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首页> 外文期刊>Mathematical Problems in Engineering >Reproducing Kernel Algorithm for the Analytical-Numerical Solutions of Nonlinear Systems of Singular Periodic Boundary Value Problems
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Reproducing Kernel Algorithm for the Analytical-Numerical Solutions of Nonlinear Systems of Singular Periodic Boundary Value Problems

机译：奇异周期边值问题非线性系统的解析数值解的再生核算法

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摘要

The reproducing kernel algorithm is described in order to obtain the efficient analytical-numerical solutions to nonlinear systems of two point, second-order periodic boundary value problems with finitely many singularities. The analytical-numerical solutions are obtained in the form of an infinite convergent series for appropriate periodic boundary conditions in the space W-2(3)[0, 1], whilst two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed approach. The main characteristic feature of the utilized algorithm is that the global approximation can be established on the whole solution domain, in contrast with other numerical methods like onestep and multistep methods, and the convergence is uniform. Two numerical experiments are carried out to verify the mathematical results, whereas the theoretical statements for the solutions are supported by the results of numerical experiments. Our results reveal that the present algorithm is a very effective and straightforward way of formulating the analytical-numerical solutions for such nonlinear periodic singular systems.
机译：描述了再现核算法，以便获得具有有限多个奇点的两点，二阶周期边值问题的非线性系统的有效解析数值解。对于空间W-2（3）[0，1]中的适当周期性边界条件，以无限收敛序列的形式获得了解析数值解，同时在算法的整个演化过程中使用了两个平滑的再生核函数，以获得所需的节点值。提供了一种有效的计算算法来保证该过程并确认所提出方法的性能。与单步法和多步法等其他数值方法相比，所用算法的主要特征是可以在整个解域上建立全局逼近，并且收敛是均匀的。进行了两个数值实验以验证数学结果，而数值实验的结果则为该解决方案提供了理论依据。我们的结果表明，本算法是为此类非线性周期奇异系统建立解析数值解的非常有效且直接的方法。

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