Some Classes of Second Derivative Trigonometrically Fitted Block Schemes for the Numerical Integration of Oscillatory Initial Value Problems
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This study develops two classes of Second Derivative Trigonometrically-Fitted Block Schemes for the numerical integration of oscillatory IVPs using collocation techniques. The two classes of methods are the Second Derivative Trigonometrically-Fitted Block Backward Differentiation Formula (TFBBDF) and Second Derivative Trigonometrically-Fitted Block Scheme of Simpson Type (TFBSST). The Trigonometrically-Fitted Methods for each scheme depend on the step size and frequency that are constructed using trigonometric basis function. The continuous Second Derivative Trigonometrically-Fitted Method for each scheme is used to generate the main method. The additional 𝑘−1 complementary methods for TFBBDF are obtained from the second differentiation of its continuous form, while the complementary methods of TFBSST are obtained from the same continuous method as main method. The main and complimentary methods in their converted power series form are combined and applied in block form as simultaneous numerical integrators. The stability properties for both classes are investigated using boundary locus plot. It is found that both classes are zero stable, consistent and convergent. The class of 𝑘`-step TFBBDF is of order 2𝑘+1 while that of 𝑘`-step TFBSST of order 2𝑘+2. Both classes of the methods are applied on a number of numerical examples and the results showed that they are more accurate and more efficient for oscillatory problems when compared with existing methods in the literature reviewed in this work.
A Thesis Submitted to the School of Postgraduate Studies, University of Lagos
Trigonometrically-Fitted methods , Backward Differentiation Formula , Collocation technique , Oscillatory Problems
Abdulganiy, R.I (2017). Some Classes of Second Derivative Trigonometrically Fitted Block Schemes for the Numerical Integration of Oscillatory Initial Value Problems. A Thesis Submitted to University of Lagos School of Postgraduate Studies Phd Thesis, 227p.