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Reproducing Kernel Method for Piecewise Smooth Boundary Value Problems

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Abstract:

In this paper, an efficient reproducing kernel method(RKM) is proposed for solving the piecewise smooth BVP. By defining an operator ${\cal L}:{W_{2}^{3}[0, 1]}\rightarrow {L_{2}[0, 1]}$ and applying the reproducing kernel property, we have skillfully solved the complex piecewise smooth BVP. The theorems show the accuracy of the algorithm. Furthermore, we get that the approximate solution $u_{n}(x)$ convergence to the exact one with order $O(h^2)$ . That is, $u_{n}(x)$ has second order convergence in the sense of norm $\left\|\displaystyle. \right\|_{W_2^{3}}$ . Finally, some numerical experiments are given to illustrate the algorithm is accuracy, simple and efficient.

Key words: Reproducing kernel method, Piecewise smooth, Convergence order

CLC Number:

O241

Zhihong Zhao,Yingzhen Lin. Reproducing Kernel Method for Piecewise Smooth Boundary Value Problems[J].Acta mathematica scientia,Series A, 2020, 40(5): 1333-1340.

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$n$	$\|u_{n}(x)-u(x)\|_{\max}^{[14]}$	$\|u_{n}(x)-u(x)\|_{\max}$	$\|u'_{n}(x)-u'(x)\|_{\max}$	$\|u''_{n}(x)-u''(x)\|_{\max}$
$10$	$4.0\times10^{-4}$	$5.877\times10^{-5}$	$2.867\times10^{-4}$	$1.263\times10^{-5}$
$100$	$4.0\times10^{-6}$	$5.894\times10^{-7}$	$2.868\times10^{-6}$	$1.260\times10^{-7}$
$1000$	$/$	$5.893\times10^{-9}$	$2.867\times10^{-8}$	$1.263\times10^{-9}$
阶	$/$	$2.000$	$2.000$	$2.000$

$n$	$\|u_{n}(x)-u(x)\|_{\max}$	$\|u'_{n}(x)-u'(x)\|_{\max}$	$\|u''_{n}(x)-u''(x)\|_{\max}$
$100$	$1.868\times10^{-6}$	$1.051\times10^{-5}$	$1.839\times10^{-5}$
$1000$	$1.868\times10^{-8}$	$1.051\times10^{-7}$	$1.839\times10^{-7}$
阶	$2.000$	$2.000$	$2.000$

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