Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equations

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

Computational Treatment of First Order Delay Differential Equations Using Hybrid Extended Second Derivative Block Backward Differentiation Formulae

This paper considers the computational solution of first order delay differential equations (DDEs) using hybrid extended second derivative backward differentiation formulae method in block form without the implementation of interpolation techniques in estimating the delay term. By matrix inversion approach, the discrete schemes were obtained through the linear multistep collocation approach from the continuous form of each step number which after implementation strongly revealed the convergence and region of absolute stability of the proposed method. Computational results are presented and compared to the exact solutions and other existing method to demonstrate its efficiency and accuracy.

Improved Two-Point Block Backward Differentiation Formulae for Solving First Order Stiff Initial Value Problems of Ordinary Differential Equations

Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.

Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae

Variable order block backward differentiation formulae (VOHOBBDF) method is employedfor treating numerically higher order Ordinary Differential Equations (ODEs). In this respect, the purpose of this research is to treat initial value problem (IVP) of higher order stiff ODEs directly. BBDF method is symmetrical to BDF method but it has the advantage of producing more than one solutions simultaneously. Order three, four, and five of VOHOBBDF are developed and implemented as a single code by applying adaptive order approach to enhance the computational efficiency. This approach enables the selection of the least computed LTE among the three orders of VOHOBBDF and switch the code to the method that produces the least LTE for the next step. A few numerical experiments on the focused problem were performed to investigate the numerical efficiency of implementing VOHOBBDF methods in a single code. The analysis of the experimental results reveals the numerical efficiency of this approach as it yielded better performances with less computational effort when compared with built-in stiff Matlab codes. The superior performances demonstrated by the application of adaptive orders selection in a single code thus indicate its reliability as a direct solver for higher order stiff ODEs.