一种求解Bessel型高振荡积分的数值方法
发布时间：2018-08-06 16:18
【摘要】：高振荡函数的积分问题在电磁计算,量子力学,信号处理等实际应用中是一个核心的研究方向,问题的关键就是如何给出高振荡函数积分的高效数值算法。因为用Gauss、Newton-Cotes等传统的积分法则来求解高振荡积分是失效的,因此我们必须寻找新的高效的数值方法。近年来有很多高效的高振荡积分数值算法相继被提出来,例如:Filon法,Levin法,数值最速下降法等。其有关中Bessel类型振荡积分是高振荡积分的核心问题之一,目前Bessel类型高振荡积分有以下三种主要方法:Filon法、Levin法、最速下降法。本文旨在基于Filon法和最速下降法的思想给出Bessel型函数积分的一种高效的数值算法,最后给出相应的数值实验,从而来验证本文方法的高效性。第一章,讨论了一些常用的高振荡函数积分算法,并分析了它们之间的优缺点和相互联系。第二章,介绍了有限区间和无限区间的Fourier型积分,重点介绍了最速下降法和Filon型方法,以及将两者结合在一起求解Fourier型积分。然后将上述方法推广到Bessel型积分,并给出了相应的误差分析。最后将该方法推广到Airy型积分。第三章,数值实验。我们用第二章中提出的方法来计算Fourier型和Bessel型高振荡积分,并通过实验结果来验证方法的收敛性和误差阶的准确性。最后我们通过数值实验来说明本文方法要比Filon法求解Bessel型高振荡积分更高效。
[Abstract]:The integration problem of high oscillation function is a core research direction in electromagnetic calculation, quantum mechanics, signal processing and other practical applications. The key of the problem is how to give an efficient numerical algorithm for the integration of high oscillation function. Because the traditional integral law such as Gaussfield Newton-Cotes is invalid to solve the high oscillatory integral, we must find a new and efficient numerical method. In recent years, many efficient numerical algorithms for high oscillatory integrals have been proposed, such as the Levin method, the most rapid descent method and so on. The Bessel type oscillation integral is one of the core problems of the high oscillation integral. At present, there are three main methods of the Bessel type high oscillation integral: the Bessel method and the Levin method, and the steepest descent method. Based on the idea of Filon method and the steepest descent method, this paper presents an efficient numerical algorithm for the integration of Bessel type functions. Finally, the corresponding numerical experiments are given to verify the efficiency of the method in this paper. In the first chapter, we discuss some common integration algorithms of high oscillation function, and analyze their advantages and disadvantages. In the second chapter, the Fourier type integral of finite interval and infinite interval is introduced, the steepest descent method and Filon type method are introduced, and the Fourier type integral is solved by combining the two methods. Then the above method is extended to Bessel type integral and the corresponding error analysis is given. Finally, the method is extended to Airy type integrals. Chapter three, numerical experiment. We use the method proposed in Chapter 2 to calculate the Fourier and Bessel type high oscillatory integrals and verify the convergence of the method and the accuracy of the error order by the experimental results. Finally, numerical experiments show that the proposed method is more efficient than the Filon method in solving the Bessel type high oscillation integral.
【学位授予单位】：华中科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.8
本文编号：2168275
[Abstract]:The integration problem of high oscillation function is a core research direction in electromagnetic calculation, quantum mechanics, signal processing and other practical applications. The key of the problem is how to give an efficient numerical algorithm for the integration of high oscillation function. Because the traditional integral law such as Gaussfield Newton-Cotes is invalid to solve the high oscillatory integral, we must find a new and efficient numerical method. In recent years, many efficient numerical algorithms for high oscillatory integrals have been proposed, such as the Levin method, the most rapid descent method and so on. The Bessel type oscillation integral is one of the core problems of the high oscillation integral. At present, there are three main methods of the Bessel type high oscillation integral: the Bessel method and the Levin method, and the steepest descent method. Based on the idea of Filon method and the steepest descent method, this paper presents an efficient numerical algorithm for the integration of Bessel type functions. Finally, the corresponding numerical experiments are given to verify the efficiency of the method in this paper. In the first chapter, we discuss some common integration algorithms of high oscillation function, and analyze their advantages and disadvantages. In the second chapter, the Fourier type integral of finite interval and infinite interval is introduced, the steepest descent method and Filon type method are introduced, and the Fourier type integral is solved by combining the two methods. Then the above method is extended to Bessel type integral and the corresponding error analysis is given. Finally, the method is extended to Airy type integrals. Chapter three, numerical experiment. We use the method proposed in Chapter 2 to calculate the Fourier and Bessel type high oscillatory integrals and verify the convergence of the method and the accuracy of the error order by the experimental results. Finally, numerical experiments show that the proposed method is more efficient than the Filon method in solving the Bessel type high oscillation integral.
【学位授予单位】：华中科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.8
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1 ;A universal solution to one-dimensional oscillatory integrals[J];Science in China(Series F:Information Sciences);2008年10期
相关博士学位论文 前1条
1 王海永;高振荡问题的高效数值方法研究[D];中南大学;2010年
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