Third Order Backward Difference Formula for a First Order Stiff System in Piecewise Uniform Mesh

B. Sumithra

doi:https://doi.org/10.14445/22315373/IJMTT-V33P535

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 33 | Number 4 | Year 2016 | Article Id. IJMTT-V33P535 | DOI : https://doi.org/10.14445/22315373/IJMTT-V33P535

Third Order Backward Difference Formula for a First Order Stiff System in Piecewise Uniform Mesh

B. Sumithra

Citation :

B. Sumithra, "Third Order Backward Difference Formula for a First Order Stiff System in Piecewise Uniform Mesh," International Journal of Mathematics Trends and Technology (IJMTT), vol. 33, no. 4, pp. 268-275, 2016. Crossref, https://doi.org/10.14445/22315373/IJMTT-V33P535

Abstract

Stiff system of Ordinary Differential Equations can be exemplified by problems in chemical kinetics, fluid dynamics, quantum mechanics, electrical networks, etc. In this paper, a third order Backward Differentiation Formula (BDF-3) is suggested on a piecewise uniform mesh(PUM) to solve a system of first order stiff Ordinary Differential Equations(ODEs). It is proved that the numerical approximations generated by this method with PUM produce numerical solutions with less computational effort and less error as compared to the method without PUM. Numerical results are presented in support of the theory.

Keywords

System of stiff differential equations, Initial value problem, BDF-3, Piecewise uniform mesh.

References

[1] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley, (2003).
[2] J. R. Cash, The integration of stiff initial value problems in O.D.E.S using modified extended backward differentiation formulae, Comp. and Maths. With Applics. (1983).
[3] D. J. L. Chen, The effective order of singly - implicit methods for stiff differential equations, Thesis submitted for the degree of Doctor of Philosophy at the University of Auckland (1998).
[4] C. Clavero, J. L. Gracia and F. Lisbona, High order methods on Shishkin meshes for singular perturbation problems of convection diffusion type, Numerical Algorithms, (1999).
[5] C. F. Curtiss and J. O. Hirschfelder, Integration of stiff equations, Acta. Math., (1952).
[6] P. Deuflhard, F. Bornemann, Scientific Computing with Ordinary Differential Equations, Springer, (2002).
[7] C. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice Hall, Englewood N J. Cliffs, (1971).
[9] C. W. Gear,The Numerical integration of ordinary differential equations, Math. Comp, (1967).
[9] E. Hairer and G. Wanner, Solving ordinary differential equations. II: Stiff and differential-algebraic problems, Second revised edition., Springer, Berlin, (1996).
[10] P. Henrici, Discrete variable methods in ordinary differential equations, New York, Wiley (1962)
[11] J. P. Keener, J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, (1998).
[12] M. K. Jain, Numerical solution of differential equations, Wiley Eastern, (1984).
[13] Kailash C. Patidar, High order parameter uniform numerical Method for Singular Perturbation problems, Appl. Math and comp, (2007).
[14] J. D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, NewYork, (1973).
[15] Lawrence F. Shampine and Skip Thompson, Stiff systems, Scholarpedia, (2007).
[16] J. J. H. Miller, E. O’ Riordan, G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing Co.pvt.Ltd, Dublin, (2012).
[17] J. D. Murray, Mathematical Biology: I. An Introduction, Third Edition, Springer, (2002).
[18] J. D. Murray, Mathematical Biology: II. Spatial Models and Biomedical Applications, Third Edition, Springer, (2004).
[19] Natalia Kopteva and Eugene O’ Riordan Shishkin meshes in the numerical solution of singularly perturbed differential equations, International Journal of Numerical Analysis and Modeling, (2010).
[20] Nor Ain Azeany Mohd Nasir, Zarina Bibi Ibrahim, Khairil Iskandar Othman and Mohamed Suleiman Numerical Solution of First Order Stiff Ordinary Differential Equations using Fifth Order Block Backward Differentiation Formulas, Sains Malaysiana, (2012).
[21] L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman Hall, (1994).
[22] L. F. Shampine, C. W. Gear, A users view of solving stiff ordinary differential equations, SIAM Review, (1979).
[23] L. F. Shampine, M. K. Gordon, Computer Solution of Ordinary Differential Equations: the Initial Value Problem, Freeman, (1975).
[24] B. Sumithra, A. Tamilselvan, Numerical Solution Of Stiff System By Trapezoidal Method, International J. of Math. Sci. and Engg. Appls. (IJMSEA)., Vol. 9, No. I, pp. 1-11, (March 2015).
[25] B. Sumithra, A. Tamilselvan, Numerical Solution of Stiff System by Backward Euler Method, Applied Mathematical Sciences., Vol. 9, No. 67, pp. 3303 - 3311, (April 2015).
[26] B. Sumithra, A. Tamilselvan, Numerical Solution of Stiff System by Second Order Backward Difference Formula, Global Journal of Pure and Applied Mathematics., Volume 11, Number 5, pp. 3035-3046, (October 2015).