Family of singly diagonally implicit block backward differentiation formulas for solving stiff ordinary differential equations

A new family of singly diagonally implicit block backward differentiation formulas (SDIBBDF) for solving first and second order stiff ordinary differential equations (ODEs) are developed. Motivation in developing the SDIBBDF method arises from the singly diagonally implicit properties that are widely used by researchers in Runge-Kutta (RK) families to improve efficiency of the classical methods. The strategy is to reduce a fully implicit method to lower triangular matrix with equal diagonal elements. In order to achieve a particular order of accuracy, error norm minimization strategy is implemented based on the error constant of the formulas. Although the derived methods have proven to solve stiff ODEs efficiently, the extended SDIBBDF (ESDIBBDF) methods are introduced by adding extra function evaluation to further improve accuracy. As some of the applied problems available in the literature are modeled as second order ODEs thus, 2ESDIBBDF method is constructed to meet the requirement. Numerical algorithm of the method is designed to solve the second order stiff ODEs directly. Subsequently, the constant step size methods are implemented with the variable step size scheme. The scheme is proposed to optimize the total steps taken by the methods to approximate solutions which later displays a better performance in solving the problems. Necessary conditions for convergence are studied to ensure that the derived methods are able to approximate solution of a differential equation to any required accuracy. Since absolute stability is a crucial characteristic for a method to be useful therefore, stability graphs of the methods derived are constructed by MAPLE programming. The stability properties of the methods are discussed to justify their ability for solving stiff problems. Performance of the methods are verified from the numerical results executed via the C++ programming by comparing them with existing methods of the same nature. Finally, the applications of developed methods in the field of applied sciences, life sciences and engineering are presented. From the numerical experiments conducted, it can be concluded that the proposed methods can serve as an alternative solver for solving stiff ODEs of first and second order directly, and applied problems.

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