In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.

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For the parabolic problems in an infinite space, previous methods basically focused on the one‐ and two‐dimensional artificial boundary. Here, a high‐order local absorbing boundary condition (ABC) used for the fluid seepage and heat transfer in unbounded one‐ and two‐dimensional domains is extended to the relative three‐dimensional analysis. The local ABCs are first derived for the problem in an isotropic media and then stretched to the case in an orthotropic media. The function including time‐related variables in Laplace‐Fourier space is approximated through the Gauss‐Legendre quadrature formula. By using the inverse Laplace‐Fourier transformation, the local ABCs in Laplace‐Fourier space are inverted into the ones in time space. The numerical examples indicate that the local ABCs can provide satisfactory results with high computational efficiency, especially for the long‐term analysis. Moreover, the relationship among the diffusion coefficient, maximum simulation time and approximation order value is also investigated.