A New Shifted Chebyshev Galerkin Operational Matrix of Derivatives: Highly Accurate Method for a Nonlinear Singularly Perturbed Problem with an Integral Boundary Condition

Abstract The main goal of this paper is to come up with a new numerical algorithm for solving a first-order nonlinear singularly perturbed differential equation (SPDE) with an integral boundary condition (IBC). This paper builds a modified shifted Chebyshev polynomials of the first kind (CPFK) basis function that meets a homogeneous IBC, named IMSCPFK. It has established an operational matrix (OM) for derivatives of IMSCPFK. The numerical solutions are spectral, obtained by applying the spectral collocation method (SCM). This approach transforms the problem by their IBC into a set of algebraic equations, which may be resolved using any suitable numerical solver. Ultimately, we substantiate the proposed theoretical analysis by providing an example to verify the correctness, efficiency, and applicability of the developed method. We compare the acquired numerical findings with those derived from other methodologies. Tables and figures display the method’s highly accurate agreement with the residual error.