scholarly journals A convergent convex splitting scheme for a nonlocal Cahn--Hilliard--Oono type equation with a transport term

2020 ◽  
Author(s):  
Hussein Fakih ◽  
Laurence Cherfils ◽  
Alain Miranville ◽  
Maurizio Grasselli
Keyword(s):  
Type Equation ◽  
Neumann Boundary ◽  
Crystal Nucleation ◽  
Splitting Scheme ◽  
Time Step ◽  
Interaction Kernel ◽  
Convex Splitting ◽  
The Stability ◽  
Theoretical Results ◽  
Transport Term

We devise a first-order in time  convex splitting scheme for a nonlocal Cahn--Hilliard--Oono type equation with a transport term and subject to homogeneous Neumann boundary conditions. However, we prove the stability of our scheme when the time step is sufficiently small,   according to the velocity field and the interaction kernel. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which confirm our theoretical results and demonstrate the performance of our scheme not only for phase separation, but also for crystal nucleation, for several choices of the interaction kernel.

scholarly journals A Time-Stepping DPG Scheme for the Heat Equation

2017 ◽  
Vol 17 (2) ◽  
pp. 237-252 ◽  
Author(s):  
Thomas Führer ◽  
Norbert Heuer ◽  
Jhuma Sen Gupta
Keyword(s):  
Heat Equation ◽  
Order Approximation ◽  
Test Functions ◽  
Optimal Test ◽  
Time Step ◽  
Time Stepping ◽  
Backward Euler ◽  
The Stability ◽  
Convergence Orders ◽  
Theoretical Results

AbstractWe introduce and analyze a discontinuous Petrov–Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.

Stability and Bifurcation Analysis in the Photosensitive CDIMA System with Delayed Feedback Control

2017 ◽  
Vol 27 (11) ◽  
pp. 1750177 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Positive Equilibrium ◽  
Stability And Bifurcation Analysis ◽  
The Stability ◽  
Theoretical Results

A diffusive photosensitive CDIMA system with delayed feedback subject to Neumann boundary conditions is considered. We derive the conditions of the occurrence of Turing instability. We also investigate the influence of delay on the stability of the positive equilibrium of the system, and prove that delay induces the occurrence of Hopf bifurcation. By computing the normal form on the center manifold, we give the formulas determining the properties of the Hopf bifurcation. Finally, we give some numerical simulations to support and strengthen the theoretical results. Our study shows that diffusion and delayed feedback can effect the stability of the equilibrium of the system.

A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

2013 ◽  
Vol 3 (4) ◽  
pp. 333-351 ◽  
Author(s):  
Lizhen Chen ◽  
Chuanju Xu
Keyword(s):  
Lower Order ◽  
Neumann Boundary ◽  
Mass Conservation ◽  
Spectral Element ◽  
Stability And Convergence ◽  
Time Step ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
The Stability

AbstractWe propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation

1971 ◽  
Vol 38 (2) ◽  
pp. 467-476 ◽  
Author(s):  
W.-Y. Tseng ◽  
J. Dugundji
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Type Equation ◽  
Harmonic Excitation ◽  
Harmonic Motion ◽  
Buckled Beam ◽  
Second Mode ◽  
The Stability ◽  
Supporting Base ◽  
Theoretical Results

A buckled beam with fixed ends, excited by the harmonic motion of its supporting base, was investigated analytically and experimentally. Using Galerkin’s method the governing partial differential equation reduced to a modified Duffing equation, which was solved by the harmonic balance method. Besides the solution of simple harmonic motion (SHM), other branch solutions involving superharmonic motion (SPHM) were found experimentally and analytically. The stability of the steady-state SHM and SPHM solutions were analyzed by solving a variational Hill-type equation. The importance of the second mode on these results was examined by a similar stability analysis. The Runge-Kutta numerical integration method was used to investigate the snap-through problem. Intermittent, as well as continuous, snap-through behavior was obtained. The theoretical results agreed well with the experiments.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

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.

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

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