Extension of hybrid block methods for solving partial differential equations

Abstract

The thesis comprehensively explores hybrid block methods for solving PDEs, traversing through diverse numerical methodologies to develop innovative solutions suited for complex scientific and engineering PDEs. newlineChapter 2 presents an optimized hybrid block method adept at solving stiff systems of first-order initial value problems, demonstrating stability and convergence properties with both fixed and adaptive step-size approaches through rigorous analysis and numerical experiments. newlineIn Chapter 3, the hybrid block method is extended to tackle nonlinear PDEs by combining it with a modified cubic B-spline collocation method. This chapter confronts the trade-off between time step-size optimization and accuracy, striving to balance computational efficiency with solution accuracy. newlineChapter 4 presents a novel approach for solving the Kuramoto-Sivashinsky equation, leveraging a fusion of differential quadrature method (DQM) and reformulated hybrid block techniques to yield accurate solutions with reduced computational effort. newlineChapter 5 investigates the rate of convergence of the hybrid block method when applied to PDE. From the mixed derivative type Hunter-Saxton equation to the formidable Camassa-Holm and Degasperis- Procesi equations, each challenge is met with a tailored solution grounded in rigorous analysis and computational prowess. The subsequent chapters continue this trajectory of innovation and refinement of the algorithm using unified extended splines as the basis of DQM in chapter 6. newlineIn conclusion, the thesis makes a significant contribution to advancing numerical methods for solving PDEs. By expanding hybrid block methods and tackling complex challenges, it fosters innovation, benefiting diverse scientific and engineering fields. newline newline