Highly Accurate Method for Poisson’s and the Helmholtz’s Equations Subject to Robin Boundary Conditions

This paper presents three novel algorithms for solving numerically Poisson’s and Helmholtz’s Eqs subject to RBCs (Robin boundary conditions). Basis functions in terms of GSCPFKs (Generalized shifted Chebyshev polynomials of the first kind) and GSLPs (Generalized shifted Legendre polynomials), named RMCPFKs (Robin-Modified Chebyshev polynomials of the first kind) and RMLPs (Robin-Modified Legendre polynomials), that satisfy four homogeneous RBCs are introduced. It has utilized OMs (Operational matrices) for derivatives of RMCPFKs and RMLPs. Applying the spectral collocation method (SCM) yields solutions exhibiting spectral accuracy. This technique effectively transforms the original problem, defined with RBCs, into a system of linear algebraic equations. We establish the theoretical validity through rigorous convergence analysis and error estimates for the proposed algorithms. To demonstrate practical performance, three numerical examples are presented, confirming the accuracy, efficiency, and applicability of our approach. These numerical results are benchmarked against ExaSs and outcomes from other methods found in the literature. Our algorithms yield significant results, demonstrating exceptional agreement with ExaSs, as presented in tables and figures. The novelty of this work lies in the construction of basis functions RMCPFKs and RMLPs and their OMs that are integrated alongside SCM, advancing beyond previous approaches in the literature that do not leverage these basis functions. This study not only enhances the accuracy and efficiency of NUMSs for these Eqs but also broadens the applicability of spectral methods in solving complex boundary value problems.