摘要
高振荡问题广泛出现于科学工程计算诸多领域.它同时也是一类被公认为非常难的国际热点前沿研究课题,存在许多挑战性问题.近几十年来,这类问题也获得了众多专家学者的极大关注.本篇博士论文由两部分组成.第一部分研究几类高振荡奇异积分的计算问题,为了克服奇异性和高振荡特征带来的困难,我们在现有方法的基础上采用不同的变换或技巧,设计可行的方法来获取高效的积分法则；第二部分我们利用微分方程技巧推导了几类超几何函数关于参数的导数,这些导数不仅在高振荡奇异积分的计算中有着重要的应用,而且在数学、物理或相关领域中扮演着非常重要的角色.鉴于此,本文具体工作组织如下：
第一章分别从高振荡积分、高振荡常微分或偏微分方程、高振荡积分方程几个方面简单介绍了高振荡问题的研究背景和研究意义.
第二章对于几类高振荡积分,我们回顾了迄今为止发展的数值积分方法,如渐近方法、Filon方法、Filon型方法、Levin方法、Levin型方法、广义积分法则、数值最速下降法以及其它数值方法.
第三章,我们给出三种方法计算一类高振荡代数奇异积分.一种是先把这类积分通过换元转化为区间[1,∞)上的无穷积分,然后根据极限和复积分法的思想,将数值最速下降法推广到这类无穷积分的计算,并作出了误差分析和给出了相应的渐近阶.这种方法具有成本低收敛速度快的优势.然而,它的不足在于要求函数f(x)在某个区域内解析.接下来我们考虑放宽这种限制条件.我们首先把原积分经过两次代换转化到标准区间[-1,1]上,然后将转化后的积分展开为关于w的负幂次的渐近级数.基于这个渐近级数,我们提出两种方法.一种是Filon型方法.另一类是借用特殊Hermite插值(只需在端点1上导数插值,其它Clenshaw-Curtis点上线性插值)以及递推关系式的Clenshaw-Curtis-Filon型方法,并讨论了这类方法的收敛性和误差阶.这种Clenshaw-Curtis-Filon型方法可经过O(N log N)次运算量快速稳定实现.后两种方法只需f(x)在区间[0,1]上光滑即可.这些方法具有共同特点：精度会随着频率w的增加迅速提高.
第四章,针对含有端点弱奇异性的高振荡积分,我们分别提出了三种数值计算方法.第一种是在渐近展开式的基础上,以x-a,b-x或两者结合为底的幂函数作为基函数的Filon型方法.第二种,我们利用Chebyshev多项式作为基函数,应用特殊的Hermite插值(需要在两个端点α，b上导数插值,其它Clenshaw-Curtis点上线性插值),给出相应的Clenshaw-Curtis-Filon型方法,并作出了收敛性分析和给出了误差界.第三种方法是基于解析延拓,首先设计合适的最速下降路径,然后利用广义Gauss Lagurre积分法则来高效实现的数值最速下降方法.这三种方法各有优劣,相互补充.
第五章,我们设计一种Chebyshev展开式法计算一类振荡或奇异积分.它们的核函数G(x)是代数型或对数型奇异性因式的乘积.这类方法是基于f的Chebyshev展开式,Chebyshev多项式的一些性质以及振荡因子eiwx的Bessel-Chebyshev展开式提出来的.由于修正矩的递推关系的向前递推和向后递推都是数值稳定的,这使得算法的实现相当简单.特别地,我们考虑了许多不同的核函数G(x),这凸显了这种方法应用广泛的优势.
第六章,基于对高斯超几何微分方程和合流超几何函数微分方程分别取参数的导数,我们提出一种有效的微分方程方法来推导高斯超几何函数和Kummer合流超几何函数关于参数的导数公式.特别地,借助这种微分方程方法,通过使用对s的数学归纳法,我们推导出了对参数的任意s阶导数的通用解析表达式.而且,我们获得两个非齐次线性微分方程,再联合一些作为初始条件的低阶混合导数,可递归产生高阶混合导数.进一步来说,我们将这种微分方程方法推广到更为复杂的广义超几何函数mFn(a1,...,am；b1,...,bn；z)的形式,并推导出其对参数的任意s阶导数的公式.最后简要介绍了这些导数在数学、物理中的一些应用.
Highly oscillatory problems arise in wide and many areas of sci-ence and engineering. They are widely perceived as not only extremely challenging significant research topics, but also frontier and hot sub-jects around the world. And, they receive significant attention of the experts and scholars in recent decades. This doctoral thesis consists of two parts. The first one is devoted to the problems of computing several types of highly oscillatory singular integrals. The singularities and possible high oscillations of the integrands make these integrals very difficult to approximate accurately. In order to overcome some problems that usually appear in the case when the integrands are both singular and oscillating, we shall design feasible methods and obtain efficient integration rules, owing to existing methods and dif-ferent techniques. In the second part, based on differential equations, we will propose a general method to derive some derivatives of lower or high order of several types of hypergeometric functions with respect to parameters. Such derivatives play important roles in the calculation of highly oscillatory singular integrals and other mathematics, physics and related fields. The outline of this thesis is organized as follows.
In Chapter1, the research background and significance is intro-duced on highly oscillatory problems from aspects of highly oscillatory integrals, ODEs, PDEs and integral equations.
In the second chapter, for several types of highly oscillatory inte-grals, we mainly review some efficient numerical methods developed so far, such as Asymptotic methods, Filon methods, Filon-type methods, Levin methods, Levin-type methods, generalized quadrature rules, nu-merical steepest descent methods and other numerical methods.
Chapter3is concerned with the numerical evaluation of a class of oscillatory singular integrals. We presents some quadrature meth-ods for such integrals. The integrals can be first transformed into the infinite integral without algebraic singularity by a change of variable. A numerical steepest descent method can be generalized to the highly oscillatory integral on a infinite range by choosing limit of proper inte-gral. Meanwhile, we provide error analysis for the first method. The method has several merits, such as low cost and fast convergent rate. However, the method requires that f(x) is analytic in a small neigh-borhood. Then, we relax the strict requirement until more general applicable approaches can be obtained for just sufficiently smooth f on [0,1]. We first expand such integrals derived by two transforma-tions, into asymptotic series in inverse powers of the frequency ω. Then, based the asymptotic series, two methods are presented. One is the Filon-type method. The other is the Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polyno-mial in the Clenshaw-Curtis points and can be evaluated efficiently in O(N log N) operations, where N+1is the number of Clenshaw-Curtis points in the interval of integration. Also, we give error and convergence analysis of the latter two methods. All these methods complement each other but share the advantageous property that their accuracy improves greatly when w increases.
In Chapter4, we design some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polynomial. In addition, we derive the corresponding error bound in inverse powers of the frequency ω and show uniform convergence for the Clenshaw-Curtis-Filon-type method for the class of highly oscillatory integrals. The third method is a numerical steepest descent method based on substituting the original interval of integration by the paths of steepest descent, which can be efficiently computed by using the generalized Gauss Laguerre quadrature rule. These methods have relative merits and disadvantages and can complement each other.
In the fifth chapter, we present a general method for computing oscillatory integrals, whose kernel function G is a product of singu-lar factors of algebraic or logarithmic type. Based on a Chebyshev expansion of f and the properties of Chebyshev polynomials, the pro-posed method for such integrals is constructed with the help of the expansion of the oscillatory factor eιωχ. What is more important, the required moments satisfy recurrence relations that are stable in either forward or backward direction, which makes the whole algorithm quite simple. We consider many different kernel functions G(x), which is a big advantage of the presented approach.
Chapter6first shows a method to derive some differentiation for-mulas of both the Gauss hypergeometric function2F1(μ,ν;λ;z) and the Kummer confluent hypergeometric function1F1(μ;ν;z) with re-spect to all parameters. A differential equation method can be con-structed, which is based on differentiating the hypergeometric differen-tial equation and the confluent hypergeometric (Kummer) differential equation with respect to parameters. In particular, thanks to the differential equation method, some general analytical expressions of any s-th derivatives with respect to single parameter can be deduced by induction in s. Moreover, we obtain the resulting two nonhomo-geneous linear differential equations which together with the initial conditions being supplied by some mixed derivatives of lower order, can recursively generate all the mixed derivatives of higher order very conveniently. Furthermore, the differential equation method can be extended to obtain any s-th derivatives of generalized hypergeomet-ric functions mFn(α1,...,αm;b1,...,bn;z) with respect to parameters. Meanwhile, a study of such derivatives is motivated by the occurrence of these problems in mathematics, physics and other related fields.
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