A septic B-spline collocation method for solving nonlinear singular boundary value problems arising in physiological models

Document Type : Article

Authors

1 Department of Mathematics, Faculty of Science, University of Menoufia, Shebein El-Koom, Egypt.

2 Department of Mathematics, Faculty of Science, University of Al-Azhar, Cairo, Nasr-City, Egypt.

3 Department of Basic Science, Higher Technological Institute, 10th of Ramadan City, Egypt

Abstract

In this paper, we present a numerical method based on septic B-spline function for nonlinear singular second-order two-point boundary value problems, which depend on different physiological processes as thermal explosions problem and the steady state oxygen diffusion in a spherical cell with Michaelis–Menten uptake kinetics and distribution of heat sources in the human head. Septic B-spline method has a truncation error of O(h^8) and converges to the exact solution with O(h^6). The numerical problems show that our method is very effective. The resulting sets of differential equations are modified at the singular point and are treated by using septic B-spline for finding the numerical solution. The maximum absolute errors and the absolute residual errors are acceptable.

Keywords

Main Subjects


1. Chawla, M.M. and Shivkumar, P.N. On the existence of solutions of a class of singular two-point nonlinear boundary value problems", J. Comput. Appl. Math., 19, pp. 379{388 (1987). 2. Russell, R.D. and Shampine, L.F. Numerical methods for singular boundary value problems", SIAM J. Numer. Anal., 12, pp. 13{36 (1975). 3. Ford, W.F. and Pennline, J.A. Singular non-linear two-point boundary value problems: Existence and uniqueness", Nonlinear Anal., 71, pp. 1059{1072 (2009). 4. Khuri, S. and Sayfy, A. A novel approach for the solution of a class of singular boundary value problems arising in physiology", Math. Comput. Modelling, 52, pp. 626{636 (2010). 5. Singh, R. and Kumar, J. An e_cient numerical technique for the solution of nonlinear singular boundary value problems", Comput. Phys. Comm., 185, pp. 1282{1289 (2014). 6. Caglar, H., Caglar, N., and Ozer, M. B-spline solution of non-linear singular boundary value problems arising in physiology", Chaos Solitons Fractals, 39(3), pp. 1232{1237 (2009). 7. Sahlan, M.N. and Hashemizadeh, E. Wavelet Galerkin method for solving nonlinear singular boundary value problems arising in physiology", Applied Mathematics and Computation, 250, pp. 260{269 (2015). 8. Niu, J., Xu, M., Lin, Y., et al. Numerical solution of nonlinear singular boundary value problems", Journal of Computational and Applied Mathematics, 331, pp. 42{51 (2018). 9. Lin, S.H. Oxygen di_usion in a spherical cell with nonlinear oxygen uptake kinetics", J. Theor. Biol., 60, pp. 449{457 (1976). 10. McElwain, D.L.S. A re-examination of oxygen di_usion in a spherical cell with Michaelis-Menten oxygen uptake kinetics", J. Theor. Biol., 71, pp. 255{263 (1978). 11. Wazwaz, A. The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models", Commun. Nonlinear Sci. Numer. Simul., 16, pp. 3881{3886 (2011). 12. Simpson, M.J. and Ellery, A.J. An analytical solutions for di_usion and nonlinear uptake of oxygen in a spherical cell", Applied Mathematical Modelling, 36, pp. 3329{3334 (2012). 13. Hiltmann, P. and Lory, P. On oxygen di_usion in a spherical cell with Michaelis-Menten oxygen uptake kinetics", Bull. Math. Biol., 45, pp. 661{664 (1983). 14. Flesch, U. The distribution of heat sources in the human head: a theoretical consideration", J. Theor. Biol., 54, pp. 285{287 (1975). 15. Duggan, R. and Goodman, A. Pointwise bounds for a nonlinear heat conduction model of the human head", Bull. Math. Biol., 48(2), pp. 229{236 (1986). 16. Garner, J.B. and Shivaji, R. Di_usion problems with mixed nonlinear boundary condition", J. Math. Anal. Appl., 148, pp. 422{430 (1990). 17. Rashidinia, J., Mohammadi, R., and Jalilian, R. The numerical solution of nonlinear singular boundary value problems arising in physiology", Applied Mathematics and Computation, 185, pp. 360{367 (2007). 18. Khuri, S.A. and Sayfy, A. A mixed decompositionspline approach for the numerical solution of a class of singular boundary value problems", Applied Mathematical Modelling, 40, pp. 4664{468 (2016). 19. Roul, P. and Thula, K. A new high-order numerical method for solving singular two-point boundary value problems", Journal of Computational and Applied Mathematics, 343, pp. 556{574 (2018). https://doi.org/10.1016/j.cam.2018.04.056 20. Khuri, S.A. and Sayfy, A. Numerical solution for the nonlinear Emden-Fowler type equations by a fourthorder adaptive method", Int. J. Comput. Methods, 11(1) (2014). 21. Roul, P. and Ujwal, W. A novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems", J. Compt. Appl. Math., 296, pp. 661{676 (2016). 22. Roul, P. and Kiran, T. A fourth order B-spline collocation method and its error analysis for Bratutype and Lane-Emden problems", Int. J. Comp. Math., 96(1), pp. 85{104 (2017). 23. Pirabaharan, P. and Chandrakumar, R.D. A computational method for solving a class of singular boundary value problems arising in science and engineering", Egyptian Journal of Basic and Applied Sciences, 3, pp. 383{391 (2016). 24. Nasab, A.K., K_l_cman, A., Babolian, E., et al. Wavelet analysis method for solving linear and nonlinear singular boundary value problems", Applied Mathematical Modelling, 37, pp. 5876{5886 (2013). 25. Chang, S.H. Taylor series method for solving a class of nonlinear singular boundary value problems arising 1684 A.R. Hadhoud et al./Scientia Iranica, Transactions E: Industrial Engineering 27 (2020) 1674{1684 in applied science", Applied Mathematics and Computation, 235, pp. 110{117 (2014). 26. Goh, J. and Ali, N.H.M. New high-accuracy nonpolynomial spline group explicit iterative method for two-dimensional elliptic boundary value problems", Scientia Iranica D, 24(6), pp. 3181{3192 (2017). 27. Rashidinia, J., Mohammadi, R., and Jalilian, R. Quintic spline solution of boundary value problems in the plate deection theory", Scientia Iranica D, 16(1), pp. 53{59 (2009). 28. Rashidinia, J., Jalilian, R., and Mohammadi, R. Convergence analysis of spline solution of certain twopoint boundary value problems", Scientia Iranica D, 16(2), pp. 128{136 (2009). 29. Ak, T., Karakoc, S.B.G., and Biswas, A. Application of Petrov-Galerkin _nite element method to shallow water waves model: Modi_ed Korteweg-deVries equation", Scientia Iranica B, 24(3), pp. 1148{1159 (2017). 30. Triki, H., Ak, T., Moshokoa, S., et al. Soliton solutions to KdV equation with spatio-temporal dispersion", Ocean Engineering, 114, pp. 192{203 (2016). 31. AK, T., Aydemir, T., Saha, A., et al. Propagation of nonlinear shock waves for the generalised Oskolkov equation and its dynamic motions in the presence of an external periodic perturbation", Pramana-J. Phys., 90(6) (2018). 32. Ak, T. and Karakoc, S.B.G. A numerical technique based on collocation method for solving modi_ed Kawahara equation", Journal of Ocean Engineering and Science, 3(1), pp. 67{75 (2018). 33. Ak, T., Dhawan, S., Karakoc, S.B.G., et al. Numerical study of Rosenau-KdV equation using _nite element method based on collocation approach", Mathematical Modelling and Analysis, 22(3), pp. 373{388 (2017). 34. Sastry, S.S., Introductory Methods of Numerical Analysis, PHI Learning Pvt, Ltd (2012).