Finite Difference Method for Approximate Solution of a Boundary Value Problem with Interior Singular Point

Year 2021, Volume: 9 Issue: 1, 40 - 48, 28.04.2021

Semih Çavuşoğlu , Oktay Mukhtarov , Hayati Olgar

Abstract

This study is aimed at finding the approximate solutions of some boundary value problems with interior singular points. The considered problems differs from the standart boundary value problems in that they contains additional transmission conditions at the points of singularity. Naturally, such type of problems are much more complicate to solve than regular boundary value problems ones. Moreover, it is not clear how to apply the classical numerical methods to problems, involving not only boundary conditions at the end points of the considered interval, but also additional transmission conditions at some interior points. By modifying the Finite Diferrence Method we present a new numerical algoritm to find approximate solutions of the considered boundary value transmission problems. The obtained approximate solutions are compared with the exact solutions for some illustrative singular boundary value problems, involving additional transmission conditions. To justify the proposed modification some graphical illustrations of the approximate solutions are also presented.

Keywords

Finite difference method, boundary value problems, transmission conditions, interior singular point

References

• [1] Abu Zaid, I. T. and El-Gebeily, M. A., A finite-difference method for the spectral approximation of a class of singular two-point boundary value problems, IMA Journal of Numerical Analysis 14, 4 (1994), 545-562.
• [2] Andrew, A. L., Correction of finite difference eigenvalues of periodic Sturm-Liouville problems, The Journal of the Australian Mathematical Society, Series B, Applied Mathematics 30, 04 (1989), 460-469.
• [3] Aydemir, K., Olgar, H., Mukhtarov, O. Sh., and Muhtarov, F. S., Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces, Filomat, 32,3, (2018), 921-931.
• [4] Baxley, J., V. , Numerical Solutions of Singular Nonlinear Boundary Value Problems. Proceedings of the Third International Colloquium on Numerical Analysis. De Gruyter, 2020.
• [5] Burden, R. L. and Faires, J. D., Numerical Analysis PWS-Kent Publ. Co. Brooks/Cole Cengage Learning, Boston, MA, (2010), 9th edition.
• [6] Chawla, M. M., Katti, C. P. , A finite-difference method for a class of singular two-point boundary-value problems. IMA journal of numerical analysis, 4(4), (1984), 457-466.
• [7] Duru, H., Baransel G., The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems. Journal of Applied Mathematics and Computational Mechanics 19.4 (2020): 45-56.
• [8] Fulton, C. T.,Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. of Edin., 77A (1977), 293-308.
• [9] El-Gebeily, M. A., Abu-Zaid, I. T, On a finite difference method for singular two-point boundary value problems. IMA journal of numerical analysis, 18(2),(1998), 179-190.
• [10] Kumar, M. , A new finite difference method for a class of singular two-point boundary value problems. Applied Mathematics and Computation, 143(2-3),(2003), 551-557.
• [11] Mukhtarov, O. Sh, Yucel¨ M., A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique. Mathematics 8.3 (2020): 415.
• [12] Mukhtarov, O., Aydemir, K., The eigenvalue problem with interaction conditions at one interior singular point, Filomat, (2017), 31(17).
• [13] Mukhtarov, O., C¸avus¸oglu, S., Olgar, H. Numerical Solution of One Boundary Value Problem Using Finite Difference Method, Turkish Journal of Mathematics and Computer Science, 11, 85-89.
• [14] Niu, J., Xu, M., Lin, Y., Xue, Q, Numerical solution of nonlinear singular boundary value problems. Journal of Computational and Applied Mathematics, 331,(2018), 42-51.
• [15] Pandey, P. K. , Finite Difference Method for a Second-Order Ordinary Differential Equation with a Boundary Condition of the Third Kind, Computational Methods In Applied Mathematics, 10(1),(2010), 109-116 .
• [16] Roul, P., Goura, V. P., Agarwal, R. , A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions. Applied Mathematics and Computation, 350,(2019), 283-304.
• [17] Snell, C., Vesey, D. G., Mullord, P. , The application of a general finite difference method to some boundary value problems. Computers Structures, 13(4),(1981), 547-552.