IJFANS International Journal of Food and Nutritional Sciences

ISSN PRINT 2319 1775 Online 2320-7876

REVIEW OF SPLINE METHODS FOR SOLVING SINGULAR PERTURBATION BOUNDARY VALUE PROBLEMS

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Prajendra Singh Raghuvanshi, Dr Shoyeb Ali Sayyed

Abstract

Spline methods have gained prominence in recent years for their efficacy in solving singular perturbation boundary value problems (SPBVPs). These problems, characterized by a boundary layer or a small parameter multiplying a derivative term, are challenging due to rapid variations in solutions near the boundary. Spline methods, particularly piecewise polynomial interpolations, offer a robust framework for approximating solutions to SPBVPs with high accuracy and computational efficiency. This review synthesizes current literature on spline methods applied to SPBVPs, emphasizing their ability to capture boundary layer phenomena and resolve steep gradients effectively. Key advantages of spline methods include their flexibility in handling irregular domains and their ability to accommodate higher-order accuracy through adaptive refinement strategies. By leveraging local interpolation properties and global continuity constraints, splines mitigate numerical artifacts and ensure stable solutions across varying spatial scales. Challenges such as optimal knot placement, convergence analysis, and computational cost are also discussed, highlighting ongoing research efforts to enhance spline methodologies' applicability to complex SPBVPs. Future directions focus on integrating spline methods with adaptive mesh refinement techniques and exploring hybrid approaches that combine spline interpolations with other numerical schemes for improved accuracy and efficiency.

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