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We have proposed a continuous third derivative formula CTDF which is used to produce main and additional methods.These methods are combined as a single block called Block Third Derivative Formulas (BTDFs) that are implemented as a boundary value methods to solve second order boundary value problems directly without first adapting the second order ordinary differential equation into an equivalent first order system.
where y, f [member of] [[Real part].sup.m], and the given boundary conditions are not a restriction on CTDFs. It has been shown in Keller  that the general conditions which ensure that the solution of (1) subject to boundary conditions exist and is unique.
In section two, we derive an approximation U (x) for y (x) which is used to obtain the main and additional methods from the CTDFs. The convergence analysis of the methods are given in Section 2.
Therefore, the CTDFs generate eighth-order and a Tenth-order convergent methods respectively.
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