Study On Efficient Numerical Methods And Implementation For Highly Oscillatory Integrals

The quadrature of highly oscillatory integrals is a computational problem of overarching importance in a wide range of applications, ranging from quantum chemistry to image analysis, electrodynamics, positron emission tomography, single photon emission tomography and fluid mechanics…. The key to these problems is to efficiently compute highly oscillatory integrals. The classical integration formulae are allegedly difficult to calculate highly oscillatory integrals, such as generalized Fourier transformations and Bessel transformations when the frequency is significantly larger than the number of quadrature points. The aim of this paper is to explore some new efficient numerical methods for evaluating highly oscillatory integrals.In Chapter one, we first give the definition of highly oscillatory function and some general examples. We introduce some applications and some efficient methods for highly oscillatory integrals, such as Filon method,Filon-type method,Levin method,Levin-type method,asymptotic method,generalized quadrature rule and numerical steepest descent method.In Chapter two, a new efficient parameter method is presented for integration of highly oscillatory function with an irregular oscillator by using the expression of multiple Fourier integrals with high frequency. If there exists not the stationary points for the function g(x), the stationary points can be omitted by a simple transformation. The effectiveness and accuracy are tested by numerical examples for the case that g(x) has stationary points.Based on the asymptotic formula of Bessel function, in Chapter three we transform Bessel transformations into Fourier transformations. We get an efficient, higher order quadrature of Bessel transformations by using numerical steepest descent method presented by Huybrechs et al..Applying the relation d(x^vJ_v(x))/dx = x^vJ_v-1(x) and integration by parts, Chapter four presents a simplier and efficient asymptotic method.Chapter five considers the integrals including Bessel functions by using the homotopy perturbation method. The integrals are first transformed into the problems of equations. The solutions are writtenas the series form u=∑_k=0^∞p^ku_k by using the perturbation techniques andall series terms can be obtained by constructing a simple homotopy, that is, the solutions of equation can be written as u=((p→1)|(lim))∑_k=0^∞p^ku_k .Chapter six considers vector-value highly oscillatory integrals. Based on the homotopy perturbation method, we obtain the vector-value homotopy perturbation numerical method for highly oscillatory integrals. New method is identical to Levin iteration method for the case that initial value is a constant.