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This paper proposed 2-point block backward differentiation formulas (BBDF) of order 3, 4, and 5 for direct solution of second order ordinary differential equations. These methods were derived via backward difference interpolation polynomial with two solutions are produced simultaneously at each step. All the three different orders of 2-point BBDF is implemented in variable order scheme. The scheme utilizes the local truncation error, which is generated by the single order of 2-point BBDF method. Numerical results are presented to illustrate the validity of the proposed scheme.

In this paper, Adapative Order of Block Backward Differentiation Formulas (ABBDFs) are formulated using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs). These ABBDF methods are of order four, five and six. The benefit of the ABBDF methods is the computation time in the computation of solutions. Numerical results are presented to demonstrate the advantage of implementing adaptive order selection in a single code.

In this paper, a new fifth order Direct Block Backward Differentiation Formula (5-DBBDF) for solving second order stiff ordinary differential equations (ODEs) is presented. Five backwards steps are used to generate the coefficients of the 5-DBBDF. This method approximates the solutions at two points concurrently by using fixed step size and solved the second order ODEs directly without reducing the ODEs to a system of first order. Numerical results on some standard problems found in the literature are presented to validate the accuracy of the proposed method.

Intervalwise partitioning is a strategy to solve stiff ordinary differential equations (ODEs). This strategy using on 3-point block method will initially starts solving ODE using Adams method, and switch the system to Backward Differentiation Formula (BDF) when there is an indication of stiffness. Indication of stiffness will be based on hacc > hiter and the trace of the Jacobian. The comparison with existing method reveals that this partitioning strategy can be an alternative method to solve stiff ODEs.

Direct Two Point Block Backward Differentiation Formula, (BBDF2) for solving second order ordinary differential equations (ODEs) will be presented throughout this paper. The method is derived by differentiating the interpolating polynomial using three back values. In BBDF2, two approximate solutions are produced simultaneously at each step of integration. The method derived is implemented by using fixed step size and the numerical results that follow demonstrate the advantage of the direct method as compared to the reduction method.

The three point block method for solving second order ordinary differential equations (ODEs) directly using constant step size is derived. The reliability of this new method is verified in the numerical results with the improved performance in terms of computation time while maintaining the accuracy. The comparison is presented between the new method and classical backward differentiation formulas (BDF) of order 3.

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.

Linear array of permutations is hard to be factorised. However, by using a starter set, the process of listing the permutations becomes easy. Once the starter sets are obtained, the circular and reverse of circular operations are easily employed to produce distinct permutations from each starter set. However, a problem arises when the equivalence starter sets generate similar permutations and, therefore, willneed to be discarded. In this paper, a new recursive strategy is proposed to generate starter sets that will not incur equivalence by circular operation. Computational advantages are presented that compare the results obtained by the new algorithm with those obtained using two other existing methods. The result indicates that the new algorithm is faster than the other two in time execution.

In this paper, 3-point Block Backward Differentiation Formulas (3BBDF) is used for the numerical solution of Fuzzy Differential Equations (FDEs). Implementation of 3BBDF using Newton iteration is discussed. Numerical results obtained by the 3BBDF are presented and compared with the Modified Simpson method to illustrate the ability of the 3BBDF method for solving FDEs.

In this paper, a class of Block Backward Differentiation Formulas (BBDFs) are developed using fixed step size with automatic order selection for the numerical solution of stiff ordinary differential equations (ODEs). The method is derived by combining the BBDF methods of order four, five and six as single code with strategy on order selection. The proposed BBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with single order BBDF. Numerical results shows the advantage of using BBDF with automatic order selection for solving ODEs.

In this paper, we consider the numerical solution of first order stiff ordinary differential equations (ODEs) by a class of variable step size three points Block Backward Differentiation Formulas (BBDFs) methods. The basic formulation using three back values and the step size controlling strategy is described. In order to optimize the performance in terms of precision and the computational time, all the coefficients of the method will be stored in simplified form to avoid the repetitive computation of the coefficients as the step changes in the integration interval. Numerical results show the 3-point BBDF method gives better accuracy compared to the existing ODE solver, ode15s when solving ODEs problems.

In this paper, we focus on the stability region of the variable step size of 3-point block backward differentiation formula (VSBBDF) method. The graphs are plotted using MAPLE software. To show the performance of the method, the accuracy is presented in the numerical results to solve first order stiff ordinary differential equations (ODEs).

In this paper, we derive a second order Variable Step Variable Order Block Backward Differentiation Formulae (VSVO-BBDF (2)). This method pertains to the study of solving stiff Ordinary Differential Equations (ODEs) of second order (y"). The pertinent findings in term of total number of steps, accuracy and computational time are displayed and discussed through distinct tables.

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

A comprehensive research on the existing Block Backward Differentiation Formulae (BBDF) was done. Based on the suitability in solving stiff ordinary differential equations (ODEs), BBDF of order 3 up 5 is collected using simplified strategy in controlling the step size and order of the method. Thus, Extended Block Backward Differentiation Formulae (EBBDF) is derived with the intention of optimizing the performance in terms of precision and computation time. The accuracy of the method are investigated using linear and non linear stiff initial value problems and its performance is compared with MATLAB’s suite of ODEs solvers namely ode15s and ode23s.

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.

Parallel implementation of fourth order Block Backward Differentiation Formulas (BBDF(4)) is presented for a numerical solution of first order stiff ordinary differential equations (ODEs). This method computes the numerical solution at two points concurrently in each block. The sequential and parallel codes are developed on Message Passing Interface (MPI) library and run on High Performance Computing (HPC). The performance of this new method is measured in terms of Speedup (Sp) and Efficiency (Ef). It is shown that the parallel implementation BBDF(4) produced a better speed up compared to the sequential timing.

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.

In this paper, the 2-point Block Backward Differentiation Formulas (BBDF) methods proposed by Zarina et al. [8] for the numerical solution of initial value problems in Ordinary Differential Equations (ODEs) are reviewed. The zero stability and consistency which are necessary conditions for convergence of the BBDF methods are studied. The order of the method is also determined.

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

The new method derived is called the fifth order two-point Block Backward Differentiation Formulas (BBDF(5)) method for solving first order stiff Ordinary Differential Equations (ODEs). This method possesses the requirement for stiffly stable and suitable to solve stiff problems. We also discussed further the implementation of the method using Newton Iteration. The numerical results are presented to verify the efficiency of this method as compared to the Classical Backward Differentiation Formula (BDF) method and ode15s in Matlab. The BBDF(5) method outperformed the BDF method and ode15s in terms of maximum error and execution time.

This paper presents a parallel implementation of a new permutation generation method.This permutation generation method is based on the starter sets for listing all the n! permutations.The sequential algorithm is developed and parallelized for parallel computing by integrating with Message Passing Interface (MPI) libraries. The performance of the parallel algorithms is presented to demonstrate the validation of the works.

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.

Kertas kerja ini membincangkan kaedah Blok Formulasi Beza Ke Belakang (BFBB) bagi menyelesaikan persamaan pembezaan biasa (PPB) jenis kaku. Umumnya, persamaan jenis kaku diselesaikan dengan kaedah tersirat yang melibatkan lelaran Newton yang mengambil masa pengiraan yang panjang. Dengan yang demikian, suatu kaedah blok dua titik menggunakan saiz langkah berubah dibangunkan berdasarkan Formulasi Beza Ke Belakang (FBB). Sebelum ini pengiraan beza pembahagi yang berulangkali dan rumit dilakukan pada setiap langkah dalam pengiraan pekali pembezaan,tetapi dalam kod yang dibangunkan, pengiraan pekali pembezaan hanya dilakukan pada langkah awal sahaja. Dipaparkan perbandingan keputusan berangka antara BFBB dan FBB yang menunjukkan keberkesanan dalam mengurangkan masa pengiraan dan kejituan penyelesaian yang lebih baik apabila kaedah BFBB di gunakan dalam menyelesaikan masalah nilai awal PPB kaku.

This paper focuses on the derivation of implicit 2-point block method based on Backward Differentiation Formula (BDF) which will be called BBDF of variable step size for solving first order stiff initial value problems (IVPs) for Ordinary Differential Equations (ODEs). The method presented is similar to the form of standard BDF. This allows us to store the coefficients of the y values and thus avoiding calculating the differentiation coefficients at each step but robust enough to allow for step size variation.Plots of their regions of absolute stability for the method are also presented. The efficiency of the 2-point BBDF is compared with the conventional variable step variable order BDF(VSVOBDF) method. Numerical results indicate that the resulting 2-point BBDF method outperform the VSVOBDF method in both execution time and accuracy.

This paper focuses on solving Ordinary Differential Equations (ODEs) using partitioning technique known as Intervalwise Partitioning Block (IPB) method which is based on Block Backward Differentiation Formulas (BBDF) and Block Adams Formulas. Method based on BBDF requires Newton-like iteration involving the Jacobian matrix of the ODEs. These matrix operations in the iteration scheme are expensive and require a considerable amount of computational effort. Therefore, IPB technique is developed to reduce the cost of these iterations. Numerical results indicate that the IPB strategy reduce the execution time while accuracy is maintained.