Abstract
A novel rational spectral collocation method is presented combined with the singularity-separated technique for a system of singularly perturbed boundary value problems. The solution is expressed as u=w+v, where w is the solution of the corresponding auxiliary boundary value problem and v is a singular correction with explicit expressions. The rational spectral collocation method in barycentric form with the sinh transformation is applied to solve the auxiliary third boundary problem. The parameters of the singular correction can be determined by the boundary conditions of the original problem. Numerical experiments are carried out to support theoretical results and provide a favorable comparison with research results of other work.
Highlights
B0m)T layer behaviors can be examined in two different cases: Case 1
A conformal map which maps the collocation points clustered near the poles of [− 1, 1] into a new set of collocation points. e parameters of the mapping are determined by the position and width of the boundary layer
To weaken the singularity and improve the accuracy of numerical simulation, the singularity-separated technique (SST) for singular perturbation problem with constant coefficients was proposed by Chen and Yang [13], and finite element methods with SST were used to solve a singular perturbation problem with a single boundary layer
Summary
For the construction of the RSC-SSM, it is necessary to understand the properties of exact solution, especially the position and width of the boundary layer. Let A have an eigenvalue of λ 0, according to the Gerschgorin disc theorem, ∃i0, |ai0i0| ≤ Ri0(A), which is a contradiction. The above theorem suggests that the boundary layer regions of the solution u(x) to the reaction-diffusion problems are [0, x∗] and [1 − x∗, 1], that is, the boundary layers are located at the two endpoints of the u√n derly√in g interval [0, 1] and each of their width is x∗ − (ln ε /α) ε. To ensure that there is a unique solution to this problem, the following assumptions need to be satisfied: B ≠ 0 is strictly diagonally dominant and invertible and define β min(bii), i 1, . For the convection-diffusion problems, there is a boundary layer region [0, x∗] in the solution u of (12) and (2). For the convection-diffusion problems, there is a boundary layer region [0, x∗] in the solution u of (12) and (2). at is, the boundary layers are located at the left endpoint, and its width is x∗ − (ln ε/β)ε
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