关于若干非线性问题算法的研究
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摘要
许多自然现象都是借助于线性、非线性方程来描述的,这些方程作为重要的数学模型在物理学、生物学、控制科学等很多研究领域中有着广泛的应用,而对这些现象的分析一般可归结为微分方程的求解问题,由于获得该类方程精确解的解析表达式是非常困难的,所以发展适用的数值方法就成为既有理论意义又有实际价值的研究课题。因此如何求解这些有实际意义的方程也就变得越来越重要了。
本文运用再生核空间的技巧,给出了几类线性和非线性微分方程的求解算法。
文中详细地叙述了再生核空间的应用背景和研究历史,回顾了再生核空间发展状况。并且在文中的每一部分都给出了若干具体的再生核空间。在所有给出的再生核空间中都有具体的再生核函数表达式,并且每一章都进行了相应的数值试验。这些数值试验验证了理论上推出的结论的正确性。
首先,通过改进原有再生核空间的内积定义,我们简化了再生核函数的表达式,并且通过再生核表达式得到了一组标准正交基,在这组基上进行了Fourier级数展开,从而在第二章中得到了五阶线性方程的解。同时,在第三章又构造了一种适合在计算机上实施的逼近序列来求解奇异线性问题,它的结构简单,对于逼近节点数目很大的离散函数颇为有效。逼近序列的误差在Sobolev范数意义下是递减的并且能保证逼近过程的一致收敛性。
其次,在再生核空间中构造了一种收敛的迭代序列,通过截断级数的形式得到了方程的近似解,同时若方程解存在不唯一的情况下,可以进一步求出满足一定附加条件的特解。这种迭代法适用于一般的非线性方程,本文运用此方法在第四章和第五章分别对奇异非线性方程组和非线性无限比例延迟方程进行了求解。
最后,在第六章中求解二阶非线性偏微分方程。主要利用再生核空间中的再生性质,将其转化为线性算子方程进行求解。将边界条件齐次化后融入到二维再生核空间中,运用本文的方法可以求得一个带有未知量的解的表达式,然后通过最小二乘法的技巧,最终获得非线性算子方程的解。
本文所提的求解算法在理论上和实际试验模拟来看有以下优点:算法简单,通过数值模拟我们也验证了每个模型的各阶导数之间的逼近效果非常好,而且不同于以往的数值算法,本文算法是连续逼近,即对空间任意点均可以算。
Many natural phenomena can be modeled by linear and nonlinear differential equations. These equations as mathematical models have important applications in physics, biology, control science and so on. In general, the analysis of these phenomena can reduce to solving differential equations. It is difficult to obtain the analytical representation of exact solutions, therefore, the research of an efficient numerical method for differential equations is of theoretical and applicable significance and how to solve these significant equations becomes more and more important.
In this thesis, several numerical methods of solving some classes of linear and nonlinear differential equations are presented by using reproducing kernel theory.
This thesis introduces application background and history of reproducing kernel spaces and runs back over the development of reproducing kernel spaces. Moreover, we give the concrete representation of the reproducing kernel spaces in which there are the corresponding reproducing kernel functions in the every chapter. Some numerical tests are given in every chapter and numerical results verify the validity of conclusion.
Firstly, the representation of reproducing kernel is simplied by improving definition of inner product in the original reproducing kernel spaces. The orthonormal basis can be obtained from the representation of reproducing kernel functions. Through Fourier series expanding on this basis, the solutions of fifth-order linear equations can be obtained in Chapter 2. In the mean time, an approximate sequence that is implemented easily on computers is constructed to solve singular linear problems in Chapter 3. Its construction is simple and it can approximate effectively the function with large number of nodes. The error of approximate sequence is decreasing in the sense of Sobolev norm and the uniform convergence can be guaranteed.
Next, a convergent iterative sequence is constructed. The approximate solutions can be obtained by truncating series. If the solutions of equations are not unique, the particular solutions satisfying addictive conditions can be given. This iterative method is suitable for solving general nonlinear equations. Using this method, we solve singular nonlinear systems and nonlinear infinite-delay-differential with proportion delay respectively in Chapter 4 and Chapter 5.
Finally, second order nonlinear partial differential equations are solved in Chapter 6. In terms of the reproducing property of reproducing kernel spaces, we convert them to operator equations .After homogenizing the boundary conditions , we put them into two-dimensional reproducing kernel spaces. By using this method in this thesis, we can obtain the presentation of solutions with a unknown. Then by using least squares algorithm, we can give the solutions of operator equations .
In summary, the algorithms in this thesis have the following advantages: First, they are very simple; Second, the rate of convergence is fast; Third, the derivatives of approximate solutions can also approximate the derivatives of exact solutions well respectively; Fourth, the algorithms are continuous approximation, that is , the value of arbitrary point can be obtained, which is different from numerical algorithms.
本文运用再生核空间的技巧,给出了几类线性和非线性微分方程的求解算法。
文中详细地叙述了再生核空间的应用背景和研究历史,回顾了再生核空间发展状况。并且在文中的每一部分都给出了若干具体的再生核空间。在所有给出的再生核空间中都有具体的再生核函数表达式,并且每一章都进行了相应的数值试验。这些数值试验验证了理论上推出的结论的正确性。
首先,通过改进原有再生核空间的内积定义,我们简化了再生核函数的表达式,并且通过再生核表达式得到了一组标准正交基,在这组基上进行了Fourier级数展开,从而在第二章中得到了五阶线性方程的解。同时,在第三章又构造了一种适合在计算机上实施的逼近序列来求解奇异线性问题,它的结构简单,对于逼近节点数目很大的离散函数颇为有效。逼近序列的误差在Sobolev范数意义下是递减的并且能保证逼近过程的一致收敛性。
其次,在再生核空间中构造了一种收敛的迭代序列,通过截断级数的形式得到了方程的近似解,同时若方程解存在不唯一的情况下,可以进一步求出满足一定附加条件的特解。这种迭代法适用于一般的非线性方程,本文运用此方法在第四章和第五章分别对奇异非线性方程组和非线性无限比例延迟方程进行了求解。
最后,在第六章中求解二阶非线性偏微分方程。主要利用再生核空间中的再生性质,将其转化为线性算子方程进行求解。将边界条件齐次化后融入到二维再生核空间中,运用本文的方法可以求得一个带有未知量的解的表达式,然后通过最小二乘法的技巧,最终获得非线性算子方程的解。
本文所提的求解算法在理论上和实际试验模拟来看有以下优点:算法简单,通过数值模拟我们也验证了每个模型的各阶导数之间的逼近效果非常好,而且不同于以往的数值算法,本文算法是连续逼近,即对空间任意点均可以算。
Many natural phenomena can be modeled by linear and nonlinear differential equations. These equations as mathematical models have important applications in physics, biology, control science and so on. In general, the analysis of these phenomena can reduce to solving differential equations. It is difficult to obtain the analytical representation of exact solutions, therefore, the research of an efficient numerical method for differential equations is of theoretical and applicable significance and how to solve these significant equations becomes more and more important.
In this thesis, several numerical methods of solving some classes of linear and nonlinear differential equations are presented by using reproducing kernel theory.
This thesis introduces application background and history of reproducing kernel spaces and runs back over the development of reproducing kernel spaces. Moreover, we give the concrete representation of the reproducing kernel spaces in which there are the corresponding reproducing kernel functions in the every chapter. Some numerical tests are given in every chapter and numerical results verify the validity of conclusion.
Firstly, the representation of reproducing kernel is simplied by improving definition of inner product in the original reproducing kernel spaces. The orthonormal basis can be obtained from the representation of reproducing kernel functions. Through Fourier series expanding on this basis, the solutions of fifth-order linear equations can be obtained in Chapter 2. In the mean time, an approximate sequence that is implemented easily on computers is constructed to solve singular linear problems in Chapter 3. Its construction is simple and it can approximate effectively the function with large number of nodes. The error of approximate sequence is decreasing in the sense of Sobolev norm and the uniform convergence can be guaranteed.
Next, a convergent iterative sequence is constructed. The approximate solutions can be obtained by truncating series. If the solutions of equations are not unique, the particular solutions satisfying addictive conditions can be given. This iterative method is suitable for solving general nonlinear equations. Using this method, we solve singular nonlinear systems and nonlinear infinite-delay-differential with proportion delay respectively in Chapter 4 and Chapter 5.
Finally, second order nonlinear partial differential equations are solved in Chapter 6. In terms of the reproducing property of reproducing kernel spaces, we convert them to operator equations .After homogenizing the boundary conditions , we put them into two-dimensional reproducing kernel spaces. By using this method in this thesis, we can obtain the presentation of solutions with a unknown. Then by using least squares algorithm, we can give the solutions of operator equations .
In summary, the algorithms in this thesis have the following advantages: First, they are very simple; Second, the rate of convergence is fast; Third, the derivatives of approximate solutions can also approximate the derivatives of exact solutions well respectively; Fourth, the algorithms are continuous approximation, that is , the value of arbitrary point can be obtained, which is different from numerical algorithms.
引文
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