Direct Integration Block Method for Solving Higher Order Ordinary Differential Equations - Azmi, Nurul Asyikin (2010)
In this thesis, the implicit block methods presented as in the simple form of Adams Moulton method are developed for solving higher order systems of Ordinary Differential Equations (ODEs). This method will solve the Initial Value Problems (IVPs) of second and third order ODEs using variable step size (VS) and variable step or variable order (VSVO) techniques. The proposed block methods will approximate the solutions at two distinct points on the x-axis simultaneously in a block. A system of higher order can also be reduced to a system of first order equations and then solved using any numerical method. This approach is very well established but it obviously will enlarge the dimension of the equations. However, the developed block method will solve the system of higher order ODEs directly without reducing it to first order. The formulae of the block method involve Lagrange’s interpolation formulae in order to compute the integration coefficients. All the coefficients will be stored in the code and there will be no computation involved for the integration coefficients. The codes were executed on UNIX operating system and the algorithms were written in C language. The numerical results showed that the performance of the developed methods gave better results in terms of total number of steps, maximum error, and total function calls compared to the existing block methods. In conclusion, the proposed implicit block methods in this thesis are appropriate for solving the second and third order ODEs.