Information-based complexity
The complexity of two-point boundary-value problems with piecewise analytic data
Journal of Complexity
Rigorous computational shadowing of orbits of ordinary differential equations
Numerische Mathematik
Solving Ordinary Differential Equations with Discontinuities
ACM Transactions on Mathematical Software (TOMS)
Iterated Defect Correction for the Solution of Singular Initial Value Problems
SIAM Journal on Numerical Analysis
Almost optimal solution of initial-value problems by randomized and quantum algorithms
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
The randomized complexity of initial value problems
Journal of Complexity
Optimal adaptive solution of initial-value problems with unknown singularities
Journal of Complexity
Uniform Approximation of Piecewise r-Smooth and Globally Continuous Functions
SIAM Journal on Numerical Analysis
Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach
SIAM Journal on Numerical Analysis
Adaptive Itô-Taylor algorithm can optimally approximate the Itô integrals of singular functions
Journal of Computational and Applied Mathematics
Taylor Approximations for Stochastic Partial Differential Equations
Taylor Approximations for Stochastic Partial Differential Equations
A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
| Hi-index | 7.29 |
In this paper we are interested in a rigorous analysis of the solution of a class of initial-value problems with singularities. We consider scalar non-autonomous IVPs with separated variables and one unknown singularity in each variable (which leads to four unknown 'events' in the two-dimensional space). Many algorithms proposed in the literature for singular IVPs are practically oriented. They do not avoid heuristic arguments, and are often checked for efficiency by numerical experiments. We design an adaptive algorithm with no heuristic steps for solving the considered problems, and provide rigorous bounds on the error. We show that in spite of the presence of singularities the algorithm preserves the (optimal) error known for the regular case. Lower bounds on the error of any algorithm in the case of a larger number of singularities are also discussed.