数理方程的分类研究报告.ppt

Hyperbolic type: Parabolic type: Or: 只能求出一个通解 Elliptic type: Or: 简化下面二阶线性方程： Principle of superposition of linear equations 1-1 Partial Differential Equations (PDE) ? 1-1-1 Partial Differential Equations and Their Orders Introduction Ordinary differential equation (the ODE for short) is a differential equation that contains one or more derivatives of the dependent variable (in addition to the dependent variable and the independent variable). 含自变量，未知函数及其导数 应变量 dependent variable 自变量 independent variable The order (阶) of an ordinary differential equation is the order of the highest-ordered derivative appearing in the equation. A differential equation that contains, in addition to the dependent variable and the independent variables, one or more partial derivatives of the dependent variable is called a partial differential equation, the PDE for short （偏微分方程） The order of the highest-ordered partial derivative appearing in the equation is called the order of the PDE Independent and dependent variables may not appear in a PDE . A PDE must contain, however, at least one partial derivative of the dependent variable. For an unknown function of two independent variables, are the first-order, the second-order and the third-order PDE, respectively. The k-th order PDE of an unknown function of independent variables can be written in a general form Three-dimensional wave equation : two-dimensional heat-conduction equation: two-dimensional and the three-dimensional Laplace equations: two-dimensional and the three-dimensional Poisson equations the dual-phase-lagging heat-conduction equation: 双相滞热传导方程 1-1-2 Linear, Nonlinear and Quasi-Linear (拟线性) Equations  An ODE can be linear, nonlinear A PDE can be linear, nonlinear , quasi-linear. A PDE is said to be linear if it is linear in the unknown function and all its derivatives; nonlinear equation ; A nonlinear equation is said to be quasi-linear if it is linear i