多维第二类Fredholm积分方程高精度数值算法研究

发布时间：2018-12-28 16:15

【摘要】：积分方程广泛应用在信号传送、生物数学、断裂力学、原子物理、神经网络、交通运输等众多工程科学的数学模型中,所以对积分方程的研究是非常有意义的,特别是第二类Fredholm积分方程。现阶段的研究都还主要集中在一维和二维,然而三维问题,是一个热点问题,也是一个难点问题,现在研究文献还比较欠缺。本文针对二维和三维的第二类Fredholm积分方程分别提出了基于Chebyshev多项式逼近理论的配置法和求积法的快速数值算法。对二维第二类Fredholm积分方程,结合Chebyshev多项式的性质,利用配置法提出了针对二维第二类Fredholm积分方程的一种快速有效的高精度算法,即用Chebyshev多项式来近似未知函数和核函数,再选取Gauss-Chebyshev-Lobatto结节点作为配置点,生成离散代数方程组,最后用Newton迭代法求解方程组得到原方程的数值近似解。对三维第二类Fredholm积分方程,本文先用常见的数值积分公式Gauss公式来离散方程,并在Chebyshev-Gauss网格中用多项式来近似核函数,最终转化成矩阵向量的计算,最后用Jacobi迭代法求解线性方程组,最终得到方程的近似数值解。其实,上述方法还有很多可以改进的地方,如最后采用余弦向量迭代法求解方程组,采用分裂外推对网格加密,提高近似解的精度等。对于这两种方法,本文都给出了相关的误差和收敛性分析。最后,数值算例验证理论分析。
[Abstract]:Integral equations are widely used in many mathematical models of engineering science, such as signal transmission, bio-mathematics, fracture mechanics, atomic physics, neural network, transportation and so on, so the study of integral equations is of great significance. In particular, the second kind of Fredholm integral equation. At present, the research is mainly focused on one and two dimensions. However, the three dimensional problem is a hot and difficult problem. In this paper, a fast numerical algorithm of collocation method and quadrature method based on Chebyshev polynomial approximation theory is proposed for two-dimensional and three-dimensional Fredholm integral equations of the second kind, respectively. For the two-dimensional Fredholm integral equation of the second kind, combined with the properties of the Chebyshev polynomial, a fast and efficient algorithm for the two-dimensional Fredholm integral equation is proposed by using the collocation method, that is, the Chebyshev polynomial is used to approximate the unknown function and the kernel function. Then the Gauss-Chebyshev-Lobatto nodule point is selected as the collocation point to generate the discrete algebraic equations. Finally, the numerical approximate solution of the original equation is obtained by Newton iterative method. For the Fredholm integral equation of the second kind, this paper first uses the common numerical integral formula Gauss formula to discretize the equation, and approximates the kernel function with polynomial in the Chebyshev-Gauss mesh, and finally transforms it into the calculation of matrix vector. Finally, the Jacobi iterative method is used to solve the linear equations, and the approximate numerical solutions of the equations are obtained. In fact, the above methods can be improved, such as using cosine vector iterative method to solve the equations, using split extrapolation to encrypt the mesh, and improving the accuracy of approximate solution. For these two methods, this paper gives the error and convergence analysis. Finally, numerical examples verify the theoretical analysis.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.8

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