A comparative study of numerical methods for singularly perturbed boundary value problems

‎Singularly perturbed boundary value problems (SPBVPs) plays an important role in modeling various phenomena in engineering and other science fields‎. ‎The aim of this article is to compare the performance of numerical methods applied to solve the SPBVPs‎, ‎which are challenging due to the presence of small‎ ‎perturbation parameters leading to boundary or interior layers‎. ‎Four numerical methods are compared‎, ‎namely‎, ‎the local radial basis function method‎, ‎the‎ ‎improved Talbot's method‎, ‎the Euler's method‎, ‎and the Weeks method‎. ‎The main objective of this work is to improve the stability and accuracy of solutions for‎ ‎SPBVPs‎, ‎which are important for many applications in science and engineering‎. ‎We consider two-point SPBVPs with a boundary layer‎ ‎at one endpoint‎. ‎To evaluate the performance and effectiveness of the presented numerical techniques‎, ‎numerical approximations of four SPBVPs are derived and‎ ‎compared with the analytical solutions‎. ‎The results presented in the tables and figures validate that the proposed numerical schemes are highly accurate and‎ ‎clearly outperform the outcomes of other methods‎. ‎Furthermore‎, ‎results from the functional analysis were used to study the existence of the solution to the considered‎ ‎model and generate sufficient requirements for Ulam-Hyers stability‎.