New Computational Methods Using Seventh Derivative Type for the Solution of First Order Initial Value Problems

Abstract

In this research, a class of implicit block methods of a seventh derivative type are examined through interpolation and collocation techniques using finite power series as the basis function. The discrete schemes, which are implicit two-point block methods, are obtained by carefully and unevenly choose collocation points that ensure better methods’ stability via test. However, these schemes require seventh derivative functions unlike other existing numerical formulae. The new methods are found, investigated and proven to be convergent and A-stable. The implementation of methods is achieved by using Newton Raphson’s method. Experiments show the efficiency and accuracy of the developed formulae on different class of first-order initial value problems, including SIR, growth models and Prothero-Robinson oscillatory problem and with comparison to such existing methods. In addition, it is observed that uneven and positioning of collocation points greatly influence the efficiency and accuracy of numerical methods.