scholarly journals Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Nikos I. Kavallaris ◽  
Raquel Barreira ◽  
Anotida Madzvamuse
Keyword(s):  
Stationary Solution ◽  
Analytical Methods ◽  
Blow Up ◽  
Turing Instability ◽  
Numerical Approach ◽  
Local Equation ◽  
Shadow System ◽  
Non Local ◽  
Stability Results ◽  
Theoretical Results

AbstractThe main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

scholarly journals SOME INTERESTING SPECIAL CASES OF A NON-LOCAL PROBLEM MODELLING OHMIC HEATING WITH VARIABLE THERMAL CONDUCTIVITY

2001 ◽  
Vol 44 (3) ◽  
pp. 585-595 ◽  
Author(s):  
D. E. Tzanetis ◽  
P. M. Vlamos
Keyword(s):  
Stationary Solution ◽  
Dirichlet Boundary ◽  
Ohmic Heating ◽  
Variable Thermal Conductivity ◽  
Dirichlet Boundary Conditions ◽  
Heaviside Function ◽  
Local Equation ◽  
Special Cases ◽  
Primary 35B30 ◽  
Non Local

AbstractThe non-local equation$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99

Singularities in complex interfaces

1990 ◽  
Vol 333 (1631) ◽  
pp. 379-389 ◽  
Keyword(s):  
Unit Circle ◽  
Blow Up ◽  
The Self ◽  
Conformal Map ◽  
Local Equation ◽  
Material Interface ◽  
Two Fluids ◽  
Non Local ◽  
Self Similar ◽  
Similar Solutions

We analyse an equation describing the motion of the material interface between two fluids in a pressure field. The interface can be expressed as the image of the unit circle under a certain time depending conformal map. This conformal transformation maps the exterior of the unit circle onto the region occupied by one of the fluids. The conformal map has singularities in the unit disc. As long as these singularities are close to the origin, the complicated non-local equation governing the evolution of the conformal map can be approximated by a somewhat simpler, local equation. We prove that there exist self-similar solutions of this equation, that they have singularities away from the origin, that these singularities hit in finite time the unit circle and that the self-similar blow up is stable to perturbations that respect the symmetry of the self-similar profile.

scholarly journals On the optimal control of coronavirus (2019-nCov) mathematical model; a numerical approach

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
N. H. Sweilam ◽  
S. M. Al-Mekhlafi ◽  
A. O. Albalawi ◽  
D. Baleanu
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Weighted Average ◽  
Necessary Optimality Conditions ◽  
Numerical Approach ◽  
Population Number ◽  
Infected People ◽  
The Stability ◽  
Novel Coronavirus ◽  
Theoretical Results

Abstract In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald–Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

scholarly journals Uniqueness and Stability Results on Non-local Stochastic Random Impulsive Integro-Differential Equations

2021 ◽  
Vol 2 (3) ◽  
pp. 9-20
Author(s):  
VARSHINI S ◽  
BANUPRIYA K ◽  
RAMKUMAR K ◽  
RAVIKUMAR K
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Local Conditions ◽  
Non Local ◽  
Inequality Techniques ◽  
Stability Results

The paper is concerned with stochastic random impulsive integro-differential equations with non-local conditions. The sufficient conditions guarantees uniqueness of mild solution derived using Banach fixed point theorem. Stability of the solution is derived by incorporating Banach fixed point theorem with certain inequality techniques.

scholarly journals A non-local formulation of rotational water waves

2011 ◽  
Vol 689 ◽  
pp. 129-148 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Velocity Potential ◽  
Vries Equation ◽  
Travelling Waves ◽  
Explicit Dependence ◽  
Local Equation ◽  
Constant Vorticity ◽  
Small Parameters ◽  
Non Local

AbstractThe classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.

Causality Violation and Non-local Generalization of the Lorentz-Dirac Equation

1977 ◽  
Vol 32 (3-4) ◽  
pp. 319-326
Author(s):  
P. Alber ◽  
W. Heudorfer ◽  
M. Sorg
Keyword(s):  
Dirac Equation ◽  
Equation Of Motion ◽  
Local Theory ◽  
Constant Force ◽  
Dirac Theory ◽  
Local Equation ◽  
Causality Violation ◽  
Finite Duration ◽  
Non Local

AbstractIt is demonstrated by a concrete example (constant force of finite duration) that the recently proposed, non-local equation of motion for the radiating electron does exhibit the effect of causality violation. This phenomenon, which occurs in the non-local theory in form of self-oscillations, is however less severe than in the Lorentz-Dirac theory, if only physically reasonable forces are admitted.

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