【摘要】：高振蕩函數(shù)的積分問題在電磁計(jì)算,量子力學(xué),信號(hào)處理等實(shí)際應(yīng)用中是一個(gè)核心的研究方向,問題的關(guān)鍵就是如何給出高振蕩函數(shù)積分的高效數(shù)值算法。因?yàn)橛肎auss、Newton-Cotes等傳統(tǒng)的積分法則來(lái)求解高振蕩積分是失效的,因此我們必須尋找新的高效的數(shù)值方法。近年來(lái)有很多高效的高振蕩積分?jǐn)?shù)值算法相繼被提出來(lái),例如:Filon法,Levin法,數(shù)值最速下降法等。其有關(guān)中Bessel類型振蕩積分是高振蕩積分的核心問題之一,目前Bessel類型高振蕩積分有以下三種主要方法: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給出Bessel型函數(shù)積分的一種高效的數(shù)值算法,最后給出相應(yīng)的數(shù)值實(shí)驗(yàn),從而來(lái)驗(yàn)證本文方法的高效性。第一章,討論了一些常用的高振蕩函數(shù)積分算法,并分析了它們之間的優(yōu)缺點(diǎn)和相互聯(lián)系。第二章,介紹了有限區(qū)間和無(wú)限區(qū)間的Fourier型積分,重點(diǎn)介紹了最速下降法和Filon型方法,以及將兩者結(jié)合在一起求解Fourier型積分。然后將上述方法推廣到Bessel型積分,并給出了相應(yīng)的誤差分析。最后將該方法推廣到Airy型積分。第三章,數(shù)值實(shí)驗(yàn)。我們用第二章中提出的方法來(lái)計(jì)算Fourier型和Bessel型高振蕩積分,并通過(guò)實(shí)驗(yàn)結(jié)果來(lái)驗(yàn)證方法的收斂性和誤差階的準(zhǔn)確性。最后我們通過(guò)數(shù)值實(shí)驗(yàn)來(lái)說(shuō)明本文方法要比Filon法求解Bessel型高振蕩積分更高效。
[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.
【學(xué)位授予單位】：華中科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O241.8

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本文編號(hào)：2168275