Fractal Burgers Equations

Abstract

The paper studies local and global in time solutions to a class of multidimensional generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and with general algebraic nonlinearity. Such equations naturally appear in continuum mechanics. Our results include existence, uniqueness, regularity and asymptotic behavior of solutions to the Cauchy problem as well as a construction of self-similar solutions. The role of critical exponents is also explained. Academic Press

Map of discrete system into continuous

“…If we consider J 2 (n, m) with fractional α 2 = α, then we get the fractional Burgers equation that is suggested in Ref. [48].…”

Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order

“…The space-and time-fractional Burgers equations have been of considerable interest in recent literature [1,19]. These equations have many applications in science and engineering.…”

“…In [20], Sugimoto developed an asymptotic and numerical analysis of the generalized Burgers equation with a space-fractional derivative. ElShahed [2] obtained an analytical solution of the spacefractional Burgers equation with β = 1 2 by considering the Adomian decomposition scheme. Biler, Funaki and Woyczynski considered the existence and uniqueness and self-similar properties of solutions of the space-fractional Burgers equation in [1].…”

“…ElShahed [2] obtained an analytical solution of the spacefractional Burgers equation with β = 1 2 by considering the Adomian decomposition scheme. Biler, Funaki and Woyczynski considered the existence and uniqueness and self-similar properties of solutions of the space-fractional Burgers equation in [1]. In [15], Momani used the decomposition method to obtain approximate solutions for the generalized Burgers equation with time-and space-fractional derivatives.…”

Exact solutions of supersymmetric Burgers equation with Bosonization procedure

“…This equation is also applied to other phenomenon such as physics, chemistry, mathematical biology, etc. In view of the fractional calculus theory, the fractional Burgers equation was proposed and several properties were discovered [22][23][24][25]. The N D 1 supersymmetric version is established with usual temporal variable t and super spatial variables .x; Â/, where  is a Grassmann variable satisfying  2 D 0, and the field u leads to a fermionic superfield .Â; x; t / D .x; t / C  u.x; t /:…”