Abstract
In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective.
Highlights
Perturbed problems SPPs arise frequently in applications including geophysical fluid dynamics, oceanic and atmospheric circulation, chemical reactions, and optimal control
The inner product is given by u x, v x W23 u av a u av a b a u xv x dx
Using Iterative Reproducing Kernel Method (IRKM), we can get the solution of the outer region problem
Summary
Perturbed problems SPPs arise frequently in applications including geophysical fluid dynamics, oceanic and atmospheric circulation, chemical reactions, and optimal control. I We define the inner product space W21 a, b {u | u is one-variable absolutely continuous function, u ∈ L2 a, b }. W21 a, b is a reproducing kernel space, and its reproducing kernel is R{x1} y . In order to solve 1.1 , we first give the analytical and approximate solutions of the following operator equation: Lu x F x, u x , 2.1 where L : H a, b → H1 a, b is a bounded linear operator and L−1 is existent. We define an approximate solution ni un x βikF xk ψi x. Let εn[2] u x − un x 2; the sequence of real numbers εn is monotonously decreasing and εn → 0 and the sequence un x is convergent uniformly to u x , k 0, 1, 2. A u − A v ≤ λ u − v , λ < 1; 2.8 un,∗ x is convergent
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