Paper Info
Title
Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions.
Abstract
We present a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to describe the time evolution of solutions. Nonlocality was modeled with a space-time fractional differential equation of order 1 <= alpha <= 2 in time, and order 1 <= beta <= 2 in space. We adopted the notion of Caputo for the time derivative and the Riesz pseudo-differential operator for the space derivative. The corresponding Cauchy problem was solved for zero initial velocity and initial disturbance, represented by either the Dirac delta or the Gaussian distributions. Well-known results for the conventional partial differential equations of wave propagation, diffusion, and (modified) transport processes were recovered as particular cases. In addition, regular solutions were found for the partial differential equation that arises from alpha = 2 and beta = 1. Unlike the above conventional cases, the latter equation permits the presence of nodes in its solutions.
Year
DOI
Venue
2018
10.3390/sym10100524
SYMMETRY-BASEL
Keywords
Field
DocType
partial differential equations,fractional differential equations,Cauchy problem,H-Fox functions,Dirac delta distribution,Gaussian distribution
Cauchy problem,Differential equation,Nonlinear system,Wave propagation,Mathematical analysis,Dirac delta function,Gaussian,Wave equation,Partial differential equation,Mathematics
Journal
Volume
Issue
ISSN
10
10
2073-8994
Citations 
PageRank 
References 
0
0.34
0
Authors
2
Name
Order
Citations
PageRank
Fernando Olivar-Romero100.34
Oscar Rosas-Ortiz200.68