Search

A ninth order block hybrid collocation method is proposed for solving general second order ordinary differential equations directly. The derivation involves interpolation and collocation of basic polynomial that generates the main and additional methods. These methods are applied simultaneously to provide approximate solutions at five main points and three off-step points. The stability properties of the block method are discussed. Some illustrative examples are given to demonstrate the efficiency of the method.

In this paper, we propose an algorithm of two-point block method to solve the nonlinear system of third-order boundary value problems directly. The proposed method is presented in a simple form of Adams type and two approximate solutions will be obtained simultaneously with the block method using variable step size strategy. The method will be implemented with the multiple shooting technique via the three-step iterative method to generate the missing initial value. Most of the existence method will reduce the third-order boundary value problems to a system of first order equations where the systems of six equations need to be solved. The method we proposed in this paper will solve the third-order boundary value problems directly. Two numerical examples are given to illustrate the efficiency of the proposed method.

We present a new rapidly convergence procedure called the interval repeated zoro symmetric single-step IRZSS1-5D which is an extension of the procedures IZSS1-5D and IRSS1. Theoretical analysis shows that the rate of convergence of IRZSS1-5D is at least 3r + 2 (r ≥ 1), whereas the rates of convergence of IZSS1-5D and IRSS1 are at least five and 2r + 1 (r ≥ 1), respectively. With r = 1, the procedure IRZSS1-5D is identical to IZSS1-5D. The procedures IRSS1 and IZSS1-5D are the basic references for the establishment of IRZSS1-5D, where the forward-backward-forward (FBF) step of IZSS1-5D are repeated r times in IRZSS1-5D, so that some elements of FBF of IZSS1-5D can be saved and reused for the next inner iterations r = 2,3, … in IRZSS1-5D.

The block hybrid-like method is presented and implemented in predictor-corrector mode to solve delay differential equations. The Lagrange interpolation polynomial is implemented to obtain the approximation for the delay terms. The Q-stability of the method is investigated. The illustrative examples are presented to demonstrate the efficiency of the block hybrid-like method.

Recently, subspace quasi-Newton (SQN) method has been widely used in solving large scale unconstrained optimization. Besides constructing sub-problems in low dimensions so that the storage requirement as well as computational cost can be reduced, it can also be implemented extremely fast when the objective function is a combination of computationally cheap non-linear functions. However, the main deficiency of SQN method is that it can be very slow on certain type of non-linear problem. Hence, a preconditioner which is computationally cheap and is a good approximation to the actual Hessian is constructed to speed up the convergence of the quasi-Newton methods since the evaluation of the actual Hessian is considered as impractical and costly. For this purpose, a diagonal updating matrix has been derived to replace the identity matrix in approximating the initial inverse Hessian. The numerical results show that the preconditioned SQN method performs better than the standard SQN method that without preconditioning.

A new Runge-Kutta-Nyström (RKN) method with phase-lag of order infinity for solving second order periodic initial value problems possessing oscillating solutions is derived in this paper. The derivation of phase-fitted four stage explicit RKN method with algebraic order five is discussed. The numerical results show that the new method is much more efficient than the existing methods in the literature.

In this paper we derived a new diagonally implicit Runge-Kutta method of order four with minimum phase-lag for solving first order linear ordinary differential equation. The stability polynomial of the method is obtained and the stability region is presented. A set of problems are tested upon and numerical results proved that the method is more accurate compared to other well known methods in the scientific literature.

In this paper, we present the analysis of the rate of convergence of the interval repeated midpoint zoro symmetric single-step procedure (IRMZSS) which is the extension of the interval midpoint zoro symmetric single-step procedure (IMZSS). The results show that the procedure IRMZSS has R-order of convergence at least 7r + 1 (r ≥ 1) or 0R (IRMZSS) ≥ 7 r + 1 (r ≥ 1), whereas the procedure IMZSS R-order of convergence at least 8 or 0R (IMZSS) ≥ 8. In fact 0R (IRMZSS) ≥ 0R (IMZSS) and 0R (IRMZSS) = 0R (IRMZSS) = 0R (IMZSS) when r = 1.

In this paper the improved Runge-Kutta method of order four with 4-stages for solving first order ordinary differential equation is proposed. The method is based on classical Runge-Kutta (RK) method also can be considered as special class of two-step method. Here, the coefficients of the method are obtained using the minimization of the error norm up to order five. The improved method with only 4-stages is more accurate than fourth order 4-stages RK method. Therefore it is computationally more efficient than the existing RK method. A number of test problems are solved and the numerical results compared with the existing RK method are given.

In this paper, a new three-stage Runge-Kutta-Nyström (RKN) method with two variable coefficients is constructed. The new method is based on Garcia's RKN method of algebraic order four. By using the idea of dispersion and dissipation of order infinity, a new method for solving second-order ordinary differential equations (ODE) with periodic solutions is derived. Numerical tests show that the new method is more accurate than the other existing methods that were derived in the past.

A similarity solution of the steady magnetohydrodynamic (MHD) mixed convection boundary layer flow due to a stretching vertical heated sheet in a power law nanofluid with thermal radiation effect is theoretically studied. The governing system of partial differential equations is first transformed into a system of ordinary differential equations. The transformed equations are solved numerically using the shooting method. The influence of pertinent parameters such as the nanoparticle volume fraction parameter, the magnetic parameter, the buoyancy or mixed convection parameter and the radiation parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results are also presented.

We present a new form of fourth-order explicit Runge-Kutta method in which the terms f and f′ are used in the formulae to increase the order as well as to enhance the performance of the method. The proposed method is then used to solve first order ordinary differential equation of the form y' = f(t,y(t)). From the numerical results, it is observed that the new method is more efficient in terms of time needed for solving the problems while maintaining the same accuracy as the existing Runge-Kutta method of the same order. Stability of the method is discussed the region of absolute stability is shown.

In this paper, two-step fourth order semi implicit hybrid method (SIHM) with dispersion of order six and zero dissipation is constructed for solving second order ordinary differential equations (ODEs). Numerical results show that SIHM is more accurate as compared to the existing hybrid method, Runge-Kutta Nyström (RKN) method, Runge-Kutta (RK) method and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same order. The interval of absolute stability of SIHM for ODE is presented. The comparison of time for solving the test problems for the various methods is also given.

A new block method that generates two values simultaneously is developed for the integration of stiff initial value problems. The method is proven to be A – stable and is a super class of the 2 – point block backward differentiation formula (BBDF). A comparison is made between the method, 1 point backward differentiation formula (BDF) and the 2 point BBDF methods. The numerical results indicate that the new method outperformed the 1 point BDF and the 2 point BBDF methods in terms of accuracy and stability. The total number of steps to complete the integration by the 1 point BDF method is reduced to half. Computation time for the method is also competitive.

In this paper, a new Runge-Kutta-Nyström (RKN) method of fourth algebraic order is constructed. The new method is based on Dormand's Runge-Kutta-Nyström method of algebraic order four, by using the idea of phase-lag of order infinity, a new method for solving second-order ordinary differential equations (ODE) with oscillatory solutions is derived. Numerical tests show that the new method is more accurate than other existing methods that based on the minimal phase-lag theory.

In this paper, a fourth-order explicit hybrid method with constant step-size is developed. The free parameters are chosen so that the error constant is small. Numerical comparisons are carried out to show the advantage of the new method for solving special second order ordinary differential equations.

A new 5(4) embedded explicit Runge-Kutta-Nystrom (RKN) method with dissipation of high order is developed to solve integration of initial-value problems for second-order ordinary differential equations possessing oscillating solutions. The fifth order formula has dispersive order eight and dissipative order nine. The fourth-order embedded formula is obtained to control the local truncation errors. Numerical experiments indicate that the new method is more efficient than the existing embedded explicit RKN methods.