A diffusion equation in transparent media

Andreu, F.; Caselles, Vicent; Mazón, José M.; Moll, Salvador

doi:10.1007/s00028-007-0249-3

Cited by 25 publications

(34 citation statements)

References 16 publications

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“…(14) tends to the classical heat equation [9]. Moreover, (as proved in [4]) if m ! þ1, the relativistic heat equation tends to the transparent media diffusion equation…”

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 88%

“…Andreu et al in [1][2][3][4][5] have performed an extended analytical study on the relativistic heat equation and its solution for a particular class of initial data. However there are not at the moment explicit solutions of the relativistic heat equation.…”

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 99%

“…Rosenau proposes to use a different velocity fieldṽ R instead ofṽ in (4) to enforce the magnitude of the velocity field not to exceed the speed of light c > 0. Then the modified heat equation…”

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 99%

“…would approach classical heat equation (4) for small jṽ R j and propagate the solution with limited speed for large jṽ R j.…”

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 99%

“…The diffusion of relativistic heat equation is confined to a region limited by a front that propagates at the speed of light. Theory of existence and uniqueness of the solution of the relativistic heat equation has been proved by Andreu and collaborators in [1,2,4,5] as a particular case of a generalized class of the Fokker-Planck equations. These authors develop a mathematical theory of the entropy solutions associated to (2).…”

Section: Introductionmentioning

confidence: 98%

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Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

Marquina¹

2010

Journal of Computational Physics

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“…(14) tends to the classical heat equation [9]. Moreover, (as proved in [4]) if m ! þ1, the relativistic heat equation tends to the transparent media diffusion equation…”

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 88%

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 99%

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 99%

“…would approach classical heat equation (4) for small jṽ R j and propagate the solution with limited speed for large jṽ R j.…”

Section: A Heuristic Approach To the Relativistic Heat Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations

Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

Marquina¹

2010

Journal of Computational Physics

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Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Rahimkhani

Ordokhani

2018

Optim Control Appl Methods

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In this paper, a method for finding an approximate solution of a class of 2D fractional optimal control problems with fractional-order dynamical system is discussed. In the proposed method, the fractional derivative is expressed in the Caputo sense. The method consists of expanding unknown functions as the elements of two-dimensional (2D) Müntz-Legendre wavelets. The 2DMüntz-Legendre wavelets are constructed and their properties are presented.The operational matrix of fractional-order integration for these wavelets is utilized to reduce the solution of 2D fractional optimal control problem to an optimization problem, which can then be solved easily. Some results concerning the error analysis are obtained. Finally, two illustrative test problems are included to demonstrate the validity and applicability of the technique. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method. KEYWORDSCaputo derivative, fractional optimal control problem, numerical solution, Riemann-Liouville fractional integral operator, wavelet method INTRODUCTIONNowadays, fractional calculus have a wide range of applications in fields of fluid-dynamic traffic, 1 frequency dependent damping behavior of various viscoelastic materials, 2 colored noise, 3 solid mechanics, 4 economics, 5 signal processing, 6 and control theory. 7 Therefore, many researchers have been interested in finding the approximate solutions of these problems, and some of these methods include spectral Tau method, 8 finite difference method, 9 wavelet method, 10 least square approximation method, 11 finite element method, 12 spline collocation method, 13 fractional wavelet method, 14 shifted fractional-order Jacobi orthogonal functions, 15 Jacobi Tau method, 16,17 and Legendre Tau method. 18 An optimal control problem (OCP) is defined to minimize the functional over an admissible set of control functions subject to dynamic constraints on the state and control variables. A fractional OCP (FOCP) is an optimal control problem in which the performance index or the differential equations governing the dynamics of the system or both contains at least one fractional-order derivative term. In recent the years, several numerical methods have been devoted to solve of one-dimensional FOCPs. We list here some of these numerical methods, as follows: • Eigen functions method (Agrawal 19 ); • Quadratic numerical scheme (Agrawal 20 ); • Rational approximation method (Tricaud and Chen 21 ); • Legendre multiwavelet collocation method (Yousefi et al 22 ); 1916

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