Abstract
In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary differential equations using collocation and interpolation approach.To derive this method, the approximate solution power series is interpolated at the first and off-step points and its second and third derivatives are collocated at all points in the given interval.Besides having good numerical method properties, the new developed method is also superior to the existing methods in terms of accuracy when solving the same problems.
Highlights
Introduction[16] introduced second derivative methods which are special types of hybrid methods (referred by [14] as Obrechkoff methods) to enhance the accuracy of the approximation which shown to reach an order k + 2
This article proposes a general one-step third derivative implicit hybrid block method (GOHBM) for the direct solution of the second order ODEs in the form y = f (x, y, y ), y(a) = y0, y0(a) = y0, a x b (1)with the assumption that f is differentiable and satisfies Lipchitz’s condition which guarantees the existence and uniqueness of the solution ([10]).Block methods which are widely used by many scholars for solving (1) were first introduced by [14] and later by [9] mainly to provide starting values for predictor-corrector algorithms
Hybrid block methods were used to circumvent Dahlquists barrier conditions which stipulate that the order of a k-step Linear Multistep Method (LMM) cannot exceed k + 1 for k is odd or k + 2 for k is even for the method to be zero-stable ([6])
Summary
[16] introduced second derivative methods which are special types of hybrid methods (referred by [14] as Obrechkoff methods) to enhance the accuracy of the approximation which shown to reach an order k + 2 Some scholars such as [5], [11] proposed a Simpson’s-type second derivative method for the solution of stiff system of first order IVPs. Their work motivated us to propose a new generalized one step third derivative implicit hybrid block method for solving second order ODEs directly using interpolation and collocation in the form αityn+it = h2[ βitfn+it +β1fn+1]+h3[ γitgn+it +γ1gn+1], x ∈ [xn, xn+1].
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