A numerical block hybrid algorithm for solving systems of first-order initial value problems

Attracted by the importance of ordinary differential equations in many physical situations like, engineering, business and health care in particular, an effective and successful numerical algorithm is needed in order to explain many of the ambiguities about the phenomena in many fields of human endeavor. In this study, an interpolation and collocation technique are adopted in deriving a Block Hybrid Algorithm (BHA) for the numerical solution of systems of first-order Initial Value Problems (IVPs). To derive the BHA, the shifted Legendre polynomials was interpolated at two selected points and its derivative was collocated at seven selected points. This led to a continuous scheme which was eventually evaluated at some points to obtain the discrete schemes used in the numerical computation. Furthermore, some illustrative examples are introduced to show the applicability and validity of the proposed algorithm. It was observed that the proposed algorithm has the desired rate of convergence to the exact solution. The suggested method utilizes data at points other than the step numbers which is viewed as an important landmark; another major advantage of this algorithm is that it possesses remarkably small error constants (Table 2). Some graphical representations of the exact and numerical results are presented to show how accurate the numerical results agree with the exact solutions.