Block method for third order ordinary differential equations

The problem of third order ordinary differential equations (ODEs) is solved directly by using the block backward differentiation formula. The block method is constructed by utilizing three back values and by differentiating the interpolating polynomial once, twice and thrice. Two approximated solutions are generated concurrently for each integration step. Numerical results indicate the efficiency of the direct method than the usual approach of transforming it into the first order ODEs

Parallel block backward differentiation formulas for solving large systems of ordinary differential equations.

In this paper, parallelism in the solution of Ordinary Differential Equations (ODEs) to increase the computational speed is studied. The focus is the development of parallel algorithm of the two point Block Backward Differentiation Formulas (PBBDF) that can take advantage of the parallel architecture in computer technology. Parallelism is obtained by using Message Passing Interface (MPI).Numerical results are given to validate the efficiency of the PBBDF implementation as compared to the sequential implementation

Block backward differentiation formulas for solving second order fuzzy differential equations

In this paper, we study the numerical method for solving second order Fuzzy Differential Equations (FDEs) using Block Backward Differential Formulas (BBDF) under generalized concept of higher-order fuzzy differentiability. Implementation of the method using Newton iteration is discussed. Numerical results obtained by BBDF are presented and compared with Backward Differential Formulas (BDF) and exact solutions. Several numerical examples are provided to illustrate our methods

Componentwise block partitioning: a new strategy to solve stiff ordinary differential equations

Componentwise Block Partitioning is a new strategy to solve stiff ODEs, based on Block Backward Differentiation Formulas (BBDFs), and block of Adam type formulas. In this partitioning technique, the ODEs system is initially solved by Adam formulas until the equation that cause instability and stiffness is identified. Then, the equations that caused instability are placed into stiff subsystem and solved using BBDF. Numerical comparisons with code in the literature such as ode15s show the efficiency of the proposed partitioning technique

Modified block Runge-Kutta methods with various weights for solving stiff ordinary differential equations

A modified block Runge-Kutta (MBRK) methods for solving first order stiff ordinary differential equations (ODEs) are developed. Three sets of weight are chosen and implemented to the proposed methods. Stability regions of the MBRK methods are analyzed. Performances of the MBRK methods in terms of accuracy and computational time are compared with the classical third order Runge-Kutta (RK3) method and modified weighted RK3 method based on Centroidal mean (MWRK3CeM). The numerical results show that the proposed methods outperformed the comparing methods. Comparisons between the sets of weight used are also examined

Development of a-stable block method for the solution of stiff ordinary differential equations

A fixed step-size multistep block method for stiff Ordinary Differential Equations (ODEs) using the 2-point Block Backward Differentiation Formulas (BBDF) with improved efficiency is established. The method is developed using Taylor’s series expansion. The order and the error constant of the method are determined. To validate the new method is suitable for solving stiff ODEs, the stability and convergence properties are discussed. Numerical results indicate that the new method produced better accuracy than the existing methods when sloving the same problems

Block backward differentiation formulas for solving fuzzy differential equations under generalized differentiability

In this paper, the fully implicit 2-point block backward differentiation formula and diagonally implicit 2-point block backward differentiation formula were developed under the interpretation of generalized differentiability concept for solving first order fuzzy differential equations. Some fuzzy initial value problems were tested in order to demonstrate the performance of the developed methods. The approximated solutions for both methods were in good agreement with the exact solutions. The numerical results showed that the diagonally implicit method outperforms the fully implicit method in term of accuracy

Derivation of BBDF-α for solving ordinary differential equation

In this paper, the block backward differentiation formulas with parameter α (BBDF-α) of order three is derived in a constant step size for solving system of first order ordinary differential equations (ODEs). The coefficients of formula are generated using Maple software package. The influence of parameter α is considered to produce better approximate solutions at two points simultaneously. Numerical experiment is included to show the capability of the derived method in solving ODEs. Numerical results indicate that the BBDF-α outperforms the existing methods in term of accuracy

On the convergence of two point block backward differentiation formula for second order ODEs

The Two Point Block Backward Differentiation Formula (BBDF2) is a direct solver for second order Ordinary Differential Equations (ODEs). It had shown its efficiency by having less total number of steps and less computational time over the first order ODEs solver. In this paper, the convergence of the BBDF2 is justified by its consistency and zero-stability properties

Convergence properties of pth order diagonally implicit block backward differentiation formulas

This paper investigates the convergence properties for diagonally implicit 2-point block backward differentiation formulas of order two, three and four. The formulation of the method is reviewed from the literature. The order of the method is verified. The concepts of consistency and zero stability are considered to prove the convergence of the method