Year: 2016
Type: preprint
Abstract: We consider the following BVP $\ddot{x}\left( t\right) =f\left( t,\dot{x}\left( t\right) ,x\left( t\right) \right) -h\left( t\right) $, $% x\left( 0\right) =x\left( 1\right) =0$, where $f$ is continuous and satisfies some other conditions, $h\in H_{0}^{1}\left( 0,1\right) $ together with its discretization $$-Δ^{2}x(k-1)+\frac{1}{n^{2}}f\left(\frac{k}{n}, nΔx\left(k-1\right), x\left(k\right)\right)=\frac{1}{n^{2}}h\left(\frac{k}{n}\right), k\in \left\{1, 2, \ldots,n \right\}.$$ Using monotonicity methods we obtain the convergence of a solutions to a family of discrete problems to the solution of a continuous one, i.e. the existence of non-spurious solutions to the above problems is considered. Continuous dependence on parameters for the continuous problem is also investigated.
Source: arXiv (Cornell University)
Author: Pietrusiak, Filip
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2471222892