Different convergences in approximation of evolution equations
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CitationPiskarev, S. (2009). Different convergences in approximation of evolution equations. Maltepe Üniversitesi. s. 352.
Consider the semilinear equation in Banach space E α u 0 (t) = Au(t) + f(u(t)), t ≥ 0, u(0) = u 0 ∈ E α, (0.1) where f(·) : E α ⊆ E → E, 0 ≤ α < 1, is assumed to be continuous, bounded and continuously Fr´echet differentiable function. The problem (0.1) in the neighborhood of the hyperbolic equilibrium can be written in the form, where Au∗ = A+f 0 (u ∗ ), Fu∗ (v(t)) = f(v(t)+u ∗ )−f(u ∗ )−f 0 (u ∗ )v(t). We consider approximation of (0.2) by the following scheme, with initial data Vn(0) = v 0 n. The solution of such problem is given by formula Vn(t + τn) = (In − τnAu∗ n,n) −1Vn(t) + τn(In − τnAu∗ n,n) −1Fu∗ n,n(Vn(t)) = = (In − τnAu∗ n,n) −kVn(0) + τnΣ k j=0(In − τnAu∗ n,n) −(k−j+1)Fu∗ n,n(Vn(jτn)), t = kτn, where Vn(0) = v 0 n. We consider different kind of consistency of generators under which one can get convergence of solutions in the vicinity of hyperbolic stationary point.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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