A second order of accuracy finite difference scheme for the integral-differential equation of the hyperbolic type

2014 ◽  
Author(s):  
Zilal Direk ◽  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Second Order ◽  
Hyperbolic Type ◽  
Order Of Accuracy ◽  
Integral Differential Equation

Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-Up Problem

2017 ◽  
Vol 7 (4) ◽  
pp. 679-696
Author(s):  
Chien-Hong Cho ◽  
Chun-Yi Liu
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Time ◽  
Finite Difference Scheme ◽  
Blow Up ◽  
Nonlinear Ordinary Differential Equation ◽  
Grid Size ◽  
Second Order ◽  
Finite Difference Schemes

AbstractWe consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, 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.

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