In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.

The main contribution in this paper is to construct an implicit fixed coefficient Block Backward Differentiation Formulas denoted as $A\left(\alpha \right)$ -BBDF with equal intervals for solving stiff ordinary differential equations (ODEs). To avoid calculating the differentiation coefficients at each step of the integration, the coefficients of the formulas will be stored, with the intention of optimizing the performance in terms of precision and computational time. The plots of their $A\left(\alpha \right)$ stability region are provided, and the order of the method is also verified. The necessary conditions for convergence, such as the consistency and zero stability of the method, are also discussed. The numerical results clearly showed the efficiency of the method in terms of accuracy and execution time as compared to other existing methods in the scientific literature.

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed $A$ –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

This paper aims to select the best value of the parameter $\rho$ from a general set of linear multistep formulae which have the potential for efficient implementation. The $\rho$ -Diagonally Implicit Block Backward Differentiation Formula ( $\rho$ -DIBBDF) was proposed to approximate the solution for stiff Ordinary Differential Equations (ODEs) to achieve the research objective. The selection of $\rho$ for optimal stability properties in terms of zero stability, absolute stability, error constant and convergence are discussed. In the diagonally implicit formula that uses a lower triangular matrix with identical diagonal entries, allowing a maximum of one lower-upper (LU) decomposition per integration stage to be performed will result in substantial computing benefits to the user. The numerical results and plots of accuracy indicate that the $\rho$ -DIBBDF method performs better than the existing fully and diagonally Block Backward Differentiation Formula (BBDF) methods.

The influences of feedback control and internal heat source on the onset of Rayleigh–Bénard convection in a horizontal nanofluid layer is studied analytically due to Soret and Dufour parameters. The confining boundaries of the nanofluid layer (bottom boundary–top boundary) are assumed to be free–free, rigid–free, and rigid–rigid, with a source of heat from below. Linear stability theory is applied, and the eigenvalue solution is obtained numerically using the Galerkin technique. Focusing on the stationary convection, it is shown that there is a positive thermal resistance in the presence of feedback control on the onset of double-diffusive convection, while there is a positive thermal efficiency in the existence of internal heat generation. The possibilities of suppress or augment of the Rayleigh–Bénard convection in a nanofluid layer are also discussed in detail.

In this paper, we analyze the criteria for the stability of a method suited to the ordinary differential equations models. The relevant proof that the method satisfies the condition of stiff stability is also provided. The aim of this paper is therefore to construct an efficient two-point block method based on backward differentiation formula which is A-stable and converged. The new diagonally implicit scheme is formulated to approximate the solution of the pharmacokinetics models. By implementing the algorithm, the numerical solution to the models is compared with a few existing methods and established stiff solvers. It yields significant advantages when the diagonally implicit method with a lower triangular matrix and identical diagonal elements is considered. The formula is designed in such a way that it permits a maximum of one LU decomposition for each integration stage.