New numerical methods for blow-up problems

2017/03/29 by Andrei D. Polyanin, Polyanin, Andrei D., Inna K. Shingareva +1
Mathematics · #65L05 #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1703.10202

openalex publication_date 2017/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Related

• Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints
• Numerical validation of blow-up solutions of ordinary differential equations
• Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity
• Blow-up analysis for the solutions of an equation with boundary condition
• Adapting the time-step to recover the asymptotic behavior in a blow-up problem