三类发展型偏微分方程数值解

发布时间：2018-04-17 22:43

  本文选题：偏积分微分方程 + 有限元方法　； 参考：《湖南师范大学》2011年博士论文

【摘要】：本文研究有限元方法、谱方法和有限差分法求解三类发展型偏微分方程。第一类发展型偏微分方程是带弱奇异核的偏积分微分方程,其形式为 初始条件和边界条件分别为：和其中,ut=(?)u/(?)t,核β(t)=tα-1/Г(α),0α1,当t=0时是一个奇异核,r是Gamma函数。为了有效的求解上面的积分微分方程,我们构造稳定的二阶格式,时间方向用有限中心差分法、空间方向用有限元方法逼近,得到时间半离散格式和全离散格式,证明两种格式的稳定性和收敛性,并通过数值实验验证了理论分析的正确性。研究的第二类方程是四阶带弱奇异核的偏积分微分方程,其形式为满足初始条件：和边界条件其中,I=(-1,1),K(t,s)=(t-s)α-1/Г(α).对这类方程,空间方向用谱方法、时间方向用有限中心差分和拉格朗日插值法来逼近,得到空间半离散格式和全离散格式,证明了空间半离散格式的稳定性和收敛性。 研究的第三类方程是奇异摄动对流扩散方程,其模型问题为初始条件和边界条件分别为：和其中,a≠0,ε0.对于这类方程,用有限差分方法,通过组合差商法来逼近,得到一系列最简单的、稳定性更好、应用范围更广、计算简单、效率更高、计算更省时的差分格式,并分析了这一系列离散格式的稳定性,对每一个差分格式都有数值例子,数值实验与理论分析相吻合。为了使计算更简单、更省时、更方便,另外还构造了在空间方向并行计算和在时间方向并行计算的两类并行格式,对并行格式的稳定性进行了分析,利用并行迭代算法得到了一些好的数值结果。
[Abstract]:In this paper, finite element method, spectral method and finite difference method are studied to solve three kinds of evolution partial differential equations.The first kind of evolutionary partial differential equation is a partial integro-differential equation with weak singular kernel in the form ofThe initial conditions and the boundary conditions are as follows: the initial condition and the boundary condition are as follows: among them, the nucleus 尾 ~ (t) ~ T ~ (-1) is a Gamma function when t = 0, and the kernel 尾 ~ (?) t ~ (-1) ~ (-1) is a Gamma function when t = 0, the initial condition and the boundary condition are respectively.In order to solve the above integrodifferential equation effectively, we construct a stable second-order scheme. The time direction is approximated by the finite-center difference method, the spatial direction is approximated by the finite element method, and the time semi-discrete scheme and the fully discrete scheme are obtained.The stability and convergence of the two schemes are proved, and the correctness of the theoretical analysis is verified by numerical experiments.The second kind of equation is the partial integro-differential equation with the fourth order weakly singular kernel in the form of satisfying the initial condition: and boundary condition, in which the Igni ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) K ~ (+) ~ (1) ~ (-) 伪 -1 / (伪 ~ (1)).For this kind of equation, the spatial direction is approximated by the spectral method, the time direction is approximated by the finite central difference and Lagrange interpolation method, the spatial semi-discrete scheme and the full discrete scheme are obtained, and the stability and convergence of the spatial semi-discrete scheme are proved.The third kind of equation is singularly perturbed convection-diffusion equation, whose model problems are initial conditions and boundary conditions respectively: and where a 鈮，

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