High-Accuracy Numerical Approximations to Several Singularly Perturbed Problems and Singular Integral Equations by Enriched Spectral Galerkin Methods (Journal of Mathematical Study, 2020)
Usual spectral methods are not effective for singularly perturbed problemsand singular integral equations due to the boundary layer functions or weakly singular solutions. To overcome this difficulty, the enriched spectral-Galerkin methods(ESG) are applied to deal with a class of singularly perturbed problems and singularintegral equations for which the form of leading singular solutions can be determined.In particular, for easily understanding the technique of ESG, the detail of the processare provided in solving singularly perturbed problems. Ample numerical examplesverify the efficiency and accuracy of the enriched spectral Galerkin methods.
Keywords: Singularly perturbed problems; weakly singular integral equations; boundary layers; enriched spectral Galerkin methods; Jacobi polynomials.