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 we proposed three explicit Improved Runge-Kutta (IRK) methods for solving first-order ordinary differential equations. These methods are two-step in nature and require lower number of stages compared to the classical Runge-Kutta method. Therefore the new scheme is computationally more efficient at achieving the same order of local accuracy. The order conditions of the new methods are obtained up to order five using Taylor series expansion and the third and fourth order methods with different stages are derived based on the order conditions. The free parameters are obtained through minimization of the error norm. Convergence of the method is proven and the stability regions are presented. To illustrate the efficiency of the method a number of problems are solved and numerical results showed that the method is more efficient compared with the existing Runge-Kutta method.
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.
Subspace quasi-Newton (SQN) method has been widely used in large scale unconstrained optimization problem. Its popularity is due to the fact that the method can construct subproblems in low dimensions so that storage requirement as well as the computation cost can be minimized. However, the main drawback of the SQN method is that it can be very slow on certain types of non-linear problem such as ill-conditioned problems. Hence, we proposed a preconditioned SQN method, which is generally more effective than the SQN method. In order to achieve this, we proposed that a diagonal updating matrix that was derived based on the weak secant relation be used instead of the identity matrix to approximate the initial inverse Hessian. Our numerical results show that the proposed preconditioned SQN method performs better than the SQN method which is without preconditioning.
Improvements over embedded diagonally implicit Runge-Kutta pair of order four in five are presented. Method of higher stage order with a zero first row and the last row of the coefficient matrix is identical to the vector output is given. The stability aspect of it is also looked into and a standard test problems are solved using the method. Numerical results are tabulated and compared with the existing method.
The paper reports on some theoretical studies concerning the impulsional mode of a quadrupole mass filter (QMF) supplied with a new periodic impulsional radio frequency voltagein the form of V(ac)cos(Ωt)⌊1=kcos(1/3Ωt/1-√kcos(2Ωt⌋ with 0 ≤k < 1 (⌊.⌋ means floor function) and eventually compares it to the classical sinusoidal case, k = 0. The physical properties of the confined ions in the r and z directions are illustrated and the fractional mass resolutions m/Δm of the confined ions in the first stability regions of both potential were analyzed for hydrogen isotopes and presented.