NUMERICAL SOLUTION OF SINGULAR BOUNDARY VALUE PROBLEMS USING HYBRID CUBIC B-SPLINE TECHNIQUE
DOI:
https://doi.org/10.71146/kjmr852Keywords:
B-Spline functions, Non-linear singular boundary value problems, Collocation method, Hybrid Cubic B-Spline, Differential equations, Approximate solutionsAbstract
Non-linear singular boundary value problems arise in several areas of applied mathematics, physics, and engineering, particularly in models involving heat transfer, fluid mechanics, and reaction–diffusion processes. The presence of singularities and non-linear terms make the analytical solution of such problems difficult, thereby motivating the use of reliable numerical techniques. To achieve this objective, a numerical scheme based on Hybrid Cubic B-Spline approximation is formulated. The unknown solution is represented as a linear combination of B-Spline basis functions defined over the problem domain. Using the collocation technique, the governing differential equation is discretized at selected nodal points, which converts the original boundary value problem into a system of non-linear algebraic equations. The equations are then iteratively solved to get the numerical solution. The method is tested on some benchmark problems whose exact solutions are known to evaluate the performance of the method. Proposed Technique provides good accuracy, smooth approximations and computational efficiency and can be extended to other classes of differential equations commonly used in scientific and engineering problems.
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[1] Caglar HN, Caglar SH, Twizell EH. The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions. Int J Comput Math 1999;71:373–81.
[2] Caglar HN, Caglar SH, Twizell EH. The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions. Appl Math Lett 1999;12:25–30.
[3] Caglar N, Caglar H, Cagal B. Spline solution of nonlinear beam problems. J Concr Appl Math 2003;1(3):253–9.
[4] Caglar H, Caglar N, Elfaituri K. B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems. Appl Math Comput 2006;175:72–9.
[5] Caglar N, Caglar H. B-spline solution of singular boundary value problems. Appl Math Comput 2006. doi:10.1016/ j.amc.2006.05.035.
[6] Chawla MM, Katti CP. A finite difference method for a class of singular two point boundary value problems. IMA J Numer Anal 1984;4:457–66.
[7] Kanth ASVR, Reddy YN. A numerical method for singular boundary value problems via Chebyshev economization. Appl Math Comput 2003;146:691–700.
[8] Kanth ASVR, Reddy YN. Cubic spline for a class of singular two-point boundary value problems. Appl Math Comput 2005;170:733–40.
[9] Kanth ASVR, Bhattacharya V. Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Appl Math Comput 2006;174:768–74.
[10] Kanth ASVR. Cubic spline polynomial for non-linear two-point boundary value problems. Appl Math Comput 2007. doi:10.1016/j.amc.2007.01.002.
[11] Caglar N, Caglar H. B-spline solution of singular boundary value problems. Appl Math Comput. doi:10.1016/j.amc.2006.05.035.
[12] Hiltmann P, Lory P. On oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. Bull Math Biol 1983;45:661–4.
[13] Anderson N, Arthurs AM. Analytical diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. Bull Math Biol 1983;47:145–53.
[14] Adam JA. A simplified mathematical model of tumor growth. Math Biosci 1986;81:224–9.
[15] Asaithambi NS, Goodman JB. Point wise bounds for a class of singular diffusion problem in physiology. Appl Math Comput 1989;30:215–22.
[16] Lin HS. Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J Theor Biol 1976;60:449–57.
[17] McElwain DLS. A re-examination of oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. J Theor Biol 1978;71:255–63.
[18] Flesch U. The distribution of heat sources in the human head: a theoretical consideration. J Theor Biol 1975;54:285–7.
[19] Garner JB, Shivaji R. Diffusion problems with mixed nonlinear boundary condition. J Math Anal Appl 1990;148:422–30.
[20] Duggan RC, Goodman AM. Point wise bounds for a nonlinear heat conduction model of the human head. Bull Math Biol 1986;48:229–36.
[21] de Boor C. A practical guide to splines. Springer-Verlag; 1978.
[22] Golub GH, Ortega JM. Scientific computing and differential equations. Academic Press; 1992.
[23] Fletcher R. Practical methods of optimization. A Wiley-Interscience Publicati
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Copyright (c) 2026 Syed Sarmad Hasan, Muhammad Wakeel, Muhammad Awais (Author)
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