A two point block backward difference method is established to solve Riccati differential equations directly. Based on a predictor-corrector two point block backward difference method (2PBBD), a code is developed using a set of integration coefficients that eliminates the need to be calculated at every step change. The method requires calculating the integration coefficients only once in the beginning. The 2PBBD has an added advantage of a recurrence relationship between coefficients of different orders which provides a more elegant algorithm. The recurrence relationship between coefficients also reduces the computational cost. � 2016 Author(s).

The study of chaotic motion in periodic self-excited oscillators are an area of interest in science and engineering. In the current research, a numerical solution in backward difference form is proposed for solving these chaotic motions in periodic- self excited oscillators. Study conducted in this article focuses on chaotic motions in the form of Duffing-Van Der Pol Oscillators. A backward difference formulation in predictor-corrector (PeCe) mode is introduced for solving these Duffing-Van Der Pol directly. Numerical simulations provided will show the accuracy of the PeCe backward difference formulation.

This paper studied the order and stability of a 2-Point Block Backward Difference method (2PBBD) for solving systems of nonstiff higher order Ordinary Differential Equations (ODEs) directly. The method computes the estimated solutions at two points concurrently within an equidistant block. The integration coefficients that are used in the method are calculated only once at the beginning of the programming. The numerical results obtained compare the stability region and error growth rate of the proposed method with 2-Point Block Divided Difference method (2PBDD) and 1-Point Backward Difference method (1PBD). � 2018 Author(s).

A two-point block backward difference technique is established for solving nonlinear Riccati differential equations directly. The proposed method is coded using a variable order step size (VOS) algorithm. The advantage of the two-point block method is its programmability to implement parallel programming techniques. Combination of the block method and VOS algorithm allows for a significant reduction of computation cost in comparison to conventional methods. With an added advantage of the recursive relationship between integration coefficients of different orders, the proposed two-point block method provides efficient computation without loss of accuracy. � 2018 Author(s).

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy. � Published under licence by IOP Publishing Ltd.