Document Type : Research Article
Authors
1 Department of Medical Instruments Engineering Techniques, University of Bilad Alrafidain, Iraq
2 Computer Science Department, University of Diyala, Iraq
10.30473/coam.2026.76880.1383
Abstract
This paper presents a systematic comparative study of two widely used numerical solvers --- HOFiD_bvp (high-order finite difference scheme) and bvp4c (collocation-based) --- for solving singular second-order ordinary differential equations (ODEs) with first-kind (regular) boundary singularities. Four representative benchmark problems drawn from fluid dynamics, materials science, and radially symmetric diffusion models are used to evaluate solver performance across key metrics: maximum residual, maximum error, mesh point count, and ODE/BC function call counts. Results show that HOFiD_bvp consistently achieves lower residuals and errors with fewer function evaluations, making it computationally more efficient. Conversely, bvp4c demonstrates superior robustness for nonlinear singular problems and offers better adaptive mesh refinement capabilities. These findings provide practical guidance for selecting the appropriate numerical technique in applied science and engineering contexts, with implications for optimization of computational simulation workflows.
Highlights
- HOFiD_bvp consistently achieves lower maximum residuals and errors compared to bvp4c at equivalent tolerance levels
- HOFiD_bvp requires significantly fewer ODE and boundary condition function evaluations, indicating superior computational efficiency
- bvp4c demonstrates greater robustness for nonlinear singular problems with adaptive mesh refinement capability
- Four representative benchmark problems covering radial diffusion, reaction-diffusion, phase separation, and infinite-domain fluid dynamics are systematically evaluated
- Practical solver-selection guidance is provided for singular second-order boundary value problems in applied science and engineering
Keywords
- Singular boundary value problem
- Computational performance optimization
- Finite difference method
- Collocation method
- Adaptive mesh refinement
Main Subjects
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