scholarly journals Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

2012 ◽  
Vol 11 (1) ◽  
pp. 229-241 ◽  
Author(s):  
Yuriy Golovaty ◽  
◽  
Anna Marciniak-Czochra ◽  
Mariya Ptashnyk ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Ordinary Differential Equations ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Reaction Diffusion Equation

Global existence and non-existence problem for a reaction—diffusion equation with a conserved first integral

1995 ◽  
Vol 451 (1943) ◽  
pp. 701-709
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Finite Time ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
First Integral ◽  
Stationary Solutions ◽  
Reaction Diffusion Equation ◽  
Existence Problem ◽  
Appl Math

In this paper, we prove the global existence and non-existence of solutions of the following problem: RDC{ u t = u xx + u 2 - ∫ u 2 ( x ) d x , x ϵ (0, 1), t > 0, u x (0, t ) = u x (1, t ) = t > 0, u ( x , 0) = u 0 ( x ), x ϵ (0, 1), ∫ 1 0 u ( x, t ) d x = 0, t > 0, Moreover, let u m ( x ) be a stationary solution of problem RDC with m zeros in the interval (0, 1) for m ϵ N , and if we take u 0 ( x ). Then we have that the solution exists globally if 0 < ϵ < 1, and blows up in finite time if ϵ > 1. This result verifies the numerical results of Budd et al . (1993, SIAM Jl appl. Math . 53, 718-742) that the non-zero stationary solutions are unstable.

Numerical solutions of the reaction-diffusion equation

2015 ◽  
Vol 25 (2) ◽  
pp. 265-271 ◽  
Author(s):  
G Wu ◽  
Eric Wai Ming Lee ◽  
Gao Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Nonlinear Partial Differential Equations ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Reaction Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
Content Type ◽  
Initial Boundary Value

Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

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