THANK YOU FOR
Transcript: Hindmarsh and Byrne (1987) Stiff ODEs solvers Cash (1980) Rapid growth of the studies on the extension traditional method for solving ODEs have led to somewhat competition in deriving an efficient algorithms for solving stiff and non-stiff systems. Cash (1983) The later method evolved in many ways including Modified Extended Backward Differentiation Formulae (MEBDF) [4], Z.B. Ibrahim, M.B. Suleiman and K.I. Othman (2008) Producing block approximations also known as Block Backward Differentiation Formulae (BBDF) [8]. These method will approximate solutions of stiff equations for second order systems at 2 point simultaneously using variable step approach. The technique proposed by Hall and Watt (1976) will be applied for selection of the step size and order. This method will be compared with the existing ODE solver in Matlab (15s and 23s) L.G. Birta and O. Abou-Rabiaa, “Parallel block predictor-corrector methods for odes,”,IEEE Transactions on Computers, vol. C-36(3), pp. 299-311, 1987. K. Burrage, “Efficient block predictor-corrector methods with a small number of corrections,” J. of Comp. and App. Mat,. vol. 45, pp. 139-150, 1993. J.R. Cash, “On the integration of stiff systems of odes using extended backward differentiation formulae,” Numer. Math,.vol. 34, pp. 235-246, 1980. J.R. Cash, “The integration of stiff initial value problems in odes using modified extended backward differentiation formulae,” Comput. Math. Appl., vol. 9, pp. 645-660, 1983. M.T. Chu, and H. Hamilton, “Parallel solution of odes by multi-block methods,” Siam J. Sci. Stat. Comput., vol. 8(1), pp. 342-353, 1987. S.O. Fatunla, “Block methods for second order odes,” Intern. J. Computer Math., vol. 40, pp. 55-63, 1990. C.W. Gear, “Numerical initial value problems in ordinary differential equations,” COMM. ACM., vol. 14, pp. 185-190, 1971. Z.B. Ibrahim, M.B. Suleiman and K.I. Othman, “Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations,” European Journal of Scientific Research, vol. 21, no.3, pp. 508-520, 2008. ICMMS 2014 The strategy would be to choose the order for which estimates the maximum step size. The new step size will be the maximum step size , and the order which produces the maximum step size will be the order of the new step. Having LTE2,k-1 ,LTE2,k and LTE2,k+1 the decision on the order must be taken and the estimation for the maximum step size are as follows: International Conference on Mathematics and Mathematical Sciences September, 22-23, 2014 Paris, France For all problems tested, it shows that, VS-BBDF(2) has outperformed the ode15s and ode23s in term of average error as well as maximum error. It also managed to reduce the number of total steps taken in most of the cases. Outline Introduction Objectives Methodology Formulation of general variable step BBDF Choosing the order and step size Numerical results Discussion References Methodology Siti Ainor Mohd Yatim Zarina Bibi Ibrahim Khairil Iskandar Othman Mohamed Suleiman Objectives On the Derivation of Variable Step BBDF for Solving Second Order Stiff ODEs Introduction Discussion Z.B. Ibrahim, K.I. Othman and M.B. Suleiman, “Variable stepsize block backward differentiation formula for solving stiff odes,” Proceedings of World Congress on Engineering 2007, LONDON, U.K., vol. 2, pp. 785-789, 2007. Z.B. Ibrahim, M.B. Suleiman and K.I. Othman, “Implicit r-point block backward differentiation formula for solving first- order stiff odes,” Applied Mathematics and Computation, vol. 186, pp. 558-565, 2007. Z.B. Ibrahim, “Block Multistep Methods For Solving Ordinary Differential Equations,” Ph. D. Thesis, Universiti Putra Malaysia, Selangor, 2006. P. Kaps and G. Wanner, “A study of rosenbrock-type methods of high order,” Numer. Math., vol. 38, pp. 279-298, 1981. J.D. Lambert, Numerical Methods for Ordinary Differential Equations: The Initial Value Problems, John Wiley & Sons, New York 1991. Derived Block Backward Differentiation Formula of order 2. Develop Variable Step Block Backward Differentiation Formula in single code. Numerical results are compared with Matlab’s ode solver namely ode15s and ode23s. THANK YOU FOR YOUR ATTENTION References
Prezi Present
Prezi Video
Prezi Design
