Fifth order 2−point implicit block method with an off-stage function for solving first order stiff initial value problems

Numerical solution schemes are often referred to as being explicit or implicit.However, implicit numerical methods are more accurate than explicit for the same number of back values in solving still Initial Value Problems (IVPs). Hence, one of the most suitable methods for solving still IVPs is the Backward Differentiation Formula (BDF). In this thesis, a new two point implicit block method with an X stage function (2P4BBDF) for solving first order still Ordinary Differential Equations (ODEs) is developed. This method computes the approximate solutions at two points simultaneously based on equidistant block method. The proposed new formula is different from previous studies because it has the advantage of generating a set of formulas by varying a value of the parameters within the interval (-1,1).In this thesis we use 1/2 and 1/4 as the parameter. The stability analysis for the method derived namely; + = 1/2 and + = 1/4 show that the method is almost A-stable. Numerical results are given to compare the competitiveness of the new method with an existing method. The new method is compared numerically with a fifth order 3 point BBDF method by Ibrahim et al (2007). It is seen that the new method is marginally better than the 3 point BBDF in terms of accuracy and computational times. We also investigate the convergence and order properties of the 2P4BBDF method. The zero stability and consistency which are necessary conditions for convergence of the BBDF method are established. The algorithm for implementing the method will also developed.

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