数学物理方法Chapt 5 (complete).pptx

Chapt 5: Partial Differential Equations 数理方程、方程的解;Introduction （1）;Introduction （2）;Most of the important PDEs of physics are second-order and linear. Some of the more important ones will now be briefly discussed. These equations apply to a wide variety of different physical systems. In general, the PDEs listed below describe three-dimensional situations, the independent variables are r and t, where r is the position vector and t is time.;5.1.1 The wave equation;Chapt 5: 数理方程;Wave equation: generalization 1;Wave equation: generalization 2;5.1.2 The diffusion equation;Chapt 5: 数理方程;Diffusion equation: discussion;Chapt 5: 数理方程;Chapt 5: 数理方程;Chapt 5: 数理方程;5.2 Solutions to PDE;General form of solution;General solution of second-order PDE;The exact functional form of the solution for a particular situation is determined by boundary conditions. For complete determination of solution u(x, y), the boundary conditions will take a form equivalent to specifying u(x, y) along points in the xy-plane (usually along a line). ;5.2.1 First-order PDE;Chapt 5: 数理方程;A(x, y) B(x, y) both not zero (1);Chapt 5: 数理方程;Chapt 5: 数理方程;the general solution may be written, for example, as;A(x, y) B(x, y) both not zero (2) ;(续上页 1）;Chapt 5: 数理方程;Let us discuss in a more general form the particular solutions of the equation It is clear that, if u(x, y) is a solution then so is any multiple of u(x, y) or any linear sum of separate solutions u1(x, y) + u2(x, y). However, when it comes to fitting the boundary conditions this is not so.;For example, solution fit the;Inhomogeneous problems;homogeneous boundary condition;Particular solution and General solution ;Example;Example (续);The most general second-order linear PDE (containing two variables) has the form;constant coefficients second-order PDE;form of p(x, y);二阶常系数PDE: Solution;二阶常系数PDE: Solution (续);Example: wave equation;Example: Laplace equation;B2 = 4AC;Simply let h(x, y) = x (as in ODE), ;周三复习、习题 下周一小测试