scholarly journals SCALING LIMITS FOR TIME-FRACTIONAL DIFFUSION-WAVE SYSTEMS WITH RANDOM INITIAL DATA

2010 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
GI-REN LIU ◽  
NARN-RUEIH SHIEH
Keyword(s):  
Initial Data ◽  
Initial Conditions ◽  
Scaling Limit ◽  
Reaction Diffusion ◽  
Equation System ◽  
Random Initial Data ◽  
Diffusion Type ◽  
Diffusion Wave ◽  
Decoupling Method ◽  
Time Space

Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

A BROWNIAN-TIME EXCURSION INTO FOURTH-ORDER PDES, LINEARIZED KURAMOTO–SIVASHINSKY, AND BTP-SPDES ON ℝ+ × ℝd

2006 ◽  
Vol 06 (04) ◽  
pp. 521-534 ◽  
Author(s):  
HASSAN ALLOUBA
Keyword(s):  
White Noise ◽  
Fourth Order ◽  
Applied Mathematics ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Standard Theory ◽  
Valuable Insight ◽  
Diffusion Type ◽  
Iterated Brownian Motion ◽  
Time Space

In recent articles we have introduced the class of Brownian-time processes (BTPs) and the Linearized Kuramoto–Sivashinsky process (LKSP). Probabilistically, BTPs represent a unifying class for some different exciting processes like the iterated Brownian motion (IBM) of Burdzy (a process with fourth-order properties) and the Brownian–snake of Le Gall (a second-order process); they also include many additional new and quite interesting processes. The LKSP is closely connected to the Kuramoto–Sivashinsky PDEs, one of the most celebrated PDEs in modern applied mathematics. We start by surveying the fourth-order PDE connections to BTPs and the LKSP that we uncovered in two recent articles. In the second part of this paper we introduce BTP-SPDEs, these are SPDEs in which the PDE part is that solved by running a BTP. We consider a BTP-SPDE driven by an additive spacetime white noise on the time-space set ℝ+ × ℝd; and we prove the existence of a unique real-valued, Lp(Ω,ℙ) for all p ≥ 1, BTP solution to such BTP-SPDEs for 1 ≤ d ≤ 3. This contrasts sharply with the standard theory of reaction-diffusion type SPDEs driven by spacetime white noise, in which real-valued solutions are confined to one spatial dimension. Like the PDEs case, BTP-SPDEs also provide a valuable insight into other fourth-order SPDEs of applied science. We carry out such a program in forthcoming articles.

scholarly journals A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations

2021 ◽  
Vol 46 (3) ◽  
pp. 221-233
Author(s):  
Burcu Gürbüz
Keyword(s):  
Approximate Solution ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Computational Technique ◽  
Test Case ◽  
Diffusion Type ◽  
Convection Diffusion ◽  
Laguerre Series ◽  
Delay Partial Differential Equations

Abstract In this work, a matrix method based on Laguerre series to solve singularly perturbed second order delay parabolic convection-diffusion and reaction-diffusion type problems involving boundary and initial conditions is introduced. The approximate solution of the problem is obtained by truncated Laguerre series. Moreover convergence analysis is introduced and stability is explained. Besides, a test case is given and the error analysis is considered by the different norms in order to show the applicability of the method.

The development of travelling waves in a simple isothermal chemical system III. Cubic and mixed autocatalysis

1990 ◽  
Vol 430 (1880) ◽  
pp. 509-524 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Chemical System ◽  
Mixed System ◽  
Dimensional Parameter ◽  
Diffusion Wave ◽  
Symmetrical Form ◽  
Asymptotic Speed ◽  
Mixed Systems ◽  
Cubic Autocatalysis

Autocatalytic chemical reactions can support isothermal travelling waves of constant speed and form. This paper extends previous studies to cubic autocatalysis and to mixed systems where quadratic and cubic autocatalyses occur concurrently. A + B → 2B, rate = k q ab , (1) A + 2B → 3B, rate = k c ab 2 . (2) For pure cubic autocatalysis the wave has, at large times, a constant asymptotic speed v 0 (where v 0 = 1/√2 in the appropriate dimensionless units). This result is confirmed by numerical investigation of the initial-value problem. Perturbations to this stable wave-speed decay at long times as t -3/2 e -1/8 t . The mixed system is governed by a non-dimensional parameter μ = k q / k c a 0 which measures the relative rates of transformation by quadratic and cubic modes. In the mixed case ( μ ≠ 0) the reaction-diffusion wave has a form appropriate to a purely cubic autocatalysis so long as μ lies between ½ and 0. When μ exceeds ½, the reaction wave loses its symmetrical form, and all its properties steadily approach those of quadratic autocatalysis. The value μ = ½ is the value at which rates of conversion by the two paths are equal.

On invariant subspaces for nonlinear finite-difference operators

1998 ◽  
Vol 128 (6) ◽  
pp. 1293-1308 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Finite Difference ◽  
Differential Operators ◽  
Invariant Subspaces ◽  
Reaction Diffusion ◽  
Difference Operators ◽  
Diffusion Type ◽  
Nonlinear Differential Operators ◽  
Discrete Operators ◽  
Spatial Discretisation ◽  
Lower Dimensional

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

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