Volume 8 Issue 3 Article 4

B-Spline Cubic Finite Element Method for Solving Ordinary Differential Equations

Authors: Bashder O. Hussen1 & Younis A. Sabawi2
1Department of Mathematics, Faculty of Science and Health, Koya University, Koya KOY45, Kurdistan Region, Iraq
2Department of Mathematics Education, Faculty of Education, Tishk International University, Erbil, Iraq

Abstract: This work proposes numerical solution of ordinary differential equations. The proposed method is based on applying modified cubic B-spline finite element method. The existence and uniqueness for the variational form proved. The convergence of the presented scheme is given. Numerical experiments are considered to confirm our theoretical results.

Download the PDF Document

Doi: 10.23918/eajse.v8i3p39

Published: December 21, 2022

References

Ahmed, N. (2017). Analysis of Cubic Spline Method, B-Spline Method and Finite Difference Method for Solving Boundary Value Problems. ADAMA, ETHIOPIA: Master thesis.

Al-Humedi, H. O., & jameel, Z. A. (2020). Combining Cubic B-Spline Galerkin Method with Quadratic Weight Function for Solving Partial Integro-Differential Equations. Journal of Al-Qadisiyah for computer science and mathematics, 12(1).

Cangiani, A., Georgouils, E. H., & Sabawi, Y. A. (2020). Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems. Journal of Computational and Applied Mathematics, 367.

Cangiani, Georgoulis, & Sabawi, E. &. (2018). Adaptive discontinuous Galerkin methods for elliptic interface problems. Math. Comp, 87(314), 2675– 2707.

Gockenbach, M. S. (2006). Understanding And Implementing the Finite Element Method. Society for Industrial & Applied.

Höllig, K. (2003). Finite Element Methods with B-Splines. Philadelphia: Society for Industrial Mathematics.

Hussein, Y. A. (2011). Combination Between Single Diagonal Implicit and Explicit Runge Kutta(SDIMEX-RK) Methods for solving stiff Differential equations. Tikrit Journal of Pure Science, 16(1), 7.

Jameel, H. O.-H. (2020). Cubic B-spline Least-square Method Combine with a Quadratic Weight Function for Solving Integro-Differential Equations. Earthline Journal of Mathematical Sciences, 4(1), 14.

Jinming Wu, X. Z. (2011). Finite element method by using quartic B-splines. Numerical Methods for Partial Differential Equations, 27(4), 11.

KrystynaStys, T. (2000). A higher-order finite difference method for solving a system of integro-differential equations. Journal of Computational and Applied Mathematics, 126(1-2), 13.

Li Ronglin, N. G. (1998 ). B-spline finite-element method in polar coordinates. Finite Elements in Analysis and Design, 28(4).

Manaa, S., Moheemmeed, M., & Hussien, Y. A. (2010). A Numerical Solution for Sine-Gordon Type System. Tikrit Journal of Pure Science, 15(3).

Munguia, M., & Bhatta, D. (2015). Use of Cubic B-Spline in Approximating Solutions. Applications and Applied, 10(2), 22.

Nielsen, H. (1998). Cubic Splines Tachnical Report. Department of Mathematical Modeling Technical University of Denmark.

Pirdawood, M. A., & Sabawi, Y. A. (2021). High-order solution of Generalized Burgers–Fisher Equation using compact finite difference and DIRK methods. InJournal of Physics: Conference Series 2021, 1999(1).

Rüdiger, V. (2013). A posteriori error estimation techniques for finite element methods. United Kingdom: Oxford University Press.

Sabawi, Y. A.(2017). Adaptive discontinuous Galerkin methods for interface problems. Leicester, PHD thesis: University of Leicester.

Sabawi, Y. A. (2020). A posteriori error analysis in finite element approximation for fully discrete semilinear parabolic problems. intechopen.

Sabawi, Y. A. (2021). A Posteriori L_∞ (L_2 )+L_2 (H^1 )–Error Bounds in Discontinuous Galerkin Methods For Semidiscrete Semilinear Parabolic Interface Problems. Baghdad Science Journal, 9.

Sabawi, Y. A. (2021). Posteriori Error bound For Fullydiscrete Semilinear Parabolic Integro-Differential equations. Journal of Physics, 1999(1).

Sabawi, Y. A., Pirdawood, M. A., & Sadeeq, M. I. (2021). A compact Fourth-Order Implicit-Explicit Runge-Kutta Type Method for Solving Diffusive Lotka–Volterra System. Journal of Physics: Conference Series. University of Al-Qadisiyah, Diwaniyah, Iraq: IOP Publishing Ltd.

Sabawii, Y. A. (2019). A posteriori L_∞ (H^1 ) error bound in finite element approximation of semdiscrete semilinear parabolic problems. First International Conference of Computer and Applied Sciences (CAS).

Sharma, D., Arora, S., & Kumar, S. (2013). A Comparative Study of Galerkin Finite Element and B-Spline Methods for Two Point Boundary Value Problems. International Journal of Computer Applications, 67(23).

Siddiqi, S., Shahid, Arshed, & Saima. (2014). Cubic B-spline for the Numerical Solution of Parabolic Integro-differential Equation with a Weakly Singular Kernel. Research Journal of Applied Sciences, Engineering and Technology, 7(10), 12.

Wenting Long, D. X. (2012). Quasi wavelet based numerical method for a class of partial integro-differential equation. Applied Mathematics and Computation, 218(24), 9.