In this paper, we conduct a numerical study of the Sturm–Liouville eigenvalue problem using the \(\operatorname {sinc}\) -Galerkin method. We provide a comprehensive discussion of the fundamental properties of \(\operatorname {sinc}\) methodology and its advantages. The solution is expressed as a finite expansion in terms of composite translated \(\operatorname {sinc}\) functions, whose coefficients are subsequently determined. We started by expressing the original Sturm–Liouville problem in a bilinear form relative to a predefined basis, namely the \(\operatorname {sinc}\) basis, with corresponding coefficients that can be determined. We then represented this bilinear form using a set of suitable integrals. Finally, conformal mapping and its inverse were systematically evaluated at \(\operatorname {sinc}\) grid points by employing the \(\operatorname {sinc}\) quadrature rule. We find that the result in error converges exponentially to zero. Three test cases with applications in the physical sciences were used to demonstrate the methodology. The findings demonstrate that the approach is highly effective, practical, and applicable to a wide range of issues. The study demonstrates that \(\operatorname {sinc}\) -Galerkin has an accuracy of order \(\mathcal {O}(\exp (-c \sqrt{N}))\) and rapidly converges to the exact solution.
