Stable stationary solutions for a reaction–diffusion equation with a multi-stable nonlinearity

2006 ◽  
Vol 357 (4-5) ◽  
pp. 319-322 ◽  
Author(s):  
Irene Nizhnik
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Reaction Diffusion Equation ◽  
Stable Stationary Solutions

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.

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

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