求解变系数二阶线性ODE？ [英] Solve variable coefficients second order linear ODE?

问题描述

对于可变系数二阶线性ODE

For the variable coefficients second order linear ODE

$ x''（t）+ \beta_1（t）x'（t）+ \beta_0 x （t）= 0 $

$x''(t)+\beta_1(t)x'(t)+\beta_0 x(t)=0$

我有$ \beta_1（t）$和$ \beta_0（t）$的数值（就矢量而言） ，有人知道一些R软件包可以做到这一点吗？还有一些简单的例子也很不错。

I have the numerical values (in terms of vectors) for $\beta_1(t)$ and $\beta_0(t)$, does anyone know some R package to do that? And some simple examples to illustrate would be great as well.

我用Google搜索找到了<< a href = http://cran.r-project.org/web /packages/bvpSolve/bvpSolve.pdf rel = nofollow> bvpSolve '可以求解常数系数值。

I googled to find 'bvpSolve' can solve constant coefficients value.

推荐答案

要使用 deSolve ，必须制作二阶ODE

In order to use deSolve, you have to make your second-order ODE

x''(t) + \beta_1(t) x'(t) + \beta_0 x(t)=0

进入一对耦合的一阶ODE：

into a pair of coupled first-order ODEs:

x'(t) = y(t)
y'(t) = - \beta_1(t) y(t) - \beta_0 x(t)

然后很简单：

gfun <- function(t,z,params) {
      g <- with(as.list(c(z,params)),
       {
         beta0 <- sin(2*pi*t)
         beta1 <- cos(2*pi*t)
       c(x=y,
         y= -beta1*y - beta0*x))
     list(g,NULL)
}
library("deSolve")
run1 <- ode(c(x=1,y=1),times=0:40,func=gfun,parms=numeric(0))

我任意选择了一些初始条件（ x（0）= 1 ， x'（0）= 1 ）；您可能还想向模型添加参数（即，使 parms 成为 numeric（0）以外的东西）

I picked some initial conditions (x(0)=1, x'(0)=1) arbitrarily; you might also want to add parameters to the model (i.e. make parms something other than numeric(0))

PS如果您不愿意手工转换为耦合的一阶ODE，并且想要一个可以无缝处理二阶ODE的程序包，那么我不知道答案...

PS if you're not happy doing the conversion to coupled first-order ODEs by hand, and want a package that will seamlessly handle second-order ODEs, then I don't know the answer ...

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