Singular Quadrature Research For Quadratic Oscillatory Kernel

In recent years,it is a new method in numerical integration on numerical computation for oscillatory integral that the integral with oscillatory weight in finite interval is transformed to the complex integration,and the research on numerical integration for the singular integral with oscillatory weight is just in the beginning,some fundamental research on numerical computation for the singular integral with quadratic oscillatory weight is discussed in the present thesis which consists of five chapters.Firstly,some methods on numerical computation for oscillatory integral are introduced,then the steepest descent method is specially explained,the path of steepest descent is provided and the path is parameterized,which develops a base on numerical computation for the singular integral with quadratic oscillatory;Secondly,taking advantage of the Cauchy theorem,the oscillatory singular integral is transformed to complex integration of some paths on the complex plane;the condition of the singularity on the boundary are discussed with the generalized residue theorem,with the vanishing of oscillation and singularity,the integral can be transformed into some generalized integrals which are non-oscillating and are converged exponentially on;Numerical computation are performed for these generalized integrals with the Gauss-Laguerre quardature formula,the quadrature formula are constructed and the error analysis of the quadrature formula can be obtained.Finally,numerical examples are realized with Matlab,the numerical results of the constructed quadrature fromulae are calculated,and the result of numerical examples in the experiments are accord with our theoretical analysis.