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Application of topological techniques to the analysis of asymptotic behavior of numerical solutions of a reaction-diffusion equation. (English) Zbl 0643.65054

The author studies the reaction-diffusion problem: \(u_ t=u_{xx}+f(u)\), \(u(x,0)=u^ 0(x)\), \(u(\pm L,t)=0\), \(| x| <L\), \(t>0\), where \(f(u)=-u(u-b)(u-1)\), \(0<b<1/2\). Using the method of lines he obtains semidiscrete approximation both for the above problem and for the steady-state problem \(u''+f(u)=0\), \(| x| <L\), \(u(\pm L)=0\). Then a numerical problem is formulated as an operator equation in a suitable Banach space. This approach combined with the known results considered continuous problem enables the author to prove that properties of numerical solutions are very close to those of the solutions of the differential problem [see J. Smoller’s monograph “Shock waves and reaction-diffusion equations” (1983; Zbl 0508.35002)].
Reviewer: St.Burys

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations

Citations:

Zbl 0508.35002
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