数理方程Chaper5.ppt

1、The method of separating variables on finite region,Chapter 5,5.1 Some reviews 5.2 Separating variable method,5.1 Some reviews,Second order ODEs,Consider the second order ODE of the special form,Suppose that the roots of the quadratic equation,are r1 and r2.,(1) If r1 r2 are two real numbers , then

2、the general solution for the ODE is,(1) If r1 =a+ib, r2=a-ib, then the general solution for the ODE can be also written as,(2) If r1 = r2= r, the general solution for the ODE is,As an application, the general solution for the ODE,is,And the general solution for the ODE,is,2. Fourier series,Suppose t

3、hat f is continuous on -l,l, then its Fourier series expansion is,where,If f is only defined on 0,l, then its Fourier cosine series expansion is,where,and its Fourier sine series expansion is,where,which belongs to one type among the following normal types:,hyperbolic type, parabolic type, elliptic

4、type.,Q? u(x,y)=X(x)Y(y),5.2 Separating variable method,1 Separation of variables for wave equation 2 Separation of variables for heat equation 3 Separation of variable for Laplace equation 4 Separation of variable for the problem of beam oscillation 5 Separation of variable for the equation of Helm

5、holz,Eample: mixed problem on string oscillation,设柔软细弦的长为 ，两端固定，弦的线密度为 ，忽略自身的重力作用。给定弦的初始位移和速度，则细弦作自由振动的定解问题归结为:,1.Separation of variables for wave equation,Solution,Let,Step1 Separate variables,Step2 Solve the eigenvalue problem,(i),(ii),(iii),Step3 Solve the ODE about T(t):,General solution,Step4 L

6、inear superposition,Basis:,Hence, the solution to this problem is,where an and bn are defined as above.,Step5 Determine the coefficients an and bn by the initial condition,2. Separation of variables for heat equation,设有一根长为 的导热细杆,密度为 ,侧面绝热, 在左端点 处;温度恒为0.在右端点 处;热量向周围介质传递, 即有热交换, 热交换系数为 ,周围介质温度为0. 设初始

7、温度分布为 , 导热杆的热传导系数为 ,求杆上的温度分布的变化规律. 这个问题可归结为下列定解问题,where,Solution,Let,Step1 Separate variables,Step2 Solve the eigenvalue problem,Step3 Solve the ODE about T(t):,Step4 Linear superposition,Step5 Determine the coefficients an by the initial condition,3. Separation of variable for Laplace equation,Solu

8、tion,Let,Step1 Separate variables,(i),(ii),Step2 Solve the eigenvalue problem,(iii),Step3 Solve the ODE about Y(y):,Step4 Linear superposition,where,Step5 Determine the coefficients An and Bn by the initial condition,4. The problem of beam oscillation,Solution,Let,Step1 Separate variables,Step1 Solv

9、e the eigenvalue problem,(i),Ther are not non-zero solutions of this eigenvalue problem,(ii),Set,then,So the eigenvalues are ln=(np/l)4 (n=1,2,), and the corresponding eigenfunctions are Xn(x)=sin(npx/l).,Step3 Solve the ODE about T(t):,Step4 Linear superposition,Step5 Determine the coefficients an and bn by the initial condition,So finally, we obtain the solution as:,where an and bn are defined as above.,亥姆霍兹（Helmholz）,let,5. The equation of Helmholz,Solution,Let,Summary,The method of separating the