This paper focuses on the construction of two-point and three-point implicit block
methods for solving general second order Initial Value Problems. The proposed methods
are formulated using Hermite Interpolating Polynomial. The block methods approximate
the numerical solutions at more than one point at a time directly without reducing the
equation into the first order system of ordinary differential equations. In the derivation of
the method, the higher derivative of the problem is incorporated into the formula to enhance
the efficiency of the proposed methods. The order and zero- stability of the methods are
also presented. Numerical result
In this paper, an improved trigonometrically fitted zero-dissipative explicit two-step hybrid method with fifth algebraic
order is derived. The method is applied to several problems where by the solutions are oscillatory in nature. Numerical
results obtained are compared with existing methods in the scientific literature. The comparison shows that the new
method is more effective and efficient than the existing methods of the same order.
In this article, the general form of Runge-Kutta method for directly solving a special fourth- order ordinary differential
equations denoted as RKFD method is given. The order conditions up to order seven are derived, based on the order
conditions, we construct a new explicit four-stage sixth-order RKFD method denoted as RKFD6 method. Zero-stability of
the method is proven. Comparisons are made using the existing Runge–Kutta methods after the problems are reduced
to a system of first order ordinary differential equations. Numerical results are presented to illustrate the efficiency and
competency of the new method.
In this paper, we present the absolute stability of the existing 2-point implicit block multistep step methods of step number k = 3 and k = 5 and solving special second order ordinary differential equations (ODEs). The methods are then trigonometrically fitted so that they are suitable for solving highly oscillatory problems arising from the special second order ODEs. Their explicit counterparts are also trigonometrically fitted so that in the implementation the methods can act as a predictor-corrector pairs. The numerical results based on the integration over a large interval are given to show the performance of the proposed methods. From the numerical results we can conclude that the new trigonometrically-fitted methods are superior in terms of accuracy and execution time, compared to the existing methods in the scientific literature when used for solving problems which are oscillatory in nature.
In this paper, we develop algebraic order conditions for two-point block hybrid method up to order five using the approach
of B-series. Based on the order conditions, we derive fifth order two-point block explicit hybrid method for solving
special second order ordinary differential equations (ODEs), where the existing explicit hybrid method of order five is
used to be the method at the first point. The method is then trigonometrically fitted so that it can be suitable for solving
highly oscillatory problems arising from special second order ODEs. The new trigonometrically-fitted block method is
tested using a set of oscillatory problems over a very large interval. Numerical results clearly showed the superiority
of the method in terms of accuracy and execution time compared to other existing methods in the scientific literature.
Kaedah baru pasangan benaman 4(3) tahap-empat berperingkat empat tak tersirat Runge-Kutta-Nyström (RKN) diterbitkan untuk mengamir persamaan pembezaan peringkat dua berbentuk yʺ = f (x, y) dengan penyelesaian bentuk berkala. Dipersembahkan kaedah yang bercirikan serakan berperingkat tinggi serta pekali ralat pangkasan utama yang ‘kecil’. Analisis kestabilan bagi kaedah yang diterbitkan juga diberikan. Perbandingan keputusan berangka antara kaedah yang dihasilkan dengan kaedah RK4(3) dan RKN4(3)D menunjukkan kaedah yang baru ini berkecekapan lebih baik daripada segi penilaian fungsi dan masa pelaksanaan.