New numerical methods for blow-up problems
For researchers solving ODEs with blow-up, this provides practical numerical approaches to handle singularities.
Two new numerical methods for integrating ODEs with blow-up solutions are proposed, transforming them into problems without singularities, enabling standard solvers. Test problems demonstrate efficiency.
Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are described. The first method is based on applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable. The second method is based on introducing a new non-local variable that reduces ODE to a system of coupled ODEs. Both methods lead to problems whose solutions do not have blowing-up singular points, therefore the standard numerical methods can be applied. The efficiency of the proposed methods is illustrated with several test problems.