Entropy solution theory for fractional degenerate convection–diffusion equations
01 May 2011-Annales De L Institut Henri Poincare-analyse Non Lineaire (Elsevier Masson)-Vol. 28, Iss: 3, pp 413-441
TL;DR: In this paper, a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term is studied and well-posedness results are obtained for weak entropy solutions.
Abstract: We study a class of degenerate convection–diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection–diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Levy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.
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TL;DR: In this article, a theory of existence and uniqueness for the following porous medium equation with fractional diffusion was developed, where data f ∈ L 1 (R N ) and all exponents 0 0.
Abstract: We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, @u @t + (−�) �/2 (|u| m−1 u) = 0, x ∈ R N , t > 0, u(x,0) = f(x), x ∈ R N . We consider data f ∈ L 1 (R N ) and all exponents 0 0. Existence and uniqueness of a weak solution is established for m > m∗ = (N − �)+/N, giving rise to an L 1 -contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range 0 < m ≤ m∗ existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above m∗. We also study the dependence of solutions on f,m and �. Moreover, we consider the above questions for the problem posed in a bounded domain.
235 citations
TL;DR: The results reveal that the method is very effective and simple in determination of solution of the fractional KdV-Burgers equation.
208 citations
Cites background or methods from "Entropy solution theory for fractio..."
...The reader is asked to refer to [12–23] in order to know more details about the mathematical properties of fractional erivatives and fractional integrals, including their types, their motivation for use, their characteristics, and their applicaions....
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...These equations are increasingly used to model problems in research areas as diverse as dynamical ystems, mechanical systems, control theory, heat transfer, chaos synchronization, mixed convection flows, anomalous difusive, unification of diffusion, wave propagation phenomenon, image processing, and entropy theory [10–18]....
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01 Jan 2012
TL;DR: In this article, the authors describe two models of flow in porous media including nonlocal (long-range) diffusion effects, based on Darcy's law and inverse fractional Laplacian operator, and prove existence of solutions that propagate with finite speed.
Abstract: We describe two models of flow in porous media including nonlocal (long-range) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that these special solutions describe the asymptotic behavior of a wide class of solutions.
188 citations
Cites background from "Entropy solution theory for fractio..."
...See also the paper by Cifani and Jakobsen [31] for an alternative L1 theory dealing with a more general class of nonlocal porous medium equations, including strong degeneration and convection....
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TL;DR: It is proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative, and a new construction of the generalized Taylor’s power series is obtained.
Abstract: In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative. A new construction of the generalized Taylor’s power series is obtained. Some applications including approximation of fractional derivatives and integrals of functions and solutions of linear and nonlinear fractional differential equations are also given. In the nonlinear case, the new and simple technique is used to find out the recurrence relation that determines the coefficients of the fractional power series.
158 citations
TL;DR: In this article, a numerical method for the fractional Laplacian was proposed, based on the singular integral representation for the operator, which combines finite differences with numerical quadrature.
Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.
143 citations
References
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Abstract: WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Levy processes are beyond their reach.Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by their use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations.Topics covered in this book include: jump-diffusion models, Levy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms.This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable new set of tools for modelling market fluctuations.
3,210 citations
Book•
01 Jan 2004TL;DR: In this paper, the authors present a general theory of Levy processes and a stochastic calculus for Levy processes in a direct and accessible way, including necessary and sufficient conditions for Levy process to have finite moments.
Abstract: Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs
2,908 citations
"Entropy solution theory for fractio..." refers methods in this paper
...We refer to [4,14] for the theory and applications of such processes and to [32] for a very relevant and nice discussion and many more references....
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