An Approach to Volterra and Fredholm Integro-Differential Equations via Six Non-Polynomial Cubic Spline functions
DOI:
https://doi.org/10.56714/bjrs.52.1.3Keywords:
Non-polynomial Cubic Splines, Collocation Method, Convergence Analysis, Numerical Approximation, Integro-differential equations.Abstract
This study introduces an effective numerical method for approximating solutions to Volterra and Fredholm integro-differential equations through a novel general formulation of non-polynomial cubic spline functions (NCSF). The proposed method enhances spline techniques by integrating trigonometric and exponential components with six non-polynomial cubic spline function formulations. A comprehensive convergence analysis occurs, establishing sufficient conditions for the stability and error bounds of the method. Multiple test problems are addressed, including Volterra and Fredholm integro-differential equations. The resulting numerical outcomes are compared with exact solutions and methodologies from the literature, presented in tables and figures. Comparisons illustrate the efficacy and robustness of the proposed six non-polynomial cubic spline method in addressing Volterra and Fredholm problems while obtaining higher precision with fewer grid points, as demonstrated by computational reports generated using Python 3.12.4
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