This paper discusses the numerical solution of periodic initial value problems. Two classes of methods are discussed, superimplicit and Obrechkoff. The advantage of Obrechkoff methods is that they are high-order one-step methods and thus will not require additional starting values. On the other hand they will require higher derivatives of the right-hand side. In cases when the right-hand side i...

in this paper, we present a new two-step trigonometrically fitted symmetric obrechkoff method. the method is based on the symmetric two-step obrechkoff method, with eighth algebraic order, high phase-lag order and is constructed to solve ivps with periodic solutions such as orbital problems. we compare the new method to some recently constructed optimized methods from the literature. the numeri...

In this paper, we present a new two-step trigonometrically fitted symmetric Obrechkoff method. The method is based on the symmetric two-step Obrechkoff method, with eighth algebraic order, high phase-lag order and is constructed to solve IVPs with periodic solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. The numeri...

Variable-step variable-order 3-stage Hermite–Birkhoff–Obrechkoff methods of order 4 to 14, denoted by HBO(4-14)3, are constructed for solving non-stiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. These methods use y′ and y′′ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads...

This paper presents new two-step explicit symmetric P-stable methods, including Obrechkoff and hybrid terms, of orders four and six for solving initial value problems of second order ordinary differential equations. In this paper, we improved the method written by Xinyuan Wu and Qinghong Li [11], in a way that we could increase order and accuracy of their methods. The numerical results obtained...

K e y w o r d s O b r e c h k o f f methods, Super-implicit method, Initial value problems. 1: I N T R O D U C T I O N In th is paper , we discuss t h e numer ica l so lu t ion of f i rs t -order ini t ia l va lue p rob l ems ( IVPs) y'(x) = f ( x , y(~)) , y(0) = y0, (1) and a specia l class (for which yl is miss ing) of second-order I V P s y ' ( x ) = f ( z , y ( x ) ) , y(O) = Yo, yI(O) = Y...

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.