Some properties of fractional burgers equation

Abstract

The fractional generalization of a one‐dimensional Burgers equation

with initial conditions ?(x, 0) = ?₀(x); ?_t(x,0) = ψ₀ (x), where ? = ?(x,t) ∈ C²(O): ?_t = ??/?t; _aD_x ^p is the Riemann‐Liouville fractional derivative of the order p; O = (x,t) : x ∈ E ¹, t > 0; and the explicit form of a particular analytical solution are suggested. Existing of traveling wave solution and conservation laws are considered. The relation with Burgers equation of integer order and properties of fractional generalization of the Hopf‐Cole transformation are discussed.

Pasiublytas trupmeninis vienmates Biurgerso lygties apibendrinimas

su pradinemis salygomis ?(x, 0) = ?₀(x); ?_t(x,0) = ψ₀ (x), where ? = ?(x,t) ∈ C²(O): ?_t = ??/?t; _aD_x ^p yra Rymano bei Liuvilio trupmenine p eiles išvestine; O p; O = (x,t) : x ∈ E ¹, t > 0;: bei šios lygties atskiras analitinis sprendinys. Nagrinejamas impulso bei energijos tvermes desniu atitinkami apibendrinimai, saryšis su paprasta Biurgerso lygtimi ir trupmenines Hopfo bei Koulo transformacijos savybes.