Hybrid Block Methods with Constructed Orthogonal Basis for Solution of Third-Order Ordinary Differential Equations

In this work, an orthogonal polynomial with weight function w(x) =x2 + x + 1 in the interval [-1, 1] was constructed and used as the basis function to develop block methods, using collocation and interpolation approach. An efficient class of continuous and discrete numerical integration schemes of implicit hybrid form for third-order problems were developed and successfully implemented. Three different problems were solved with these schemes and they performed favourably. The investigation, using the appropriate existing theorems, shows that the methods are consistent, zero-stable and hence, convergent.