A CLASS OF BLOCK MULTISTEP METHODS FOR SOLVING GENERAL THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS
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The numerical solutions of general third order initial value problems of ordinary di�erential
equations have been studied in this research work. A new class of block multistep methods
capable of solving general third order IVPs of ODEs using variable step size technique have
been developed. Collocation and interpolation of power series as the approximate solution is
adopted. The block multistep method was intensi�ed by the introduction of continuous scheme
in order to circumvent the limitation created by reducing to systems of �rst order ODEs.
The new class of variable step-size method has the advantage to control and minimize error,
determine and vary the step size as well as decide the prescribed tolerance level to ascertain
the maximum errors. Some theoretical properties of the block multistep methods such as order
of the scheme, zero stability, consistency and determination of the region of absolute stability
of the scheme have been conducted and presented. Numerical examples on nonsti� IVPs have
been used to test the performance of the methods, in addition, comparing the maximum error
as the prescribed tolerance parameter level is reduced in the method. The newly developed
methods have been written as mathematical program and expressed in form of mathematical
language which can run simultaneously when implemented. The newly formulated variable
step-size block multistep methods perform better when compared with other existing methods
as the prescribed tolerance parameter level got smaller and smaller.
Furthermore, the newly developed methods possess the attribute to control and decide on
the estimate of the actual step size that will guarantee an improved results with better maximum
errors. This, in particular, is seen as an advantage of the variable step size method over other
existing methods approximated with �xed step size. Finally, the idea of predictor-corrector
methods used by various researchers to predict and correct estimates has been extended in
the newly proposed method to change/decide on suitable step size, determine the prescribed tolerance level and error control/minimization.
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QA Mathematics