Direct one-step block methods for solving third and fourth order ordinary differential equations

One-step block methods are presented in this research work to solve initial value problems (IVPs) of general third and fourth order ordinary differential equations (ODEs). These methods are utilized to solve general third and fourth order ODEs using constant step size. The methods will simultaneously obtain the approximation solutions at two and three points in a block. The general third order and fourth order ODEs are solved directly. Most of existing literatures have used IVPs to reduce problems in first order ODE systems. However, the approach in the current research is more efficient than the common technique involving first order equations. This research defines the order of the derived two-point and the three-point one-step block methods. In addition, the block method adopts Lagrange’s interpolation formulae to compute the integration coefficients. Notably, a new code is developed to solve the IVPs of third order and fourth order ODEs using constant step size. In the numerical results, the performance of the developed methods generated better results in terms of the total number of steps, maximum error, and total function calls compared with existing methods. In conclusion, the proposed direct one-step block methods in this thesis are appropriate for solving third and fourth order ODEs.

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