Splines and Special Functions to Solve Boundary Value Problems in Differential Equations

Abstract

Professional applications in engineering and physics and applied sciences require Boundary value problems (BVPs) for their mathematical modeling. The traditional solution methods struggle to handle nonlinear BVPs because stability issues and accuracy limits prevent them from obtaining satisfactory results. The research explores spline-based numerical methods that use special function approximations to achieve efficient solutions of nonlinear BVPs. The combination of B-splines and high-degree splines with spectral special functions allows for building accurate smooth approximations that preserve computational stability. The performance metrics of different variational formulations and Galerkin methods and hybrid spline-special function approaches get tested through evaluation. The validation tests through computation reveal that using splines as a solver produces solutions more rapidly than conventional simulation algorithms do. Numerical solvers with spline bases prove effective for solving complex differential equations which enables crucial improvements to emerge in engineering simulation as well as scientific computing applications