SOLUTION OF A CLASS OF FOURTH ORDER SINGULAR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS BY HYBRID QUINTIC B-SPLINE METHOD
DOI:
https://doi.org/10.71146/kjmr757Keywords:
Singular boundary layers, Hybrid quintic B-spline, Fourth-order differential operators, Quasi linearization, L'Hôpital regularization, Parameter-uniform convergenceAbstract
The fourth-order differential equations with singular spatial coefficients and small perturbation parameters are highly challenging from computational point of view because of the existence of steep boundary layers and the coordinate indeterminacy near x = 0. This work constructs an adaptive hybrid quintic B-spline (HQBS) collocation algorithm designed to tackle this difficulty. By using L'Hôpital's rule at x = 0, the singular derivative term becomes well conditioned. The use of a sequence of linear problems generated by quasi linearization using the Newton-Raphson method provides an efficient way to treat non-linear source terms. The hybrid basis is a combination of splines of degree 5 and non-polynomial tension terms which eliminate unwanted numerical oscillations in the sharp boundary layers. The rigorous error analyses and numerical tests for three standard test problems – both with the single boundary layer zone and the dual boundary layer zone – show that HQBS scheme has parameter uniform accuracy up to the perturbation parameters ε = 10⁻⁸.
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Copyright (c) 2025 Tayyba Arif, Fakhar Waqqas, Mehreen Zahid, Ayesha Tanveer (Author)
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