两类力学问题导出的奇异积分方程数值方法研究

发布时间：2018-02-24 00:00

本文关键词： 奇异积分方程 Bagley-Torvik方程 Volterra积分方程 Fred-holm积分方程数值解收敛性与误差估计　出处：《广西大学》2017年硕士论文　论文类型：学位论文

【摘要】：近年来,许多学者对奇异积分方程的研究取得了丰硕的成果,而奇异积分方程也在解决弹性理论和断裂力学等数学物理问题中发挥重要的作用.本文采用新的数值方法研究两类力学问题导出的奇异积分方程的近似解,并分析数值方法的收敛性和误差估计,通过数值实例且与其他方法比较,验证方法的可行性与有效性.主要内容有:(1)研究流体力学问题导出的一类弱奇异积分方程.首先对含有变系数的分数阶微分Bagley-Torvik方程进行积分处理转化为含有弱奇异核的第二类Volterra积分方程,然后利用Banach空间的压缩算子原理研究方程解的存在性和唯一性,给出对应的充分条件.接着对含有弱奇异核的第二类Volterra积分方程构造数值解,同时给出相应的收敛性分析和误差估计.最后计算含有变系数的分数阶微分Bagley-Torvik方程的近似解并且与其他方法进行比较,验证本文的方法.特别的,对于精确解为多项式函数的积分方程,本文的方法可以获得精确解.而当待求解的积分方程的精确解未知时,通过与经典的差分法比较表明,本文的方法依然有效.(2)研究断裂力学中十字裂纹问题导出的奇异积分方程.该奇异积分方程与一般的积分方程相比主要的区别就是它存在奇异点,为此本文采用修正的数值方法构造方程的近似解,通过引进权重参量ω提高近似解的精确度,给出修正的数值方法的误差及收敛性分析,并通过数值实例验证本文的方法.以上的研究充分体现了奇异积分方程在实际问题中的应用,并为解决力学与工程学等各类相似的问题提供理论方法.
[Abstract]:In recent years, many scholars have made fruitful achievements in the study of singular integral equations. Singular integral equations also play an important role in solving mathematical and physical problems such as elastic theory and fracture mechanics. In this paper, the approximate solutions of singular integral equations derived from two kinds of mechanical problems are studied by using a new numerical method. The convergence and error estimation of numerical methods are analyzed and compared with other methods by numerical examples. The purpose of this paper is to study a class of weakly singular integral equations derived from fluid mechanics problems. Firstly, the fractional differential Bagley-Torvik equation with variable coefficients is transformed into a class with weak singular equations. Volterra integral equation of the second kind of heteronuclear, Then the existence and uniqueness of the solution of the equation are studied by using the contraction operator principle of Banach space, and the corresponding sufficient conditions are given. Then, the numerical solution of the second kind of Volterra integral equation with weakly singular kernel is constructed. At the same time, the convergence analysis and error estimate are given. Finally, the approximate solution of fractional differential Bagley-Torvik equation with variable coefficients is calculated and compared with other methods. For an integral equation whose exact solution is a polynomial function, the method in this paper can obtain an exact solution. When the exact solution of the integral equation to be solved is unknown, it is shown by comparison with the classical difference method that, The method in this paper is still valid for studying the singular integral equation derived from the cross crack problem in fracture mechanics. The main difference between the singular integral equation and the general integral equation is the existence of singular points. In this paper, the approximate solution of the equation is constructed by using the modified numerical method. By introducing the weight parameter 蠅 to improve the accuracy of the approximate solution, the error and convergence analysis of the modified numerical method are given. The above research fully embodies the application of singular integral equations in practical problems and provides a theoretical method for solving similar problems such as mechanics and engineering.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O302;O241.83

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