{'fraction_non_alphanumeric': 0.02738336713995943, 'fraction_numerical': 0.006085192697768763, 'mean_word_length': 4.875, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Derivation of an Implicit 4-Point Block Runge-Kutta for Direct Integration of Third Order IVPs and BVPs in ODEs using the Quade’s Type Multistep Method

In this paper, we extend the four-step implicit Runge-Kutta method for direct integration of second order initial value ODEs which is an extension of the Quade’s Type Multistep (QTM) method (four step block hybrid to an Implicit 4-Point Block Runge-Kutta for Direct Integration Of Third Order initial value problems (IVPs) and boundary value problems (BVPs) in ODEs via the idea as those invented by Nyström. The theory of Nyström method was adopted in the derivation of the method. The method has an implicit structure for efficient implementation and produces simultaneously approximation of the solution of IVPs and BVPs at a block of four points i n x  (i=1,2,3,4). The proposed method was tested with Numerical experiment to illustrate its efficiency and the method can be extended to solve higher order differential equations.