standard form m first-order ODEs is y1= Fi(a, 31, 32, ...,n), 91(a)=A1, F2(x,1,y yn= Fn(a, 1, 3/2, .,3n), yn(a)=A, 7 或者,用向量记号 F(,g), y(a)=A NA of ODE 2003, Fudan University, Chen Wenbin-p 1 1/65
standard form n first-order ODEs is y 0 1 = F1 (x,y 1,y 2,...,y n ), y 1 ( a) = A 1, y 0 2 = F2 (x,y 1,y 2,...,y n ), y 2 ( a) = A 2, . . . y 0 n = Fn (x,y 1,y 2,...,y n ), y n ( a) = A n, y 0 = F (x,y ), y ( a) = A NA of ODE, 2003, Fudan University, Chen Wenbin – p.11/65
阶的形式 般我们总可以把初值问题化成一阶系统,大部分的计 算机代码也是为了解决这样的情况。但是,有时我们考 虑下面二阶系统: y"=F(,,g), y(a)=A, y'a=s 9,y(0)=0,9(0 也可以转化成一阶系统 gi=y(a) (0)=0 y=-1,y2(0)=1 NA of ODE 2003, Fudan University, Chen Wenbin-p. 12/65
y 00 = F(x,y), y(a) = A, y0(a) = S. y00 = −y,y(0) = 0,y0(0) = 1 y01 = y0(x) = y2, y1(0) = 0 y02 = y00 = −y = −y1, y2(0) = 1 NA of ODE, 2003, Fudan University, Chen Wenbin – p.12/65
n阶方程转化为一阶系统 y n)=F(a, y y(a)=A1,y/(a)=A2,y-1)(a)=An 设 y2, y1(a)=A 92=33 (a)=A yn= F(a, 91, 92, 引入新的参数来表示导数不仅仅是一种技术手段,而且 还是数值计算的要求,特别是当对导数计算的精度比较 高的时候 NA of ODE 2003, Fudan University, Chen Wenbin-p. 13/65
n y(n) = F(x,y,y0,y(2),... ,y(n−1)), y(a) = A1,y0(a) = A2,... ,y(n−1)(a) = An y1 = y,y2 = y0,... ,yn = y(n−1) y01 = y2, y1(a) = A1, y02 = y3, y2(a) = A2, . . . y0n = F(x,y1,y2,... ,yn), yn(a) = An. NA of ODE, 2003, Fudan University, Chen Wenbin – p.13/65
转化的例子 基本的转化的方法是对比最高阶导数少一阶的导数引入 参数 "+ sin a U'u+u=cos C 引入孙=,y2=,3=0,可以转化成 92 2= sin x-92(cos C-31-33 93=cos-33-y NA of ODE 2003, Fudan University, Chen Wenbin-p. 14/65
u00 + u0v0 = sin x, v0 + v + u = cos x y1 = u,y2 = u0,y3 = v, y01 = y2, y02 = sin x − y2(cos x − y1 − y3), y03 = cos x − y3 − y1 NA of ODE, 2003, Fudan University, Chen Wenbin – p.14/65
存在性 如果F(x,y)在一个开区域D包含初始点(a,A),则最 少存在一个解y(x)满足 y=F(, g), y(a)=A 这里技术性的假设:D是开的,意味着(a,A)不能在D 的边界上, NA of ODE 2003, Fudan University, Chen Wenbin -p. 15/65
F(x, y) D (a, A) y(x) y 0 = F(x,y), y(a) = A D (a, A) D NA of ODE, 2003, Fudan University, Chen Wenbin – p.15/65