Efficient Numerical Methods And Implementations Of Highly Oscillatory Problems

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.