In this paper, we construct a family of high order self-starting one-block numerical methods for the solution of stiff initial value problems (IVP) in ordinary differential equations (ODE). The Reversed Adams Moulton (RAM) methods, Generalized Backward Differentiation Formulas (GBDF) and Backward Differentiation Formulas (BDF) are used in the constructions. The E-transformation is applied to the triples and a family of self-starting methods are obtained. The family is for . The numerical implementation of the methods on some stiff initial value problems are reported to show the effectiveness of the methods. The computational rate of convergence tends to the theoretical order as h tends to zero.

This paper deals with the construction of l-stable implicit one-block methods for the solution of stiff initial value problems. The constructions are done using three different multi-block methods. The first multi-block method is composed using Generalized Backward Differentiation Formula (GBDF) and Backward Differentiation Formula (BDF), the second is composed using Reversed Generalized Adams Moulton (RGAM) and Generalized Adams Moulton (GAM) while the third is composed using Reversed Adams Moulton (RAM) and Adams Moulton (AM). Shift operator is then applied to the combination of the three multi-block methods in such a manner that the resultant block is a one-block method and self-starting. These one-block methods are up to order six and  with at order ten. Numerical experiments show that they are good for solving stiff initial problems.

This work presents an efficient procedure based on Chebychev spectral collocation method for computing the 2D Laplace’s equation on a rectangular domain. The numerical results and comparison of finite difference and finite element methods are presented. We obtained a satisfactory result when compared with other numerical solutions.