Two-point block backward differentiation formula for solving higher order ordinary differential equations - Zainuddin, Nooraini (2011)
This thesis focuses on solving higher order Ordinary Differential Equations (ODEs) directly using the Block Backward Differentiation Formula (BBDF) method. The BBDF method approximates the solution at two points concurrently. Implementation of this method is done by using equidistant stepsize on the set of stiff problems. The first part of the thesis gives the derivation of the BBDF method for solving second order and third order stiff ODEs directly. The algorithms are written in C language and the numerical results of these methods are compared to that of reducing it to a system of first order ODEs and solves using the first order ODEs method. The subsequent part of the thesis discusses in detail the stability properties of the BBDF method which are given in the previous part. The stability properties justify the efficiency of the BBDF method as used in solving stiff problems. The illustrations of the stability region are provided. Finally, this thesis zooms into the implementation of the BBDF method using the variable order algorithm for the solution of second order stiff ODEs directly. The variable order strategies for the BBDF method is elaborated and the numerical result of the variable order BBDF method is compared with the variable order method which is available in MATLAB. In conclusion, the results show that BBDF method reduces the total number of steps and the time execution when compared to the nonblock first order ODEs method. Therefore, these new methods present significant alternatives for solving higher order ODEs directly.