Computational Performance Optimization in Solving Singular Boundary Value Problems: A Comparative Study of Finite Difference and Collocation Methods

Abbas, Saad Qasim; Ahmed, Wasan Saad

doi:10.30473/coam.2026.76880.1383

- فلوچارت مراحل ورود و ثبت نام در سامانه نشریات علمی وزارت عتف

- راه اندازی سیستم مدیریت نشریات علمی دانشگاه پیام نور

تعداد نشریات	49
تعداد شماره‌ها	1,264
تعداد مقالات	10,909
تعداد مشاهده مقاله	22,310,504
تعداد دریافت فایل اصل مقاله	14,992,004

	Computational Performance Optimization in Solving Singular Boundary Value Problems: A Comparative Study of Finite Difference and Collocation Methods
Control and Optimization in Applied Mathematics
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 15 فروردین 1405 اصل مقاله (495.38 K)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2026.76880.1383
نویسندگان
Saad Qasim Abbas^* ¹؛ Wasan Saad Ahmed²
¹Department of Medical Instruments Engineering Techniques, University of Bilad Alrafidain, Iraq
²Computer Science Department, University of Diyala, Iraq
چکیده
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.
تازه های تحقیق
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
کلیدواژه‌ها
Singular boundary value problem؛ Computational performance optimization؛ Finite difference method؛ Collocation method؛ Adaptive mesh refinement

مراجع
[1] Abeldinov, Y. (2025). “Numerical methods for solving direct and inverse Sturm–Liouville problems.” Proceedings of Young Scientists and Specialists of the Samara University. https://vmuis.ru/smus/article/view/27439. [2] Adams, M., Muir, P. (2024). “Differential equation software for the computation of error-controlled continuous approximate solutions.” Numerical Algorithms, 96(3), 1021–1044. DOI: https://doi.org/10.1007/S11075-024-01784-1. [3] Ahmed, H.M., Abd-Elhameed, W.M. (2024). “Spectral solutions of specific singular differential equations using a unified spectral Galerkin–collocation algorithm.” Journal of Nonlinear Mathematical Physics, 31(1), 1–22. DOI: https://doi.org/10.1007/ S44198-024-00194-0. [4] Al Arfaj, K., Levesley, J. (2023). “Lagrange radial basis function collocation method for boundary value problems in 1D.” AIMS Mathematics, 8(11), 27542–27572. DOI: https://doi.org/10.3934/MATH.20231409. [5] Al-Doori, V.S., Abbas, S.Q., Kulikov, O., Ismail, M.N. (2024). “Home automation system with a GUI that is powered by Arduino and MATLAB.” Conference of Open Innovation Association, FRUCT. https://www.scopus.com/pages/publications/85193393535. [6] Alshanti, W.G., Batiha, I.M., Hammad, M.A., Khalil, R. (2023). “A novel analytical approach for solving partial differential equations via a tensor product theory of Banach spaces.” Partial Differential Equations in Applied Mathematics, 8, 100531. DOI: https://doi.org/10.1016/j.padiff.2023.100531. [7] Amodio, P., Settanni, G. (2009). “A deferred correction approach to the solution of singularly perturbed BVPs by high order upwind methods: Implementation details”. AIP Conference Proceedings, 1168, 711–714. DOI: https://doi.org/10.1063/1.3241565 [8] Amodio, P., Sgura, I. (2005). “High-order finite difference schemes for the solution of second-order BVPs.” Journal of Computational and Applied Mathematics, 176(1), 59– 76. DOI: https://doi.org/10.1016/j.cam.2004.07.008. [9] Arifeen, S.U., Haq, S., Ghafoor, A., Ullah, A., Kumam, P., Chaipanya, P. (2021). “Numerical solutions of higher order boundary value problems via wavelet approach.” Advances in Difference Equations, 2021(1), 1–15. DOI: https://doi.org/10.1186/ S13662-021-03495-6. [10] Ascher, U., Christiansen, J., Russell, R.D. (1981). “Algorithm 569: COLSYS: Collocation software for boundary-value ODEs [D2].” ACM Transactions on Mathematical Software, 7(2), 223–229. DOI: https://doi.org/10.1145/355945.355951. [11] Auzinger, W., Fallahpour, M., Koch, O., Weinmüller, E. (2024). “Implementation of a path-following strategy with an automatic step-length control: New MATLAB package bvpsuite2.0.” TU Wien Repository. https://repositum.tuwien.at/handle/20. 500.12708/30869. [12] Balluffi, R.W., Allen, S.M., Carter, W.C. (2005). “Diffusion as a series of discrete jumps.” Kinetics of Materials, 154–159. [13] Bandyopadhyay, S., Kunkel, C.J. (2025). “Existence result for singular second-order dynamic equations with mixed boundary conditions.” DOI: https://doi.org/10.48550/ arXiv.2506.16505 [14] MATLAB Central. (n.d.). “Tutorial on solving BVPs with BVP4C.” https://www.mathworks.com/matlabcentral/fileexchange/3819-tutorial-on-solving-bvps-with-bvp4c. [15] Chen, S.B., Shahmir, N., Ramzan, M., Sun, Y.L., Aly, A.A., Malik, M.Y. (2021). “Thermophoretic particle deposition in the flow of dual stratified Casson fluid with magnetic dipole and generalized Fourier’s and Fick’s laws.” Case Studies in Thermal Engineering, 26, 101186. DOI: https://doi.org/10.1016/J.CSITE.2021.101186. [16] Ctor León, V., Scárdua, B. (2024). “Regularity, synthesis, rigidity and analytic classification for linear ordinary differential equations of second order.” Journal of Applied Mathematics, 2(4), 1698. DOI: https://doi.org/10.59400/JAM.V2I4.1698. [17] Cui, X., Xia, Y. (2022). “Second-order differential equation with indefinite and repulsive singularities.” Bulletin of the Malaysian Mathematical Sciences Society, 46(3). DOI: https://doi.org/10.1007/s40840-023-01497-z. [18] Datsko, B.Y., Kutniv, M.V. (2025). “Explicit numerical methods for solving singular initial value problems for systems of second-order nonlinear ODEs.” Numerical Algorithms, 98(2), 929–942. DOI: https://doi.org/10.1007/s11075-024-01820-0. [19] Drazin, P.G. (1983). Solitons. Cambridge University Press. DOI: https://doi.org/10. 1017/CBO9780511662843. [20] Egidi, N., Giacomini, J., Maponi, P. (2023). “A perturbative approach for the solution of Sturm–Liouville problems.” Applied and Computational Mathematics, 12(3), 46–54. DOI: https://doi.org/10.11648/J.ACM.20231203.11. [21] Ergashev, T.G. (2020). “Solving the Dirichlet and Holmgren problems for a three-dimensional elliptic equation by the potential method.” arXiv: Analysis of PDEs. DOI: https://api.semanticscholar.org/CorpusID:213005636. [22] Filipuk, G., Kecker, T. (2022). “On singularities of certain non-linear second-order ordinary differential equations.” Results in Mathematics, 77(1), 41. DOI: https://doi.org/ 10.1007/s00025-021-01577-1. [23] File, G., Gadisa, G., Aga, T., Reddy, Y.N. (2017). “Numerical solution of singularly perturbed delay reaction-diffusion equations with layer or oscillatory behaviour.” American Journal of Numerical Analysis, 5(1), 1–10. DOI: https://doi.org/10.12691/ ajna-5-1-1. [24] Hamidizadeh, K., Manaviyat, R., Mirvakili, S., Davvaz, B. (2024). “Application of soft sets to graph coloring.” Computational and Applied Mathematics, 43(4). DOI: https: //doi.org/10.1007/S40314-024-02738-Y. [25] Heinonen, J., Kilpeläinen, T., Martio, O. (2018). Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications, Mineola, New York. [26] Hohenegger, M., Settanni, G., Weinmüller, E.B., Wolde, M. (2024). “Numerical treatment of singular ODEs using finite difference and collocation methods.” Applied Numerical Mathematics, 205, 184–194. DOI: https://doi.org/10.1016/j.apnum.2024. 07.002. [27] Jajarmi, A., Baleanu, D. (2018). “Suboptimal control of fractional-order dynamic systems with delay argument.” Journal of Vibration and Control, 24(12), 2430–2446. DOI: https: //doi.org/10.1177/1077546316687936. [28] Koltape, L.T., Hojjati, G., Fazeli, S., Abdi, A. (2024). “Super implicit two-step collocation methods for ordinary differential equations.” Computational and Applied Mathematics, 43(6), 1–22. DOI: https://doi.org/10.1007/s40314-024-02848-7. [29] Kutluay, S., Murat, Y., Karakas, A.S. (2023). “A robust septic Hermite collocation technique for Dirichlet boundary condition heat conduction equation.” arXiv:2312.05042. DOI: https://arxiv.org/pdf/2312.05042. [30] Li, M.M., Parise, D., Sarnataro, L. (2024). “Boundary behavior of limit-interfaces for the Allen–Cahn equation on Riemannian manifolds with Neumann boundary condition.” Archive for Rational Mechanics and Analysis, 248(6), 1–43. DOI: https://doi.org/ 10.1007/S00205-024-02070-Z. [31] Liu, C.S., Chang, C.W., Kuo, C.L. (2024). “Numerical analysis for Sturm–Liouville problems with nonlocal generalized boundary conditions.” Mathematics, 12(8), 1265. DOI: https://doi.org/10.3390/MATH12081265. [32] Mohamed, Z., Yousif, M., Hamza, A.E. (2022). “Solving nonlinear fractional partial differential equations using the Elzaki transform method and the homotopy perturbation method.” Abstract and Applied Analysis, 2022, 4743234. DOI: https://doi.org/10. 1155/2022/4743234. [33] Nur, C. (2021). “On the estimates of periodic eigenvalues of Sturm–Liouville operators with trigonometric polynomial potentials.” Mathematical Notes, 109(5–6), 794–807. DOI: https://doi.org/10.1134/S0001434621050114. [34] Omar, S.S., Ahmed, W.S., Ismail, M.N., Sieliukov, O. (2024). “In-depth examination of a fingerprint recognition system using the Gabor filter.” Conference of Open Innovation Association, FRUCT, 532–543. DOI: https://doi.org/10.23919/fruct61870.2024. 10516364. [35] Scott, D.R., Stevenson, D.J. (1984). “Magma solitons.” Geophysical Research Letters, 11(11), 1161–1164. DOI: https://doi.org/10.1029/GL011I011P01161. [36] Seiler, W.M., Seiß, M. (2024). “Singular initial value problems for some quasi-linear second-order ODEs.” Journal of Dynamics and Differential Equations, 1–23. DOI: https://doi.org/10.1007/s10884-024-10396-1. [37] Settanni, G. (2024). “Potentiality of the HOFiD_bvp code in solving different kind of second-order boundary value problems.” Applied Numerical Mathematics, 200, 379–388. DOI: https://doi.org/10.1016/j.apnum.2023.08.008. [38] Shampine, L.F., Gladwell, I., Thompson, S. (2003). Solving ODEs with MATLAB. Cambridge University Press. [39] Stoer, J., Bulirsch, R. (1980). Introduction to Numerical Analysis (Chapter 7: Ordinary Differential Equations, pp. 404–535). Springer. DOI: https://doi.org/10.1007/ 978-1-4757-5592-3_7. [40] Trogdon, T. (2024). “The ultraspherical rectangular collocation method and its convergence.” DOI: https://doi.org/10.48550/arXiv.2401.03608 [41] Wazwaz, A.M. (2000). “The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth-order and twelfth-order.” International Journal of Nonlinear Sciences and Numerical Simulation, 1(1), 17–24. DOI: https:// doi.org/10.1515/IJNSNS.2000.1.1.17. [42] Xin, Y., Cui, X., Liu, J. (2020). “An exact expression of positive periodic solution for a first-order singular equation.” Advances in Difference Equations, 2020(1), 1–9. DOI: https://doi.org/10.1186/s13662-020-02986-2. [43] Yücel, U. (2015). “Numerical approximations of Sturm–Liouville eigenvalues using Chebyshev polynomial expansions method.” Cogent Mathematics, 2(1), 1045223. DOI: https://doi.org/10.1080/23311835.2015.1045223. [44] Zhu, H.L. (2022). “General solutions’ laws of linear partial differential equations I.” Partial Differential Equations in Applied Mathematics, 6, 100418. DOI: https://doi.org/ 10.1016/j.padiff.2022.100418
آمار تعداد مشاهده مقاله: 4

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

Computational Performance Optimization in Solving Singular Boundary Value Problems: A Comparative Study of Finite Difference and Collocation Methods