High Order Block Method for Third Order ODEs

Many initial value problems are difficult to be solved using ordinary, explicit step-by-step methods because most of these problems are considered stiff. Certain implicit methods, however, are capable of solving stiff ordinary differential equations (ODEs) usually found in most applied problems. This study aims to develop a new numerical method, namely the high order variable step variable order block backward differentiation formula (VSVO-HOBBDF) for the main purpose of approximating the solutions of third order ODEs. The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code. The order of the proposed method was then discussed in detail. The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively. In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB, particularly ode15s and ode23s, a numerical experiment was conducted on a set of stiff problems. The numerical results prove that for this particular set of problem, the use of the proposed method is more efficient than the comparable methods. VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs.