两类带弱奇异核四阶积分微分方程的高精度数值解法
发布时间：2018-10-31 09:44
【摘要】：本文针对两类带弱奇异核四阶积分微分方程,提出高精度的LegendreGalerkin谱方法进行求解。对于第一类时间方向含有一阶偏导数的四阶积分微分方程,通过采用CrankNicolson方法离散原方程,构造Jacobi数值积分和Legendre数值积分近似替代积分项;空间方向采用Legendre-Galerkin谱方法进行逼近,得到第一类方程相应的稀疏离散代数系统。数值结果表明该方法具有有效性和长时间计算稳定性。对于第二类时间方向含有二阶偏导数的四阶积分微分方程,通过对时间方向采用二阶中心差分格式离散原方程,构造Jacobi数值积分和Legendre数值积分近似替代积分项;空间方向采用Legendre-Galerkin谱方法进行逼近,得到第二类方程相应的稀疏离散代数系统。数值结果表明该方法是有效的。
[Abstract]:In this paper, two kinds of fourth order integro-differential equations with weakly singular kernels are solved by LegendreGalerkin spectral method with high accuracy. For the fourth order integro-differential equation with first order partial derivative in the first kind of time direction, the CrankNicolson method is used to discretize the original equation to construct the Jacobi numerical integral and Legendre numerical integral to replace the integral term. The space direction is approximated by Legendre-Galerkin spectrum method, and the sparse discrete algebraic system corresponding to the first kind of equation is obtained. The numerical results show that the method is effective and stable for a long time. For the fourth order integro-differential equation with second order partial derivative in the second kind, Jacobi numerical integral and Legendre numerical integral are constructed by using the second order central difference scheme to discretize the original equation in the time direction. The space direction is approximated by Legendre-Galerkin spectrum method, and the sparse discrete algebraic system corresponding to the second kind of equation is obtained. Numerical results show that the method is effective.
【学位授予单位】：华侨大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2301713
[Abstract]:In this paper, two kinds of fourth order integro-differential equations with weakly singular kernels are solved by LegendreGalerkin spectral method with high accuracy. For the fourth order integro-differential equation with first order partial derivative in the first kind of time direction, the CrankNicolson method is used to discretize the original equation to construct the Jacobi numerical integral and Legendre numerical integral to replace the integral term. The space direction is approximated by Legendre-Galerkin spectrum method, and the sparse discrete algebraic system corresponding to the first kind of equation is obtained. The numerical results show that the method is effective and stable for a long time. For the fourth order integro-differential equation with second order partial derivative in the second kind, Jacobi numerical integral and Legendre numerical integral are constructed by using the second order central difference scheme to discretize the original equation in the time direction. The space direction is approximated by Legendre-Galerkin spectrum method, and the sparse discrete algebraic system corresponding to the second kind of equation is obtained. Numerical results show that the method is effective.
【学位授予单位】：华侨大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
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相关期刊论文 前3条
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3 郭大钧;非线性算子方程的正解及其对非线性积分方程的应用[J];数学进展;1984年04期
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