卷积型奇异积分方程与边值理论

发布时间：2018-07-01 12:49

本文选题：奇异积分方程 + Fredholm方程　；参考：《中国科学技术大学》2016年博士论文

【摘要】：卷积型奇异各积分方程与边值理论在许多实际问题,如物理学、弹性力学、工程力学、空气动力学、电子光学、工程技术等领域具有广泛的应用。近年来,该领域的研究已经深入到难度极大的高维、变系数、超奇异等情形。针对这些热点问题,本文进行了系统而深刻地研究。本文的主要内容和创新点如下：(1)对于一类对偶型卷积型奇异积分方程得到了具有指数增长或哀减的解。这样的解由于其在无穷远处的指数的增长或衰减性,在物理学、辐射平衡理论中具有重要意义。该类方程的求解方法是新颖的,它是通过积分变换转化为化为带形域上具有复合边界的Riemann边值问题。(2)对于含有调和奇异算子的离散卷积型方程建立了方程解的存在性。与经典的离散卷积型方程不同,该方程的核函数的Fourier变换在单位圆周上有问断点。(3)全纯函数边值问题已有的结果大多局限于一个未知函数情形,该文研究了多个未知函数的Riemann边值问题。其方法与经典情形不同,采用了解析开拓原理。(4)变系数奇异积分方程的研究,由于其方法很少,结果口前尚不多见。本文利用局部性理论研究与全纯函数边值相关的变系数的卷积型奇异积分方程的可解性。
[Abstract]:Convolution singular integral equations and boundary value theory are widely used in many practical problems, such as physics, elasticity, engineering mechanics, aerodynamics, electron optics, engineering technology and so on. In recent years, the research in this field has gone deep into the high dimension, variable coefficient, hyper-singularity and so on. In view of these hot issues, this paper has carried on the systematic and profound research. The main contents and innovations of this paper are as follows: (1) for a class of dual convolution singular integral equations, solutions with exponential growth or mourning subtraction are obtained. Such solutions are of great significance in physics and radiation equilibrium theory because of their exponential growth or attenuation at infinity. The method of solving this kind of equation is novel, it is transformed into Riemann boundary value problem with compound boundary by integral transformation. (2) the existence of solution for discrete convolution type equation with harmonic singular operator is established. Different from the classical discrete convolution equation, the Fourier transform of the kernel function of the equation has a broken point on the unit circumference. (3) the results of the boundary value problem for Holomorphic functions are mostly confined to an unknown function case. In this paper, the Riemann boundary value problem for several unknown functions is studied. The method is different from the classical case, and the analytic expansion principle is adopted. (4) the study of singular integral equation with variable coefficients is rare because of its few methods and few results before the mouth. In this paper, the solvability of convolutional singular integral equations with variable coefficients related to the boundary value of Holomorphic functions is studied by using the theory of locality.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.8

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