A singularly perturbed second order evolution equation with nonlocal nonlinearity

1987 ◽  
Vol 9 (9-10) ◽  
pp. 969-986 ◽  
Author(s):  
Benjamin F. Esham
Keyword(s):  
Evolution Equation ◽  
Second Order ◽  
Singularly Perturbed ◽  
Nonlocal Nonlinearity

An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

2021 ◽  
pp. 2150056
Author(s):  
Sheetal Chawla ◽  
Jagbir Singh ◽  
Urmil
Keyword(s):  
Uniform Convergence ◽  
Order Parameter ◽  
Source Term ◽  
Reaction Diffusion ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Diffusion Type ◽  
Bakhvalov Mesh ◽  
Second Order Parameter

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

scholarly journals Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

2016 ◽  
Vol 7 (2) ◽  
pp. 130-149 ◽  
Author(s):  
Lidia Saluto ◽  
Maria Stella Mongioví
Keyword(s):  
Evolution Equation ◽  
Dimensional Analysis ◽  
Unit Volume ◽  
Vortex Formation ◽  
Second Order ◽  
Line Density ◽  
Superfluid Turbulence ◽  
Turbulence Flow ◽  
Vortex Diffusion

Abstract We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.

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