Solving Highly Oscillatory Problems By Laplace Transforms And Their Applications

The highly oscillatory problems arise in mathematics and many other science fields widely. It is more difficult to calculate these problems when the oscillation frequency is higher. In this doctoral thesis, we focuse on the asymptotic and Filon-type methods on approximation of these highly oscillatory problems by using Laplace transforms. These ideas are extended to solve some kinds of differential and integral equations, and used in calculation of optical diffraction. The thesis is organized as follows.In Chapter1, we review several efficient methods for highly oscillatory integrals and introduce some definitions related to Laplace and inverse Laplace transforms briefly. In the case that the singular point of image function is a pole, the inverse Laplace transform can be calculated directly according to the Cauchy residue theorem. In the case that singular point is a branch point, we should turn it into a contour integral by using the single-valued cutting method. The contour integral usually involves special functions, which is difficult to be calculated efficiently. However, if we seek to the asymptotic solution instead of accurate solution, referring to existing literature, we can obtain asymptotic theorems which are suitable for calculating high-frequency case by the special contour and W transform.In Chapter2, we devote our attention to the efficient computation of highly oscillatory Fourier integral. Firstly, according to asymptotic theorem of W transform, we obtain the asymptotic expression of the Fourier-type moment integral which has branch at two endpoints. In order to calculate the Fourier integral, we select the sufficiently smooth part of amplitude function, and use Hermite interpolations at the two end points of the integration interval and convert it into the asymptotic expression of moment integral. The method can be applied to the case that the amplitude function is composite branch function, and to the case that the generalized Fourier integral contains stationary points. In this chapter, we rewrite the Watson Lemma, and the rewritten asymptotic expression is suitable for the case of imaginary argument under the condition that some angular domain is holomorphic bounded. The quadrature method of changing the highly oscillatory integrand into the nonoscillatory function is given based on Taylor expansion.In Chapter3, we are concerned with the efficient evaluation of integrals about other kernels. These kernels include Bessel function, Airy function, the product of Fourier and Bessel function of the first kind, and the product of Fourier and Bessel function of the first second. The properties of kernels are varied. They are highly oscillatory, grow exponentially, or decay exponentially. The methods to analyze their properties include computing the residues at the poles, single-valued cutting method, and saddle point method to deal with essentially singular points or regular functions. The interpolation laws of them are of diversity as the kernel functions and amplitude functions change. There are Hermite interploation, Taylor expansion, interpolation at other points on the real axis, and also interpolation at the points which are not on the real axis, and so on.In chapter4, we discuss a certain number of numerical method of highly oscillatory differential equations and integral equations. We, by Laplace transform, convert the solution forms of these differential equations and integral equations to corresponding integral forms which can be evaluated efficiently by numerical method proposed in Chapter2and Chapter3. In addition, we consider a discretization scheme using higher order derivatives for solving ordinary differential equation in this chapter.We discuss the applications of highly oscillatory integral in diffractive optics computation in Chapter5. Taking parabola for example, we introduc the general aperture Fraunhofer diffraction calculation. Further more, we deduce two existed calculation formulas by Laplace transform when approximating integrals involved in near the focus of the three-dimensional optical state distribution of Fresnel diffraction, which may be obtained unified and complete expression in the larger range. In the end, some conclusions are presented in Chapter6. The classical integration methods approximate the values of integrals using the linear combination of the values of integrand on some certain points. This paper concern on asymptotic method and Filon-type method which are very different from the classical integration method. This paper also states the urgent need of a more unified theoretical and computational way to accommodate the higher requirements in practical computing.