Numerical Methods
This paper discusses a direct three-point implicit block multistep method for direct solution of the general third-order initial value problems of ordinary differential equations using variable step size. The method is based on a pair of explicit and implicit of Adams type formulas which are implemented in PE(CE) t mode and in order to avoid calculating divided difference and integration coefficients all the coefficients are stored in the code. The method approximates the numerical solution at three equally spaced points simultaneously. The Gauss Seidel approach is used for the implementation of the proposed method. The local truncation error of the proposed scheme is studied. Numerical examples are given to illustrate the efficiency of the method.
A new four-point implicit block multistep method is developed for solving systems of first-order ordinary differential equations with variable step size. The method computes the numerical solution at four equally spaced points simultaneously. The stability of the proposed method is investigated. The Gauss–Seidel approach is used for the implementation of the proposed method in the PE(CE)^m mode. The method is presented in a simple form of Adams type and all coefficients are stored in the code in order to avoid the calculation of divided difference and integration coefficients. Numerical examples are given to illustrate the efficiency of the proposed method.
A new 3-point three step method is developed for solving system of first order ordinary differential equations (ODEs). This method at each step approximates the solution at three points simultaneously using variable step size. The method is in a simple form as Adams Moulton method with the specific aim of gaining efficiency. The Gauss-Seidel style is used for the implementation of the proposed method in PE(CE) mode. The stability regions of the method are discussed. Numerical results show that the proposed method is more efficient than some existent block method, in terms of accuracy, total number of steps and function calls and execution times.
Block method for numerical solution of fuzzy differential equations [International Mathematical Forum 2009][PDF]
In this paper the 2 point 2 step method for solving fuzzy initial value problem is proposed. This method at each step will estimate the solutions of the fuzzy initial value problem at two points simultaneously using variable step size. The stability of the proposed method is dis- cussed. Examples are presented to illustrate the computational aspect of the method.
Parallel Solution in Space of Large ODEs Using Block Multistep Method with Step Size Controller [ European Journal of Scientific Research 2009][PDF]
The parallel 4-point implicit multistep block method is developed for solving large system of first order ODEs using variable step size. The proposed method computes the numerical solution at four points simultaneously. The Gauss Seidel style is used for the implementation of the method. The parallelism across the system is considered for the parallelization of the method and parallel version has been implemented using MPI communication environment on a High Performance Computer (HPC). The results indicate the advantage of utilizing the parallel implementation of the proposed method for solving large-scale system of ODEs.
