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

New block Runge-Kutta method with various weights for solving stiff ordinary differential equations

This paper discusses the development of a new block Runge-Kutta (NBRK) method with various weights for solving stiff ordinary differential equations (ODEs). Performance of the proposed method in terms of maximum error and computational time is compared with third order RK (RK3) method and modified weighted RK3 method based on Centroidal Mean (MWRK3CeM). Comparison between weights used is also analyzed. Numerical results show that the proposed method outperformed the MWRK3CeM and the RK3 method

Weighted block Runge-Kutta method for solving stiff ordinary differential equations

In this paper, weighted block Runge-Kutta (WBRK) method is derived for solving stiff ordinary differential equations (ODEs). Implementation of weights on the method and its stability region are shown. Numerical results of the WBRK method are presented and compared with the existing methods to prove the ability of the proposed method to solve stiff ODEs. The results show that the WBRK method has better accuracy than the comparing methods

Singly Diagonally Implicit Block Backward Differentiation Formulas for HIV Infection of CD4+T Cells

In this study, a singly diagonally implicit block backward differentiation formula (SDIBBDF) was proposed to approximate solutions for a dynamical HIV infection model of CD 4 + T cells. A SDIBBDF method was developed to overcome difficulty when implementing the fully implicit method by deriving the proposed method in lower triangular form with equal diagonal coefficients. A comparative analysis between the proposed method, BBDF, classical Euler, fourth-order Runge-Kutta (RK4) method, and a Matlab solver was conducted. The numerical results proved that the SDIBBDF method was more efficient in solving the model than the methods to be compared

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

Weighted Block Runge-Kutta methods for solving stiff ordinary differential equations

Weighted Block Runge-Kutta (WBRK) methods are derived to solve first order stiff ordinary differential equations (ODEs). The proposed methods approximate solutions at two points concurrently in a block at each step. Three sets of weight are chosen and implemented to the WBRK methods. Stability regions of the WBRK methods with each set of weight are constructed by using MAPLE14. Stability properties of the proposed methods with each weight show that the methods are suitable for solving stiff ODEs. Numerical results are presented and illustrated in the form of efficiency curves. Performances of the WBRK methods in terms of maximum error and computational time are compared with the third order Runge-Kutta (RK3) and the modified weighted RK3 method based on centroidal mean (MWRK3CeM). These methods are tested with problems of single and system of first order stiff ODEs. Comparison of the proposed methods between sets of weight is also analyzed. The numerical results are obtained by using MATLAB R2011a. Numerical results generated show that the WBRK methods obtained better accuracy and less computational time than the RK3 and MWRK3CeM method

Singly diagonally implicit block backward differentiation formulas for HIV infection of CD4+T cells

In this study, a singly diagonally implicit block backward differentiation formula (SDIBBDF) was proposed to approximate solutions for a dynamical HIV infection model of CD 4+ T cells. A SDIBBDF method was developed to overcome difficulty when implementing the fully implicit method by deriving the proposed method in lower triangular form with equal diagonal coefficients. A comparative analysis between the proposed method, BBDF, classical Euler, fourth-order Runge-Kutta (RK4) method, and a Matlab solver was conducted. The numerical results proved that the SDIBBDF method was more efficient in solving the model than the methods to be compared

Stability analysis of singly diagonally implicit block backward differentiation formulas for stiff ordinary differential equations

In this research, a singly diagonally implicit block backward differentiation formulas (SDIBBDF) for solving stiff ordinary differential equations (ODEs) is proposed. The formula reduced a fully implicit method to lower triangular matrix with equal diagonal elements which will results in only one evaluation of the Jacobian and one LU decomposition for each time step. For the SDIBBDF method to have practical significance in solving stiff problems, its stability region must at least cover almost the whole of the negative half plane. Step size restriction of the proposed method have to be considered in order to ensure stability of the method in computing numerical results. Efficiency of the SDIBBDF method in solving stiff ODEs is justified when it managed to outperform the existing methods for both accuracy and computational time