Numerical Solution of One Boundary Value Problem Using Finite Difference Method

Yıl 2019, Cilt: 11, 85 - 89, 30.12.2019

Oktay Mukhtarov , Semih Çavuşoğlu , Hayati Olğar

Öz

Many problem of physics and engineering are modelled by boundary value problems for ordinary or partial differential equations. Usually, it is impossible to find the exact solution of the boundary value problems, so we have to apply various numerical methods. There are different numerical methods (for example, the Explicit Euler method, the Runge-Kutta method, the Improved Euler method, Finite difference method and finite element method) for determining the approximate solutions of initial and boundary-value problems. One of them is the finite difference method, which is the simplest scheme. This method can be applied to higher of ordinary differential equations, provided it is possible to write an explicit expression for the highest order derivative and the system has a complete set of initial conditions. In this study, we are interested in the finite difference method for new type boundary value problems. We describe the numerical solutions of some two-point boundary value problems by using finite difference method. This method are based upon the approximations that allow to replace the differential equations by algebraic system of equations and the unknowns solutions are related to grid points. In this article, we have presented a finite difference method for solving second order boundary value problems for ordinary differential equations with an internal singularity. This method tested on several model problems for the numerical solution.

Anahtar Kelimeler

Finite difference method, boundary value problems, transmission conditions

Kaynakça

• Abu-Zaid, I.T., El-Gebeily, M.A., {\em A finite-difference method for the spectral approximation of a class of singular two-point boundary value problems}, IMA Journal of Numerical Analysis, \textbf{14(4)}(1994), 545--562.
• Aydemir, K., Ol\v{g}ar, H., Mukhtarov, O.Sh., Muhtarov, F.S., {\em Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces}, Filomat, \textbf{32(3)}(2018), 921--931.
• Burden, R.L., Faires, J.D., Numerical Analysis, PWS-Kent Publ. Co. Brooks/Cole Cengage Learning, Boston, MA, 9th edition, 2010.
• Fulton, C.T., {\em Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions}, Proc. Roy. Soc. of Edin., \textbf{77A}(1977), 293--308.
• Jamet, P., {\em On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems} ,Numerische Mathematik, \textbf{14(4)}(1970), 355--378.
• Kaw, A., Garapati, S.H., Textbook Notes for The Parabolic Differential Equations, 2011.
• Keller, H.B., Numerical methods for two-point boundary-value problems, Courier Dover Publications, 2018.
• LeVeque, R.J., Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems, Vol. 98, Siam, 98 2007.
• Mukhtarov, O., Ol\v{g}ar, H., Aydemir, K.,{\em Resolvent Operator and Spectrum of New Type Boundary Value Problems}, Filomat, \textbf{29(7)}(2015), 1671--1680.