@Article{EAJAM-7-679, author = {}, title = {Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {4}, pages = {679--696}, abstract = {

We consider the second order nonlinear ordinary differential equation $u′′(t)=u^{1+α}(α>0)$ with positive initial data $u(0)=a_0$ , $u′(0)=a_1$ , whose solution becomes unbounded in a finite time $T$. The finite time $T$ is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in this paper a finite difference scheme with a rather larger grid size and show the convergence of the numerical solution and the numerical blow-up time. Application to the nonlinear wave equation is also discussed.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.220816.300517a}, url = {http://global-sci.org/intro/article_detail/eajam/10713.html} }