Neha
Mathematics
July 2024
Adaptive spline methods have become a popular way to solve boundary value problems (BVPs) with singular perturbation issues. The fast changes in solution behavior that these problems frequently display might provide a challenge to existing numerical methods because they are not very good at capturing steep gradients. Using adaptive splines allows us to provide improved accuracy without needless computation in smoother parts by allowing us to locally refine the approximation in areas where the answer varies greatly. Through the adaptive method, boundary layers and singularities present in the problem can be more accurately represented by dynamically modifying the spline's knots in response to the behavior of the solution. This strategy is interesting for complex BVPs in a variety of scientific and engineering applications because it not only improves convergence rates but also dramatically lowers processing costs. Adaptive spline methods are shown to be more effective than conventional ways through rigorous error analysis and numerical experimentation, offering a strong foundation to address singular perturbation problems
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