Abstract
In this paper, authors present a new family of two-step explicit fourth order methods, for the numerical integration of second order periodic initial-value problems. These methods are explicit in nature and we intend to use them, in future, as a predictor for the family of direct hybrid methods. Their stability properties and the efficiency are also discussed. Considering some numerical results, authors saw that the new methods are superior to the existing explicit methods.
Highlights
Definition 1: The numerical method (3) is said to have interval of periodicity, Lambert and Watson [11].Authors are mainly concerned with the numerical integration of the special second order initial value systems of order m, ( ) 0, H, if for all ∈, r1 and r2 satisfy y y " =d= 2 y dt 2 f (t, y), y(0), ' (0) given and t>0
For the numerical integration of the second order initial value problem (1), consider the explicit fourth order method, which the authors will denote as EXP4 and is given by yn+1 = 2 yn − yn−1 + h2 fn
Authors have developed explicit fourth order method, which we will be using as a predictor for a class of direct hybrid methods
Summary
Definition 1: The numerical method (3) is said to have interval of periodicity, Lambert and Watson [11]. In which the first derivative does not appear explicitly Such problems have periodic solutions and they are common in celestial. Definition 2:The method (3) is P-stable if its interval of periodicity mechanics, quantum mechanical scattering theory, theoretical physics is (0,∞)Lambert and Watson [11]. Definition 3:The method (3) is unconditionally stable if r1 ≤ 1 and r2 ≤ 1for all values ofλh, Hairer [13]. These predictors should be explicit in nature Gladwell and Thomas [5] and Wang [6].
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