset\mathbb{R}^N$. Construction of solutions has been only done by classical compactness method in previous results. Here, we construct solutions by a simple alternative approach. More precisely, solutions are constructed by proving that approximate solutions form a Cauchy sequence in some Banach space. We discuss three different types of nonlinearities: power type nonlinearities, logarithmic nonlinearities and damping nonlinearities.Comment: minor revision; 24 pages. To appear in Nonlinear Analysi