Abstract
In the present paper, reproducing kernel method (RKM) is introduced, which is employed to solve singularly perturbed convection-diffusion parabolic problems (SPCDPPs). It is noteworthy to mention that regarding very serve singularities, there are regular boundary layers in SPCDPPs. On the other hand, getting a reliable approximate solution could be difficult due to the layer behavior of SPCDPPs. The strategy developed in our method is dividing the problem region into two regions, so that one of them would contain a boundary layer behavior. For more illustrations of the method, certain linear and nonlinear SPCDPP are solved.
Highlights
Let us consider the following singularly perturbed convection-diffusion parabolic problem, L(y(x, t)) + N (y(x, t)) = f (x, t),(x, t) ∈ D ≡ [0, 1] × [0, 1], y(0, t) = y(1, t) = 0, t ∈ [0, 1], y(x, 0) = y0(x), x ∈ [0, 1], (1.1)where Ly ≡ −ε∂x2y(x, t)+p(x, t)∂xy(x, t)+q(x, t)y(x, t)+∂ty(x, t) and 0 < ε 1, is perturbation parameter and p(x, t), q(x, t), f (x, t) are sufficiently smooth functions such that, p(x, t) ≥ α > 0, q(x, t) ≥ β ≥ 0, and N (y(x, t)) is nonlinear differential operator
Problem (1.1) with the above-mentioned conditions is of a unique solution y(x, t) with boundary layer behavior, in which the boundary layer width is
Since the reproducing kernel method (RKM) is a powerful numerical method, if these three steps are properly applied to the problem, this technique will be able to provide an appropriate approximation of the solution, even with severe singularities
Summary
Some applications of the RKM and singularly perturbed problems are introduced in [14, 16, 26, 28]. The strategy developed in order to solve these problems with layer behavior is explained in three steps. Since the RKM is a powerful numerical method, if these three steps are properly applied to the problem, this technique will be able to provide an appropriate approximation of the solution, even with severe singularities. The RKM does not provide an approximate solution for the right layer region, it is essential to shift region D2 to another region, such as D3 ≡ [−1, 0] × [0, 1]. Further details are provided in [34]
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