Fractal Burgers Equations
TL;DR: In this paper, the authors studied local and global in time solutions to a class of generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and with general algebraic nonlinearity.
About: This article is published in Journal of Differential Equations.The article was published on 1998-09-01 and is currently open access. It has received 215 citations till now. The article focuses on the topics: Nonlinear system & Continuum mechanics.
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TL;DR: In this article, a space fractional Fokker-Planck equation (SFFPE) with instantaneous source is considered, and a numerical scheme for solving SFFPE is presented.
699 citations
TL;DR: In this paper, the percolation constant ≈ 1.327 is defined as a universal parameter describing the topology of nonequilibrium (quasi)stationary states in complex nonlinear dynamical systems allowing self-organized critical behavior.
Abstract: The goal of this review is to outline some unconventional ideas behind new paradigms in the modern theory of turbulence. Application of nonstandard, topological methods to describe the structural properties of the turbulent state is considered and the transition to kinetic equations in fractional derivatives for describing the microscopic behavior of a medium is examined. Central to the discussion is the concept of the percolation constant ≈1.327..., a universal parameter describing the topology of nonequilibrium (quasi)stationary states in complex nonlinear dynamical systems allowing self-organized critical behavior. Much attention is given to the formation of power-law energy density spectra in turbulent media. A number of topical problems in modern cosmic electrodynamics, including the self-consistent fractal model of a turbulent current sheet, substorm dynamics, and the formation and dynamical evolution of large-scale magnetic fields in the solar photosphere and interplanetary space, are also discussed.
219 citations
TL;DR: In this paper, a time fractional advection-dispersion equation was obtained from the standard advective dispersion equations by replacing the first-order derivative in time by a fractional derivative in order α(0 < α<-1).
Abstract: A time fractional advection-dispersion equation is obtained from the standard advection-dispersion equation by replacing the firstorder derivative in time by a fractional derivative in time of order α(0<α<-1). Using variable transformation, Mellin and Laplace transforms, and properties of H-functions, we derive the complete solution of this time fractional advection-dispersion equation.
210 citations
TL;DR: In this paper, the authors derived the fractional generalization of the Ginzburg-Landau equation from the variational Euler-Lagrange equation for fractal media.
Abstract: We derive the fractional generalization of the Ginzburg–Landau equation from the variational Euler–Lagrange equation for fractal media. To describe fractal media we use the fractional integrals considered as approximations of integrals on fractals. Some simple solutions of the Ginzburg–Landau equation for fractal media are considered and different forms of the fractional Ginzburg–Landau equation or nonlinear Schrodinger equation with fractional derivatives are presented. The Agrawal variational principle and its generalization have been applied.
205 citations
TL;DR: In this article, the authors consider semi-linear partial differential equations involving a particular pseudo-differential operator and show the convergence of the solution towards the entropy solution of the pure conservation law and the non-local Hamilton-Jacobi equation.
Abstract: The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.
198 citations
Cites background from "Fractal Burgers Equations"
...One of the first works on this subject is probably [4], which deals with (3) when h = 0 and f(t, x, u) = f(u); using energy estimates, it states some local-in-time existence and uniqueness results of weak solutions if f has a polynomial growth....
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References
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01 Jan 1941
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Abstract: Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear partial differential equations: Sobolev spaces Second-order elliptic equations Linear evolution equations Part III: Theory for nonlinear partial differential equations: The calculus of variations Nonvariational techniques Hamilton-Jacobi equations Systems of conservation laws Appendices Bibliography Index.
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01 Jan 1969
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31 Dec 1969
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Abstract: linear and quasi linear equations of parabolic type by o a ladyzhenskaia 1968 american mathematical society edition in english, note citations are based on reference standards however formatting rules can vary widely between applications and fields of interest or study the specific requirements or preferences of your reviewing publisher classroom teacher institution or organization should be applied, we consider a hyperbolicparabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation, pris 1899 kr hftad 1968 tillflligt slut bevaka linear and quasi linear equations of parabolic type s fr du ett mejl nr boken gr att kpa igen, then u x t solves the following system of quasilinear parabolic pde where y is the infinitesimal operator generated by the diffusion process y a particular case is that of linear one dimensional backward equation where f does not contain q in this case the corresponding system of equation becomes a linear parabolic pde, we consider linear parabolic equations of second order in a sobolev space setting we obtain existence and uniqueness results for such equations on a closed two dimensional manifold with minimal assumptions about the regularity of the coefficients of the elliptic operator, linear equations of the second order of parabolic type a m il in a s kalashnikov and o a oleinik the solvability of mixed problems for hyperbolic and parabolic equations v a il in quasi linear elliptic equations and variational problems with many independent variableso a ladyzhenskaya and n n ural tseva, the first boundary value problem for a linear second order parabolic equation is studied under the assumption that the inhomogeneous term is continuous in space and time and hlder continuous only with respect to the space variables linear and quasilinear equations of parabolic type transl math monographs 23 providence amer math, collapse in finite time is established for part of the solutions of certain classes of quasilinear equations of parabolic and hyperbolic types the linear part of which has general form certain hyperbolic equations having l m pairs belong to these classes, find helpful customer reviews and review ratings for linear and quasi linear equations of parabolic type at amazon com read honest and unbiased product
7,118 citations
TL;DR: A model is proposed for the evolution of the profile of a growing interface that exhibits nontrivial relaxation patterns, and the exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations.
Abstract: A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.
4,299 citations
Book•
01 Jan 1983
TL;DR: In this paper, the basics of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley, are presented in a way accessible to a wider audience than just mathematicians.
Abstract: The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index For the second edition, typographical errors and other mistakes have been corrected and a new chapter on recent results has been added The new chapter contains discussion of the stability of travelling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for constructing travelling-wave solutions for systems of non-linear equations
3,991 citations