scholarly journals Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Luciano Abadias ◽  
Pedro J. Miana
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Infinitesimal Generator ◽  
Fractional Derivatives ◽  
Integral Inequalities ◽  
Fractional Powers ◽  
Partial Differential ◽  
Infinitesimal Generators ◽  
Positive Kernel ◽  
Quasigeostrophic Equations

In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂/∂t+u·∇f=-σ-Aαf,  0≤α≤1, where (A,D(A)) is the infinitesimal generator of a convolution C0-semigroup of positive kernel on Lp(Rn), with 1≤p<∞. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (-A)α for 0≤α≤1. We use these estimates to obtain Lp-decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking A=Δ, the Laplacian), which has been studied in the literature.

scholarly journals Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190564
Author(s):  
Zhi-Yong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Algebra ◽  
Infinitesimal Generator ◽  
Linear Time ◽  
Lie Symmetry ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Pde ◽  
Leading Position

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

Researches on a Class of Reaction-Diffusion Thermo-Plastic Material Equations

2011 ◽  
Vol 219-220 ◽  
pp. 1022-1025 ◽  
Author(s):  
Shu Xian Deng ◽  
Ming Jun Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Plastic Material ◽  
Reaction Diffusion ◽  
Fractional Powers ◽  
Partial Differential ◽  
Global Strong Solution ◽  
Material Equation ◽  
Material Equations ◽  
New Equation

This paper deals with a class of hyperbolic thermo-plastic material equation. The equation includes a reaction-diffusion-taxis partial differential equation, a reaction-diffusion partial differential equation. In the actual course of the discussion, we append a motility term in the equation. Then, the existence of unique global strong solution is proved using the theory of fractional powers of analytic semi group generators to new equation.

scholarly journals Towards a combined fractional mechanics and quantization

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Hamiltonian Formalism ◽  
Partial Differential ◽  
Caputo Fractional Derivatives ◽  
Fractional Mechanics ◽  
Fractional Calculus Of Variations

AbstractA fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type.

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