Modified Finite Difference Method for Solution of Two-interval Boundary Value Problems with Transition Conditions

Abstract

In this study, we have proposed a new modification of classical Finite Difference Method (FDM) for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transition conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To illustrate the efficiency and reliability of the proposed modification of FDM some examples are solved. The obtained results are compared with those obtained by the standart FDM and by the analytical method. Corresponding graphical illustration are also presented.

Keywords

• Finite difference method
• transition conditions
• boundary value problem

References

1. Ascher, U. M., Mattheij R. M. M., Russell R. D., Numerical solution of boundary value problems for ordinary differential equations, Vol.13, Siam, 1994.
2. Aydemir, K., Mukhtarov, O.S., Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem, Boundary Value Problems, 2016(1)(2016), 1-16.
3. Burden, R. L., Faires, J. D., Numerical Analysis, Brooks, Cole Pub. Co., Pacific Grove, California, 609, 1997.
4. Çavuşoğlu, S., O. Sh Mukhtarov, A new finite difference method for computing approximate solutions of boundary value problems including transition conditions, Bulletin of the Karaganda university Mathematics series, 102(2)(2021), 54-61.
5. Kincaid, D., Kincaid, D.R., Cheney, E.W., Numerical analysis: mathematics of scientific computing (Vol. 2), American Mathematical Soc, 2009.
6. LeVeque, R.J. , Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, Society for Industrial and Applied Mathematics, 2007.
7. Mukhtarov, O. S., Aydemir, K., Oscillation properties for non-classical Sturm-Liouville problems with additional transmission conditions , Mathematical Modelling and Analysis, 26(3)(2021), 432-443.
8. Mukhtarov, O., Çavuşoğlu, S., Olgar, H., Numerical Solution of One Boundary Value Problem Using Finite Difference Method, Turkish Journal of Mathematics and Computer Science, 11(2019), 85-89.

9. Mukhtarov, O. S., Çavuşoğlu, S., Pandey, P.K., Development of the Finite Difference Method to solve a new type Sturm-Liouville problems, Tbilisi Mathematical Journal, 14(3)(2021), 141-154.
10. Muhtarov, O., Yakubov, S., Problems for ordinary differential equations with transmission conditions, Applicable Analysis, 5(2002), 1033-1064.
11. Olgar, H., Mukhtarov, O., Aydemir, K., Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem, In AIP Conference Proceedings, 1759, (2018), No. 1, p. 020060.
12. Roul, P., Goura, V.P. Numerical solution of doubly singular boundary value problems by finite difference method , Computational and Applied Mathematics, 39(4)(2020), 1-25.
13. Yizengaw, N., Convergence analysis of finite difference method for differential equation, Journal of Physical Mathematics, 8(3)(2017), 1-3.