A New Multi-Step Approach based on Top Order Methods (TOMs) for the Numerical Integration of Stiff Ordinary Differential Equations
Awari, Yohanna Sani1, Kumleng, Geofrey Micah2
1.Department of Mathematical Sciences, Taraba State University, Jalingo Nigeria.
2.Department of Mathematics, University of Jos, Nigeria.
Abstract: This paper presents an entirely new approach to obtaining self-starting Top Order Methods (TOMs) which we shall called Extended Top Order Methods (ETOMs). ETOMs were obtained through hermite polynomial used as basis function. Stability analysis of the new approach shows a uniform order six method for k = 3, they also possess very good absolute stability regions which made them highly suitable for the numerical integration of stiff ordinary differential equations. Implementation of the method in block form eliminates the need for starters and hence, generating simultaneously approximate solutions yi, i = 1, 2, …, 6on the go. To further observe the effect of the new approach, it was implemented on four numerical initial value problems of stiff ordinary differential equations occurring in real life and was shown to compete favorably with the work of existing scientists.
Keywords: Trapezoidal Method, Hermite Polynomial, Top Order Methods, Stiff Equation, Block Method, Stiff ODEs.
Pages: 1 – 14 | Full PDF Paper
Department of Statistics, Sultan Qaboos University, Muscat, Oman
Abstract: A Bayesian sequential procedure for estimating the parameter of a class of life distributions under the squared loss is proposed. It is assumed that the cost per observation is one unit and that the parameter is the value of a random variable having the Inverse Gamma distribution with known parameters. The procedure has two components, a stopping time and the Bayes estimator based on the observations taken up to the stopping time. An upper bound is obtained for the Bayes regret, that is used as a measure of the performance of the proposed. It is found that the proposed procedure performs better than the best fixed-sample-size procedure, asymptotically.
Keywords: Bayes estimator, Bayes regret, Jensen’s inequality, Fatou’s Lemma, martingale, posterior distribution, sequential procedure, stopping time
Pages: 15 – 25 | Full PDF Paper